Financial models with interacting heterogeneous agents: modeling assumptions and mathematical tools from discrete dynamical system theory. Minicourse for the PhD Program in Methods and Models for Economic Decisions, Insubria University Marina Pireddu University of Milano-Bicocca Dept. of Mathematics and its Applications [email protected]Marina Pireddu (Univ. of Milano-Bicocca) Discrete-time heterogeneous agent models Insubria Univ., 20/03/2018 1 / 139
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Empirical observations of financial market data have highlighted, inaddition to large trading volume, some crucial stylized facts forfinancial time series at the daily frequency:
(i) asset prices follow a near unit root process;
(ii) asset returns are unpredictable with almost no autocorrelations;
(iii) the returns distribution has fat tails;
(iv) financial returns exhibit long-range volatility clustering, i.e., slowdecay of autocorrelations of squared returns and absolute returns.
Although facts (i) and (ii) are consistent with a random walk model witha representative rational agent, that kind of model has difficulty inexplaining fact (iii), (iv) and also high and persistent trading volume.In the past two decades, such limits of the traditional approach gaverise in economics and finance to a paradigm shift towards abehavioral, agent-based approach.
Empirical observations of financial market data have highlighted, inaddition to large trading volume, some crucial stylized facts forfinancial time series at the daily frequency:
(i) asset prices follow a near unit root process;
(ii) asset returns are unpredictable with almost no autocorrelations;
(iii) the returns distribution has fat tails;
(iv) financial returns exhibit long-range volatility clustering, i.e., slowdecay of autocorrelations of squared returns and absolute returns.
Although facts (i) and (ii) are consistent with a random walk model witha representative rational agent, that kind of model has difficulty inexplaining fact (iii), (iv) and also high and persistent trading volume.In the past two decades, such limits of the traditional approach gaverise in economics and finance to a paradigm shift towards abehavioral, agent-based approach.
Empirical observations of financial market data have highlighted, inaddition to large trading volume, some crucial stylized facts forfinancial time series at the daily frequency:
(i) asset prices follow a near unit root process;
(ii) asset returns are unpredictable with almost no autocorrelations;
(iii) the returns distribution has fat tails;
(iv) financial returns exhibit long-range volatility clustering, i.e., slowdecay of autocorrelations of squared returns and absolute returns.
Although facts (i) and (ii) are consistent with a random walk model witha representative rational agent, that kind of model has difficulty inexplaining fact (iii), (iv) and also high and persistent trading volume.In the past two decades, such limits of the traditional approach gaverise in economics and finance to a paradigm shift towards abehavioral, agent-based approach.
Empirical observations of financial market data have highlighted, inaddition to large trading volume, some crucial stylized facts forfinancial time series at the daily frequency:
(i) asset prices follow a near unit root process;
(ii) asset returns are unpredictable with almost no autocorrelations;
(iii) the returns distribution has fat tails;
(iv) financial returns exhibit long-range volatility clustering, i.e., slowdecay of autocorrelations of squared returns and absolute returns.
Although facts (i) and (ii) are consistent with a random walk model witha representative rational agent, that kind of model has difficulty inexplaining fact (iii), (iv) and also high and persistent trading volume.In the past two decades, such limits of the traditional approach gaverise in economics and finance to a paradigm shift towards abehavioral, agent-based approach.
Empirical observations of financial market data have highlighted, inaddition to large trading volume, some crucial stylized facts forfinancial time series at the daily frequency:
(i) asset prices follow a near unit root process;
(ii) asset returns are unpredictable with almost no autocorrelations;
(iii) the returns distribution has fat tails;
(iv) financial returns exhibit long-range volatility clustering, i.e., slowdecay of autocorrelations of squared returns and absolute returns.
Although facts (i) and (ii) are consistent with a random walk model witha representative rational agent, that kind of model has difficulty inexplaining fact (iii), (iv) and also high and persistent trading volume.In the past two decades, such limits of the traditional approach gaverise in economics and finance to a paradigm shift towards abehavioral, agent-based approach.
Empirical observations of financial market data have highlighted, inaddition to large trading volume, some crucial stylized facts forfinancial time series at the daily frequency:
(i) asset prices follow a near unit root process;
(ii) asset returns are unpredictable with almost no autocorrelations;
(iii) the returns distribution has fat tails;
(iv) financial returns exhibit long-range volatility clustering, i.e., slowdecay of autocorrelations of squared returns and absolute returns.
Although facts (i) and (ii) are consistent with a random walk model witha representative rational agent, that kind of model has difficulty inexplaining fact (iii), (iv) and also high and persistent trading volume.In the past two decades, such limits of the traditional approach gaverise in economics and finance to a paradigm shift towards abehavioral, agent-based approach.
Empirical observations of financial market data have highlighted, inaddition to large trading volume, some crucial stylized facts forfinancial time series at the daily frequency:
(i) asset prices follow a near unit root process;
(ii) asset returns are unpredictable with almost no autocorrelations;
(iii) the returns distribution has fat tails;
(iv) financial returns exhibit long-range volatility clustering, i.e., slowdecay of autocorrelations of squared returns and absolute returns.
Although facts (i) and (ii) are consistent with a random walk model witha representative rational agent, that kind of model has difficulty inexplaining fact (iii), (iv) and also high and persistent trading volume.In the past two decades, such limits of the traditional approach gaverise in economics and finance to a paradigm shift towards abehavioral, agent-based approach.
Empirical observations of financial market data have highlighted, inaddition to large trading volume, some crucial stylized facts forfinancial time series at the daily frequency:
(i) asset prices follow a near unit root process;
(ii) asset returns are unpredictable with almost no autocorrelations;
(iii) the returns distribution has fat tails;
(iv) financial returns exhibit long-range volatility clustering, i.e., slowdecay of autocorrelations of squared returns and absolute returns.
Although facts (i) and (ii) are consistent with a random walk model witha representative rational agent, that kind of model has difficulty inexplaining fact (iii), (iv) and also high and persistent trading volume.In the past two decades, such limits of the traditional approach gaverise in economics and finance to a paradigm shift towards abehavioral, agent-based approach.
Markets are populated by boundedly rational, heterogeneous agentsusing different heuristics or rule of thumb strategies.In particular, in financial market applications, simple heterogeneousagent models can mimic and explain the above-mentioned stylizedfacts observed in financial time series.Indeed, for instance:
high trading volume is mainly caused by differences in beliefs;
volatility in asset prices is driven by news about economicfundamentals, amplified by the interaction of different tradingstrategies.
Due to the presence of boundedly rational, heterogeneous agents,which progressively learn how to behave on the basis of theirinteraction with the environment and the realized values of the relevantvariables, those models are necessarily dynamic in nature.
Markets are populated by boundedly rational, heterogeneous agentsusing different heuristics or rule of thumb strategies.In particular, in financial market applications, simple heterogeneousagent models can mimic and explain the above-mentioned stylizedfacts observed in financial time series.Indeed, for instance:
high trading volume is mainly caused by differences in beliefs;
volatility in asset prices is driven by news about economicfundamentals, amplified by the interaction of different tradingstrategies.
Due to the presence of boundedly rational, heterogeneous agents,which progressively learn how to behave on the basis of theirinteraction with the environment and the realized values of the relevantvariables, those models are necessarily dynamic in nature.
Markets are populated by boundedly rational, heterogeneous agentsusing different heuristics or rule of thumb strategies.In particular, in financial market applications, simple heterogeneousagent models can mimic and explain the above-mentioned stylizedfacts observed in financial time series.Indeed, for instance:
high trading volume is mainly caused by differences in beliefs;
volatility in asset prices is driven by news about economicfundamentals, amplified by the interaction of different tradingstrategies.
Due to the presence of boundedly rational, heterogeneous agents,which progressively learn how to behave on the basis of theirinteraction with the environment and the realized values of the relevantvariables, those models are necessarily dynamic in nature.
Markets are populated by boundedly rational, heterogeneous agentsusing different heuristics or rule of thumb strategies.In particular, in financial market applications, simple heterogeneousagent models can mimic and explain the above-mentioned stylizedfacts observed in financial time series.Indeed, for instance:
high trading volume is mainly caused by differences in beliefs;
volatility in asset prices is driven by news about economicfundamentals, amplified by the interaction of different tradingstrategies.
Due to the presence of boundedly rational, heterogeneous agents,which progressively learn how to behave on the basis of theirinteraction with the environment and the realized values of the relevantvariables, those models are necessarily dynamic in nature.
Markets are populated by boundedly rational, heterogeneous agentsusing different heuristics or rule of thumb strategies.In particular, in financial market applications, simple heterogeneousagent models can mimic and explain the above-mentioned stylizedfacts observed in financial time series.Indeed, for instance:
high trading volume is mainly caused by differences in beliefs;
volatility in asset prices is driven by news about economicfundamentals, amplified by the interaction of different tradingstrategies.
Due to the presence of boundedly rational, heterogeneous agents,which progressively learn how to behave on the basis of theirinteraction with the environment and the realized values of the relevantvariables, those models are necessarily dynamic in nature.
Markets are indeed viewed as complex adaptive systems, where theevolutionary selection of expectations rules or trading strategies isendogenously coupled with the market dynamics.
Being usually highly nonlinear, for instance due to evolutionaryswitching between strategies, the heterogeneous agent models exhibita wide range of dynamical behaviors.
We will then introduce some basic mathematical tools from discretedynamical system theory, which will be applied to analyze simple (1Dand 2D) heterogeneous agent models, like those proposed inWesterhoff (2012) and in Naimzada and Pireddu (2014, 2015a,2015b).
Markets are indeed viewed as complex adaptive systems, where theevolutionary selection of expectations rules or trading strategies isendogenously coupled with the market dynamics.
Being usually highly nonlinear, for instance due to evolutionaryswitching between strategies, the heterogeneous agent models exhibita wide range of dynamical behaviors.
We will then introduce some basic mathematical tools from discretedynamical system theory, which will be applied to analyze simple (1Dand 2D) heterogeneous agent models, like those proposed inWesterhoff (2012) and in Naimzada and Pireddu (2014, 2015a,2015b).
Markets are indeed viewed as complex adaptive systems, where theevolutionary selection of expectations rules or trading strategies isendogenously coupled with the market dynamics.
Being usually highly nonlinear, for instance due to evolutionaryswitching between strategies, the heterogeneous agent models exhibita wide range of dynamical behaviors.
We will then introduce some basic mathematical tools from discretedynamical system theory, which will be applied to analyze simple (1Dand 2D) heterogeneous agent models, like those proposed inWesterhoff (2012) and in Naimzada and Pireddu (2014, 2015a,2015b).
– De Grauwe P (2012) Lectures on Behavioral Macroeconomics.Princeton University Press, New Jersey.– Hommes CH (2006) Heterogeneous Agent Models in Economicsand Finance. In: L. Tesfatsion and K.L. Judd (Eds.), Agent-BasedComputational Economics, pp. 1109–1186. Handbook ofComputational Economics, vol.2. Elsevier Science, Amsterdam.Sections 1 and 6– Hommes CH (2013) Behavioral Rationality and HeterogeneousExpectations in Complex Economic Systems. Cambridge UniversityPress, Cambridge.– Naimzada A, Pireddu M (2014) Dynamic behavior of product andstock markets with a varying degree of interaction. EconomicModelling 41, 191–197
– Naimzada A, Pireddu M (2015a) Introducing a price variation limitermechanism into a behavioral financial market model. Chaos 25,083112. doi: 10.1063/1.4927831– Naimzada A, Pireddu M (2015b) Real and financial interactingmarkets: A behavioral macro-model. Chaos Solitons Fractals 77,111–131– Westerhoff F (2012) Interactions between the real economy and thestock market: A simple agent-based approach, Discrete Dynamics inNature and Society 2012, Article ID 504840
A 1D discrete dynamical system is a sequence of numbers, xt , that aredefined recursively, i.e., there is a rule relating each number in thesequence to (some or all) the previous numbers in the sequence; wedenote such a sequence by {xt}.Time is discrete, i.e., t = 0,1,2, . . .
A first-order discrete dynamical system is a sequence of numbers xt fort = 0,1,2, . . . such that each number after the first is related just to theprevious number by the relationship xt+1 = f (xt ), where f : A ⊆ R→ R.
An mth-order discrete dynamical system takes the formxt+m = f (xt+m−1, xt+m−2, . . . , xt ), for some m ∈ N \ {0}.Since the systems above do not explicitly depend on t , they are calledautonomous.
We will consider only the case m = 1, i.e., first-order autonomousdiscrete dynamical systems.
A 1D discrete dynamical system is a sequence of numbers, xt , that aredefined recursively, i.e., there is a rule relating each number in thesequence to (some or all) the previous numbers in the sequence; wedenote such a sequence by {xt}.Time is discrete, i.e., t = 0,1,2, . . .
A first-order discrete dynamical system is a sequence of numbers xt fort = 0,1,2, . . . such that each number after the first is related just to theprevious number by the relationship xt+1 = f (xt ), where f : A ⊆ R→ R.
An mth-order discrete dynamical system takes the formxt+m = f (xt+m−1, xt+m−2, . . . , xt ), for some m ∈ N \ {0}.Since the systems above do not explicitly depend on t , they are calledautonomous.
We will consider only the case m = 1, i.e., first-order autonomousdiscrete dynamical systems.
A 1D discrete dynamical system is a sequence of numbers, xt , that aredefined recursively, i.e., there is a rule relating each number in thesequence to (some or all) the previous numbers in the sequence; wedenote such a sequence by {xt}.Time is discrete, i.e., t = 0,1,2, . . .
A first-order discrete dynamical system is a sequence of numbers xt fort = 0,1,2, . . . such that each number after the first is related just to theprevious number by the relationship xt+1 = f (xt ), where f : A ⊆ R→ R.
An mth-order discrete dynamical system takes the formxt+m = f (xt+m−1, xt+m−2, . . . , xt ), for some m ∈ N \ {0}.Since the systems above do not explicitly depend on t , they are calledautonomous.
We will consider only the case m = 1, i.e., first-order autonomousdiscrete dynamical systems.
A 1D discrete dynamical system is a sequence of numbers, xt , that aredefined recursively, i.e., there is a rule relating each number in thesequence to (some or all) the previous numbers in the sequence; wedenote such a sequence by {xt}.Time is discrete, i.e., t = 0,1,2, . . .
A first-order discrete dynamical system is a sequence of numbers xt fort = 0,1,2, . . . such that each number after the first is related just to theprevious number by the relationship xt+1 = f (xt ), where f : A ⊆ R→ R.
An mth-order discrete dynamical system takes the formxt+m = f (xt+m−1, xt+m−2, . . . , xt ), for some m ∈ N \ {0}.Since the systems above do not explicitly depend on t , they are calledautonomous.
We will consider only the case m = 1, i.e., first-order autonomousdiscrete dynamical systems.
A 1D discrete dynamical system is a sequence of numbers, xt , that aredefined recursively, i.e., there is a rule relating each number in thesequence to (some or all) the previous numbers in the sequence; wedenote such a sequence by {xt}.Time is discrete, i.e., t = 0,1,2, . . .
A first-order discrete dynamical system is a sequence of numbers xt fort = 0,1,2, . . . such that each number after the first is related just to theprevious number by the relationship xt+1 = f (xt ), where f : A ⊆ R→ R.
An mth-order discrete dynamical system takes the formxt+m = f (xt+m−1, xt+m−2, . . . , xt ), for some m ∈ N \ {0}.Since the systems above do not explicitly depend on t , they are calledautonomous.
We will consider only the case m = 1, i.e., first-order autonomousdiscrete dynamical systems.
A 1D discrete dynamical system is a sequence of numbers, xt , that aredefined recursively, i.e., there is a rule relating each number in thesequence to (some or all) the previous numbers in the sequence; wedenote such a sequence by {xt}.Time is discrete, i.e., t = 0,1,2, . . .
A first-order discrete dynamical system is a sequence of numbers xt fort = 0,1,2, . . . such that each number after the first is related just to theprevious number by the relationship xt+1 = f (xt ), where f : A ⊆ R→ R.
An mth-order discrete dynamical system takes the formxt+m = f (xt+m−1, xt+m−2, . . . , xt ), for some m ∈ N \ {0}.Since the systems above do not explicitly depend on t , they are calledautonomous.
We will consider only the case m = 1, i.e., first-order autonomousdiscrete dynamical systems.
A 1D discrete dynamical system is a sequence of numbers, xt , that aredefined recursively, i.e., there is a rule relating each number in thesequence to (some or all) the previous numbers in the sequence; wedenote such a sequence by {xt}.Time is discrete, i.e., t = 0,1,2, . . .
A first-order discrete dynamical system is a sequence of numbers xt fort = 0,1,2, . . . such that each number after the first is related just to theprevious number by the relationship xt+1 = f (xt ), where f : A ⊆ R→ R.
An mth-order discrete dynamical system takes the formxt+m = f (xt+m−1, xt+m−2, . . . , xt ), for some m ∈ N \ {0}.Since the systems above do not explicitly depend on t , they are calledautonomous.
We will consider only the case m = 1, i.e., first-order autonomousdiscrete dynamical systems.
If f is linear, i.e., f (xt ) = axt + b, for some a,b ∈ R, the system is saidto be linear; if f is nonlinear, i.e., if f is not linear, then the system issaid to be nonlinear.
If f is linear, i.e., f (xt ) = axt + b, for some a,b ∈ R, the system is saidto be linear; if f is nonlinear, i.e., if f is not linear, then the system issaid to be nonlinear.
If f is linear, i.e., f (xt ) = axt + b, for some a,b ∈ R, the system is saidto be linear; if f is nonlinear, i.e., if f is not linear, then the system issaid to be nonlinear.
If f is linear, i.e., f (xt ) = axt + b, for some a,b ∈ R, the system is saidto be linear; if f is nonlinear, i.e., if f is not linear, then the system issaid to be nonlinear.
To start the system xt+1 = f (xt ), we need to specify the initial conditionx0 ∈ R.
Then O(x0) = {x0, f (x0), f (f (x0)), f (f (f (x0))), . . . } ={x0, f (x0), f 2(x0), f 3(x0), . . . } is the (positive) orbit of x0.
Solving the system xt+1 = f (xt ) with initial condition x0 ∈ R meansfinding a sequence {yt , t ∈ N} such that yt+1 = f (yt ), for all t ∈ N, andy0 = x0.
To start the system xt+1 = f (xt ), we need to specify the initial conditionx0 ∈ R.
Then O(x0) = {x0, f (x0), f (f (x0)), f (f (f (x0))), . . . } ={x0, f (x0), f 2(x0), f 3(x0), . . . } is the (positive) orbit of x0.
Solving the system xt+1 = f (xt ) with initial condition x0 ∈ R meansfinding a sequence {yt , t ∈ N} such that yt+1 = f (yt ), for all t ∈ N, andy0 = x0.
To start the system xt+1 = f (xt ), we need to specify the initial conditionx0 ∈ R.
Then O(x0) = {x0, f (x0), f (f (x0)), f (f (f (x0))), . . . } ={x0, f (x0), f 2(x0), f 3(x0), . . . } is the (positive) orbit of x0.
Solving the system xt+1 = f (xt ) with initial condition x0 ∈ R meansfinding a sequence {yt , t ∈ N} such that yt+1 = f (yt ), for all t ∈ N, andy0 = x0.
To start the system xt+1 = f (xt ), we need to specify the initial conditionx0 ∈ R.
Then O(x0) = {x0, f (x0), f (f (x0)), f (f (f (x0))), . . . } ={x0, f (x0), f 2(x0), f 3(x0), . . . } is the (positive) orbit of x0.
Solving the system xt+1 = f (xt ) with initial condition x0 ∈ R meansfinding a sequence {yt , t ∈ N} such that yt+1 = f (yt ), for all t ∈ N, andy0 = x0.
If xt+1 = f (xt ) is a 1D discrete dynamical system, then x∗ is a fixedpoint or equilibrium point of the system ifx∗ = f (x∗)⇒ xt = x∗,∀t ∈ N⇒ O(x∗) = {x∗}.
The fixed points are found as the intersections between the graph of fand the 45-degree line xt+1 = xt .
If xt+1 = f (xt ) is a 1D discrete dynamical system, then x∗ is a fixedpoint or equilibrium point of the system ifx∗ = f (x∗)⇒ xt = x∗,∀t ∈ N⇒ O(x∗) = {x∗}.
The fixed points are found as the intersections between the graph of fand the 45-degree line xt+1 = xt .
If xt+1 = f (xt ) is a 1D discrete dynamical system, then x∗ is a fixedpoint or equilibrium point of the system ifx∗ = f (x∗)⇒ xt = x∗,∀t ∈ N⇒ O(x∗) = {x∗}.
The fixed points are found as the intersections between the graph of fand the 45-degree line xt+1 = xt .
If xt+1 = f (xt ) is a 1D discrete dynamical system, then x∗ is a fixedpoint or equilibrium point of the system ifx∗ = f (x∗)⇒ xt = x∗,∀t ∈ N⇒ O(x∗) = {x∗}.
The fixed points are found as the intersections between the graph of fand the 45-degree line xt+1 = xt .
If xt+1 = f (xt ) is a 1D discrete dynamical system, then x∗ is a fixedpoint or equilibrium point of the system ifx∗ = f (x∗)⇒ xt = x∗,∀t ∈ N⇒ O(x∗) = {x∗}.
The fixed points are found as the intersections between the graph of fand the 45-degree line xt+1 = xt .
Given xt+1 = f (xt ), with f : I ⊂ R→ R defined on the interval I, theequilibrium point x∗ ∈ I is stable if ∀ε > 0 ∃δ > 0 such that ∀x0 ∈ I with|x0 − x∗| < δ it holds that |f t (x0)− x∗| < ε, ∀t ∈ N \ {0}.
If x∗ is stable and attracting, i.e., there exists η > 0 such that for allx0 ∈ I with |x0 − x∗| < η it holds that limt→+∞ f t (x0) = x∗, for t ∈ N,then x∗ is called locally asymptotically stable.
x∗ is locally asymptotically stable
If η = +∞, then x∗ is called globally asymptotically stable.
If x∗ is stable and attracting, i.e., there exists η > 0 such that for allx0 ∈ I with |x0 − x∗| < η it holds that limt→+∞ f t (x0) = x∗, for t ∈ N,then x∗ is called locally asymptotically stable.
x∗ is locally asymptotically stable
If η = +∞, then x∗ is called globally asymptotically stable.
TheoremLet x∗ be an equilibrium point of the dynamical system xt+1 = f (xt ),with f continuously differentiable at x∗.
(i) If |f ′(x∗)| < 1, then x∗ is locally asymptotically stable;(ii) if |f ′(x∗)| > 1, then x∗ is unstable;(iii) if |f ′(x∗)| = 1, you need higher derivatives to establish the nature
of x∗.
We will focus on hyperbolic equilibrium points x∗, i.e., with |f ′(x∗)| 6= 1.
TheoremLet x∗ be an equilibrium point of the dynamical system xt+1 = f (xt ),with f continuously differentiable at x∗.
(i) If |f ′(x∗)| < 1, then x∗ is locally asymptotically stable;(ii) if |f ′(x∗)| > 1, then x∗ is unstable;(iii) if |f ′(x∗)| = 1, you need higher derivatives to establish the nature
of x∗.
We will focus on hyperbolic equilibrium points x∗, i.e., with |f ′(x∗)| 6= 1.
TheoremLet x∗ be an equilibrium point of the dynamical system xt+1 = f (xt ),with f continuously differentiable at x∗.
(i) If |f ′(x∗)| < 1, then x∗ is locally asymptotically stable;(ii) if |f ′(x∗)| > 1, then x∗ is unstable;(iii) if |f ′(x∗)| = 1, you need higher derivatives to establish the nature
of x∗.
We will focus on hyperbolic equilibrium points x∗, i.e., with |f ′(x∗)| 6= 1.
TheoremLet x∗ be an equilibrium point of the dynamical system xt+1 = f (xt ),with f continuously differentiable at x∗.
(i) If |f ′(x∗)| < 1, then x∗ is locally asymptotically stable;(ii) if |f ′(x∗)| > 1, then x∗ is unstable;(iii) if |f ′(x∗)| = 1, you need higher derivatives to establish the nature
of x∗.
We will focus on hyperbolic equilibrium points x∗, i.e., with |f ′(x∗)| 6= 1.
A) Why do we use derivatives to check the local stability of equilibria?
If f is continuously differentiable, we can linearize it in a neighborhoodof x∗ as follows:
f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),
with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.
Ignoring the remainder term, we obtain
f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).
The rhs is a linear equation in x with slope f ′(x∗).
If the absolute slope is less than that of the 45-degree line (i.e.,f ′(x∗) ∈ (−1,1)), then x∗ is locally asymptotically stable.Otherwise, x∗ is unstable.
Such stability condition can be used at any equilibrium.
A) Why do we use derivatives to check the local stability of equilibria?
If f is continuously differentiable, we can linearize it in a neighborhoodof x∗ as follows:
f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),
with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.
Ignoring the remainder term, we obtain
f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).
The rhs is a linear equation in x with slope f ′(x∗).
If the absolute slope is less than that of the 45-degree line (i.e.,f ′(x∗) ∈ (−1,1)), then x∗ is locally asymptotically stable.Otherwise, x∗ is unstable.
Such stability condition can be used at any equilibrium.
A) Why do we use derivatives to check the local stability of equilibria?
If f is continuously differentiable, we can linearize it in a neighborhoodof x∗ as follows:
f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),
with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.
Ignoring the remainder term, we obtain
f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).
The rhs is a linear equation in x with slope f ′(x∗).
If the absolute slope is less than that of the 45-degree line (i.e.,f ′(x∗) ∈ (−1,1)), then x∗ is locally asymptotically stable.Otherwise, x∗ is unstable.
Such stability condition can be used at any equilibrium.
A) Why do we use derivatives to check the local stability of equilibria?
If f is continuously differentiable, we can linearize it in a neighborhoodof x∗ as follows:
f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),
with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.
Ignoring the remainder term, we obtain
f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).
The rhs is a linear equation in x with slope f ′(x∗).
If the absolute slope is less than that of the 45-degree line (i.e.,f ′(x∗) ∈ (−1,1)), then x∗ is locally asymptotically stable.Otherwise, x∗ is unstable.
Such stability condition can be used at any equilibrium.
A) Why do we use derivatives to check the local stability of equilibria?
If f is continuously differentiable, we can linearize it in a neighborhoodof x∗ as follows:
f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),
with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.
Ignoring the remainder term, we obtain
f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).
The rhs is a linear equation in x with slope f ′(x∗).
If the absolute slope is less than that of the 45-degree line (i.e.,f ′(x∗) ∈ (−1,1)), then x∗ is locally asymptotically stable.Otherwise, x∗ is unstable.
Such stability condition can be used at any equilibrium.
A) Why do we use derivatives to check the local stability of equilibria?
If f is continuously differentiable, we can linearize it in a neighborhoodof x∗ as follows:
f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),
with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.
Ignoring the remainder term, we obtain
f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).
The rhs is a linear equation in x with slope f ′(x∗).
If the absolute slope is less than that of the 45-degree line (i.e.,f ′(x∗) ∈ (−1,1)), then x∗ is locally asymptotically stable.Otherwise, x∗ is unstable.
Such stability condition can be used at any equilibrium.
A) Why do we use derivatives to check the local stability of equilibria?
If f is continuously differentiable, we can linearize it in a neighborhoodof x∗ as follows:
f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),
with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.
Ignoring the remainder term, we obtain
f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).
The rhs is a linear equation in x with slope f ′(x∗).
If the absolute slope is less than that of the 45-degree line (i.e.,f ′(x∗) ∈ (−1,1)), then x∗ is locally asymptotically stable.Otherwise, x∗ is unstable.
Such stability condition can be used at any equilibrium.
A) Why do we use derivatives to check the local stability of equilibria?
If f is continuously differentiable, we can linearize it in a neighborhoodof x∗ as follows:
f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),
with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.
Ignoring the remainder term, we obtain
f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).
The rhs is a linear equation in x with slope f ′(x∗).
If the absolute slope is less than that of the 45-degree line (i.e.,f ′(x∗) ∈ (−1,1)), then x∗ is locally asymptotically stable.Otherwise, x∗ is unstable.
Such stability condition can be used at any equilibrium.
A) Why do we use derivatives to check the local stability of equilibria?
If f is continuously differentiable, we can linearize it in a neighborhoodof x∗ as follows:
f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),
with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.
Ignoring the remainder term, we obtain
f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).
The rhs is a linear equation in x with slope f ′(x∗).
If the absolute slope is less than that of the 45-degree line (i.e.,f ′(x∗) ∈ (−1,1)), then x∗ is locally asymptotically stable.Otherwise, x∗ is unstable.
Such stability condition can be used at any equilibrium.
A) Why do we use derivatives to check the local stability of equilibria?
If f is continuously differentiable, we can linearize it in a neighborhoodof x∗ as follows:
f (x)− x∗ = f ′(x∗)(x − x∗) + r(x − x∗),
with r(x − x∗) = o(|x − x∗|) as x − x∗ → 0.
Ignoring the remainder term, we obtain
f (x) ≈f ′(x∗)x + x∗(1− f ′(x∗)).
The rhs is a linear equation in x with slope f ′(x∗).
If the absolute slope is less than that of the 45-degree line (i.e.,f ′(x∗) ∈ (−1,1)), then x∗ is locally asymptotically stable.Otherwise, x∗ is unstable.
Such stability condition can be used at any equilibrium.
If xt+1 = f (xt ) is a discrete dynamical system, then x is a periodic pointof the system with period k if f k (x) = x for some positive integer k . Inthis case x is called k -periodic.
If in addition f i(x) 6= x , for 0 < i < k , then k is called the minimalperiod of x .
Since by definition x is k -periodic if it is a fixed point of the map f k , thek -periodic points are found as the intersections between the graph off k and the 45-degree line xt+1 = xt .
Moreover, if k is the minimal period of x , then its orbit is given byO(x) = {x , f (x), f 2(x), . . . , f k−1(x)}. This is called a k-periodic cycle.
In terms of the system, it follows that, starting from x0 = x , we findxt+k = xt , for all t ∈ N, i.e., the system admits the k -periodic solution{x , f (x), f 2(x), . . . , f k−1(x)}.
If xt+1 = f (xt ) is a discrete dynamical system, then x is a periodic pointof the system with period k if f k (x) = x for some positive integer k . Inthis case x is called k -periodic.
If in addition f i(x) 6= x , for 0 < i < k , then k is called the minimalperiod of x .
Since by definition x is k -periodic if it is a fixed point of the map f k , thek -periodic points are found as the intersections between the graph off k and the 45-degree line xt+1 = xt .
Moreover, if k is the minimal period of x , then its orbit is given byO(x) = {x , f (x), f 2(x), . . . , f k−1(x)}. This is called a k-periodic cycle.
In terms of the system, it follows that, starting from x0 = x , we findxt+k = xt , for all t ∈ N, i.e., the system admits the k -periodic solution{x , f (x), f 2(x), . . . , f k−1(x)}.
If xt+1 = f (xt ) is a discrete dynamical system, then x is a periodic pointof the system with period k if f k (x) = x for some positive integer k . Inthis case x is called k -periodic.
If in addition f i(x) 6= x , for 0 < i < k , then k is called the minimalperiod of x .
Since by definition x is k -periodic if it is a fixed point of the map f k , thek -periodic points are found as the intersections between the graph off k and the 45-degree line xt+1 = xt .
Moreover, if k is the minimal period of x , then its orbit is given byO(x) = {x , f (x), f 2(x), . . . , f k−1(x)}. This is called a k-periodic cycle.
In terms of the system, it follows that, starting from x0 = x , we findxt+k = xt , for all t ∈ N, i.e., the system admits the k -periodic solution{x , f (x), f 2(x), . . . , f k−1(x)}.
If xt+1 = f (xt ) is a discrete dynamical system, then x is a periodic pointof the system with period k if f k (x) = x for some positive integer k . Inthis case x is called k -periodic.
If in addition f i(x) 6= x , for 0 < i < k , then k is called the minimalperiod of x .
Since by definition x is k -periodic if it is a fixed point of the map f k , thek -periodic points are found as the intersections between the graph off k and the 45-degree line xt+1 = xt .
Moreover, if k is the minimal period of x , then its orbit is given byO(x) = {x , f (x), f 2(x), . . . , f k−1(x)}. This is called a k-periodic cycle.
In terms of the system, it follows that, starting from x0 = x , we findxt+k = xt , for all t ∈ N, i.e., the system admits the k -periodic solution{x , f (x), f 2(x), . . . , f k−1(x)}.
If xt+1 = f (xt ) is a discrete dynamical system, then x is a periodic pointof the system with period k if f k (x) = x for some positive integer k . Inthis case x is called k -periodic.
If in addition f i(x) 6= x , for 0 < i < k , then k is called the minimalperiod of x .
Since by definition x is k -periodic if it is a fixed point of the map f k , thek -periodic points are found as the intersections between the graph off k and the 45-degree line xt+1 = xt .
Moreover, if k is the minimal period of x , then its orbit is given byO(x) = {x , f (x), f 2(x), . . . , f k−1(x)}. This is called a k-periodic cycle.
In terms of the system, it follows that, starting from x0 = x , we findxt+k = xt , for all t ∈ N, i.e., the system admits the k -periodic solution{x , f (x), f 2(x), . . . , f k−1(x)}.
Let x be a periodic point of f with minimal period k . Then we say that:
x is asymptotically stable if it is an asymptotically stable fixed pointof f k .
x is unstable if it is an unstable fixed point of f k .
Hence, studying the stability of k -periodic solutions of xt+1 = f (xt )reduces to studying the stability of the equilibrium points of theassociated difference equation yt+1 = f k (yt ).
Let x be a periodic point of f with minimal period k . Then we say that:
x is asymptotically stable if it is an asymptotically stable fixed pointof f k .
x is unstable if it is an unstable fixed point of f k .
Hence, studying the stability of k -periodic solutions of xt+1 = f (xt )reduces to studying the stability of the equilibrium points of theassociated difference equation yt+1 = f k (yt ).
Let x be a periodic point of f with minimal period k . Then we say that:
x is asymptotically stable if it is an asymptotically stable fixed pointof f k .
x is unstable if it is an unstable fixed point of f k .
Hence, studying the stability of k -periodic solutions of xt+1 = f (xt )reduces to studying the stability of the equilibrium points of theassociated difference equation yt+1 = f k (yt ).
Let x be a periodic point of f with minimal period k . Then we say that:
x is asymptotically stable if it is an asymptotically stable fixed pointof f k .
x is unstable if it is an unstable fixed point of f k .
Hence, studying the stability of k -periodic solutions of xt+1 = f (xt )reduces to studying the stability of the equilibrium points of theassociated difference equation yt+1 = f k (yt ).
f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)),
we find the following practical criterion to check the stability of periodicpoints:
TheoremGiven the discrete dynamical system xt+1 = f (xt ), let x be a periodicpoint of f with minimal period k . If f is continuously differentiable atevery point of O(x), then it holds that:
(i) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| < 1, then x is locallyasymptotically stable;
(ii) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| > 1, then x is unstable.
f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)),
we find the following practical criterion to check the stability of periodicpoints:
TheoremGiven the discrete dynamical system xt+1 = f (xt ), let x be a periodicpoint of f with minimal period k . If f is continuously differentiable atevery point of O(x), then it holds that:
(i) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| < 1, then x is locallyasymptotically stable;
(ii) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| > 1, then x is unstable.
f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)),
we find the following practical criterion to check the stability of periodicpoints:
TheoremGiven the discrete dynamical system xt+1 = f (xt ), let x be a periodicpoint of f with minimal period k . If f is continuously differentiable atevery point of O(x), then it holds that:
(i) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| < 1, then x is locallyasymptotically stable;
(ii) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| > 1, then x is unstable.
f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)),
we find the following practical criterion to check the stability of periodicpoints:
TheoremGiven the discrete dynamical system xt+1 = f (xt ), let x be a periodicpoint of f with minimal period k . If f is continuously differentiable atevery point of O(x), then it holds that:
(i) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| < 1, then x is locallyasymptotically stable;
(ii) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| > 1, then x is unstable.
Rmk: If x is a periodic point of f with minimal period k , thenddx f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)) = f ′(x0) · f ′(x1) · . . . · f ′(xk−1),where we set x = x0, f (x) = x1, . . . , f k−1(x) = xk−1, i.e.,O(x) = {x , f (x), f 2(x), . . . , f k−1(x)} = O(x0) = {x0, x1, . . . , xk−1}.
Hence, the conclusions in the previous result can be equivalentlyrewritten as:
(i) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| < 1, then the k-periodic cycleO(x) is asymptotically stable;
(ii) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| > 1, then O(x) is unstable.
Rmk: If x is a periodic point of f with minimal period k , thenddx f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)) = f ′(x0) · f ′(x1) · . . . · f ′(xk−1),where we set x = x0, f (x) = x1, . . . , f k−1(x) = xk−1, i.e.,O(x) = {x , f (x), f 2(x), . . . , f k−1(x)} = O(x0) = {x0, x1, . . . , xk−1}.
Hence, the conclusions in the previous result can be equivalentlyrewritten as:
(i) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| < 1, then the k-periodic cycleO(x) is asymptotically stable;
(ii) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| > 1, then O(x) is unstable.
Rmk: If x is a periodic point of f with minimal period k , thenddx f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)) = f ′(x0) · f ′(x1) · . . . · f ′(xk−1),where we set x = x0, f (x) = x1, . . . , f k−1(x) = xk−1, i.e.,O(x) = {x , f (x), f 2(x), . . . , f k−1(x)} = O(x0) = {x0, x1, . . . , xk−1}.
Hence, the conclusions in the previous result can be equivalentlyrewritten as:
(i) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| < 1, then the k-periodic cycleO(x) is asymptotically stable;
(ii) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| > 1, then O(x) is unstable.
Rmk: If x is a periodic point of f with minimal period k , thenddx f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)) = f ′(x0) · f ′(x1) · . . . · f ′(xk−1),where we set x = x0, f (x) = x1, . . . , f k−1(x) = xk−1, i.e.,O(x) = {x , f (x), f 2(x), . . . , f k−1(x)} = O(x0) = {x0, x1, . . . , xk−1}.
Hence, the conclusions in the previous result can be equivalentlyrewritten as:
(i) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| < 1, then the k-periodic cycleO(x) is asymptotically stable;
(ii) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| > 1, then O(x) is unstable.
Rmk: If x is a periodic point of f with minimal period k , thenddx f k (x) = f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x)) = f ′(x0) · f ′(x1) · . . . · f ′(xk−1),where we set x = x0, f (x) = x1, . . . , f k−1(x) = xk−1, i.e.,O(x) = {x , f (x), f 2(x), . . . , f k−1(x)} = O(x0) = {x0, x1, . . . , xk−1}.
Hence, the conclusions in the previous result can be equivalentlyrewritten as:
(i) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| < 1, then the k-periodic cycleO(x) is asymptotically stable;
(ii) if |f ′(x) · f ′(f (x)) · . . . · f ′(f k−1(x))| > 1, then O(x) is unstable.
x∗ = 0 is asymptotically stable for µ < 1 and unstable for µ > 1.x∗ = 1− 1
µ is asymptotically stable for µ ∈ (1,3).
For µ = 3, x∗ = 1− 1µ becomes unstable but it gives rise to two
stable solutions {x1, x2} to f 2(x) = x , which form anasymptotically stable period-two cycle for f ⇒for µ = 3 a period-doubling bifurcation occurs for f at x∗ = 1− 1
µ .
The period-two cycle {x1, x2}, with x1,2 = 12 + 1
2µ ±12
√1− 2
µ −3µ2 ,
is asymptotically stable for µ ∈ (3,3.449).
For µ = 3.449, both x1 and x1 become unstable but each of themgives rise to two stable solutions to f 4(x) = x , which form anasymptotically stable period-four cycle for f ⇒for µ = 3.449 a double period-doubling bifurcation occurs for f 2 atx = x1 and x = x2.
x∗ = 0 is asymptotically stable for µ < 1 and unstable for µ > 1.x∗ = 1− 1
µ is asymptotically stable for µ ∈ (1,3).
For µ = 3, x∗ = 1− 1µ becomes unstable but it gives rise to two
stable solutions {x1, x2} to f 2(x) = x , which form anasymptotically stable period-two cycle for f ⇒for µ = 3 a period-doubling bifurcation occurs for f at x∗ = 1− 1
µ .
The period-two cycle {x1, x2}, with x1,2 = 12 + 1
2µ ±12
√1− 2
µ −3µ2 ,
is asymptotically stable for µ ∈ (3,3.449).
For µ = 3.449, both x1 and x1 become unstable but each of themgives rise to two stable solutions to f 4(x) = x , which form anasymptotically stable period-four cycle for f ⇒for µ = 3.449 a double period-doubling bifurcation occurs for f 2 atx = x1 and x = x2.
x∗ = 0 is asymptotically stable for µ < 1 and unstable for µ > 1.x∗ = 1− 1
µ is asymptotically stable for µ ∈ (1,3).
For µ = 3, x∗ = 1− 1µ becomes unstable but it gives rise to two
stable solutions {x1, x2} to f 2(x) = x , which form anasymptotically stable period-two cycle for f ⇒for µ = 3 a period-doubling bifurcation occurs for f at x∗ = 1− 1
µ .
The period-two cycle {x1, x2}, with x1,2 = 12 + 1
2µ ±12
√1− 2
µ −3µ2 ,
is asymptotically stable for µ ∈ (3,3.449).
For µ = 3.449, both x1 and x1 become unstable but each of themgives rise to two stable solutions to f 4(x) = x , which form anasymptotically stable period-four cycle for f ⇒for µ = 3.449 a double period-doubling bifurcation occurs for f 2 atx = x1 and x = x2.
x∗ = 0 is asymptotically stable for µ < 1 and unstable for µ > 1.x∗ = 1− 1
µ is asymptotically stable for µ ∈ (1,3).
For µ = 3, x∗ = 1− 1µ becomes unstable but it gives rise to two
stable solutions {x1, x2} to f 2(x) = x , which form anasymptotically stable period-two cycle for f ⇒for µ = 3 a period-doubling bifurcation occurs for f at x∗ = 1− 1
µ .
The period-two cycle {x1, x2}, with x1,2 = 12 + 1
2µ ±12
√1− 2
µ −3µ2 ,
is asymptotically stable for µ ∈ (3,3.449).
For µ = 3.449, both x1 and x1 become unstable but each of themgives rise to two stable solutions to f 4(x) = x , which form anasymptotically stable period-four cycle for f ⇒for µ = 3.449 a double period-doubling bifurcation occurs for f 2 atx = x1 and x = x2.
x∗ = 0 is asymptotically stable for µ < 1 and unstable for µ > 1.x∗ = 1− 1
µ is asymptotically stable for µ ∈ (1,3).
For µ = 3, x∗ = 1− 1µ becomes unstable but it gives rise to two
stable solutions {x1, x2} to f 2(x) = x , which form anasymptotically stable period-two cycle for f ⇒for µ = 3 a period-doubling bifurcation occurs for f at x∗ = 1− 1
µ .
The period-two cycle {x1, x2}, with x1,2 = 12 + 1
2µ ±12
√1− 2
µ −3µ2 ,
is asymptotically stable for µ ∈ (3,3.449).
For µ = 3.449, both x1 and x1 become unstable but each of themgives rise to two stable solutions to f 4(x) = x , which form anasymptotically stable period-four cycle for f ⇒for µ = 3.449 a double period-doubling bifurcation occurs for f 2 atx = x1 and x = x2.
x∗ = 0 is asymptotically stable for µ < 1 and unstable for µ > 1.x∗ = 1− 1
µ is asymptotically stable for µ ∈ (1,3).
For µ = 3, x∗ = 1− 1µ becomes unstable but it gives rise to two
stable solutions {x1, x2} to f 2(x) = x , which form anasymptotically stable period-two cycle for f ⇒for µ = 3 a period-doubling bifurcation occurs for f at x∗ = 1− 1
µ .
The period-two cycle {x1, x2}, with x1,2 = 12 + 1
2µ ±12
√1− 2
µ −3µ2 ,
is asymptotically stable for µ ∈ (3,3.449).
For µ = 3.449, both x1 and x1 become unstable but each of themgives rise to two stable solutions to f 4(x) = x , which form anasymptotically stable period-four cycle for f ⇒for µ = 3.449 a double period-doubling bifurcation occurs for f 2 atx = x1 and x = x2.
x∗ = 0 is asymptotically stable for µ < 1 and unstable for µ > 1.x∗ = 1− 1
µ is asymptotically stable for µ ∈ (1,3).
For µ = 3, x∗ = 1− 1µ becomes unstable but it gives rise to two
stable solutions {x1, x2} to f 2(x) = x , which form anasymptotically stable period-two cycle for f ⇒for µ = 3 a period-doubling bifurcation occurs for f at x∗ = 1− 1
µ .
The period-two cycle {x1, x2}, with x1,2 = 12 + 1
2µ ±12
√1− 2
µ −3µ2 ,
is asymptotically stable for µ ∈ (3,3.449).
For µ = 3.449, both x1 and x1 become unstable but each of themgives rise to two stable solutions to f 4(x) = x , which form anasymptotically stable period-four cycle for f ⇒for µ = 3.449 a double period-doubling bifurcation occurs for f 2 atx = x1 and x = x2.
x∗ = 0 is asymptotically stable for µ < 1 and unstable for µ > 1.x∗ = 1− 1
µ is asymptotically stable for µ ∈ (1,3).
For µ = 3, x∗ = 1− 1µ becomes unstable but it gives rise to two
stable solutions {x1, x2} to f 2(x) = x , which form anasymptotically stable period-two cycle for f ⇒for µ = 3 a period-doubling bifurcation occurs for f at x∗ = 1− 1
µ .
The period-two cycle {x1, x2}, with x1,2 = 12 + 1
2µ ±12
√1− 2
µ −3µ2 ,
is asymptotically stable for µ ∈ (3,3.449).
For µ = 3.449, both x1 and x1 become unstable but each of themgives rise to two stable solutions to f 4(x) = x , which form anasymptotically stable period-four cycle for f ⇒for µ = 3.449 a double period-doubling bifurcation occurs for f 2 atx = x1 and x = x2.
Period-doubling bifurcations (leading from a stable period-four cycle toa stable period-eight cycle; from a stable period-eight cycle to a stableperiod-sixteen cycle, and so on) occur until µ ≈ 3.57.
After this value, the system starts exhibiting aperiodic or chaoticbehavior, i.e., behavior that, although generated by a deterministicsystem, has all the characteristics of randomness.
Period-doubling bifurcations (leading from a stable period-four cycle toa stable period-eight cycle; from a stable period-eight cycle to a stableperiod-sixteen cycle, and so on) occur until µ ≈ 3.57.
After this value, the system starts exhibiting aperiodic or chaoticbehavior, i.e., behavior that, although generated by a deterministicsystem, has all the characteristics of randomness.
Period-doubling bifurcations (leading from a stable period-four cycle toa stable period-eight cycle; from a stable period-eight cycle to a stableperiod-sixteen cycle, and so on) occur until µ ≈ 3.57.
After this value, the system starts exhibiting aperiodic or chaoticbehavior, i.e., behavior that, although generated by a deterministicsystem, has all the characteristics of randomness.
For instance, we have sensitive dependence on initial conditions andthus predictions become virtually impossible.Indeed, given arbitrarily small differences in initial conditions, then thesystem will after time behave very differently.
For instance, we have sensitive dependence on initial conditions andthus predictions become virtually impossible.Indeed, given arbitrarily small differences in initial conditions, then thesystem will after time behave very differently.
For instance, we have sensitive dependence on initial conditions andthus predictions become virtually impossible.Indeed, given arbitrarily small differences in initial conditions, then thesystem will after time behave very differently.
Bifurcation diagram of fµ for µ ∈ [3.4,4] and x0 = 0.5
The chaotic (dark) region occurring for µ ∈ [3.57,4] is interrupted bysome periodicity windows.In the first window (for µ ≈ 3.62) a period-six cycle emerges;in the second window (for µ ≈ 3.74) a period-five cycle emerges;in the third window (for µ ≈ 3.83) a period-three cycle emerges.
Bifurcation diagram of fµ for µ ∈ [3.4,4] and x0 = 0.5
The chaotic (dark) region occurring for µ ∈ [3.57,4] is interrupted bysome periodicity windows.In the first window (for µ ≈ 3.62) a period-six cycle emerges;in the second window (for µ ≈ 3.74) a period-five cycle emerges;in the third window (for µ ≈ 3.83) a period-three cycle emerges.
Bifurcation diagram of fµ for µ ∈ [3.4,4] and x0 = 0.5
The chaotic (dark) region occurring for µ ∈ [3.57,4] is interrupted bysome periodicity windows.In the first window (for µ ≈ 3.62) a period-six cycle emerges;in the second window (for µ ≈ 3.74) a period-five cycle emerges;in the third window (for µ ≈ 3.83) a period-three cycle emerges.
Bifurcation diagram of fµ for µ ∈ [3.4,4] and x0 = 0.5
The chaotic (dark) region occurring for µ ∈ [3.57,4] is interrupted bysome periodicity windows.In the first window (for µ ≈ 3.62) a period-six cycle emerges;in the second window (for µ ≈ 3.74) a period-five cycle emerges;in the third window (for µ ≈ 3.83) a period-three cycle emerges.
Bifurcation diagram of fµ for µ ∈ [3.4,4] and x0 = 0.5
The chaotic (dark) region occurring for µ ∈ [3.57,4] is interrupted bysome periodicity windows.In the first window (for µ ≈ 3.62) a period-six cycle emerges;in the second window (for µ ≈ 3.74) a period-five cycle emerges;in the third window (for µ ≈ 3.83) a period-three cycle emerges.
Let f : [a,b]→ [a,b] be a continuous function which has a periodicpoint with minimal period n. If n � m in the Sharkovsky ordering, then falso has a periodic point with mimimal period m.
As a corollary, we have the following result:
Theorem (Li and Yorke)
Let f : [a,b]→ R be a continuous function which has a period-threepoint. Then f has a periodic point with minimal period m, for allm ∈ N \ {0}.
Let f : [a,b]→ [a,b] be a continuous function which has a periodicpoint with minimal period n. If n � m in the Sharkovsky ordering, then falso has a periodic point with mimimal period m.
As a corollary, we have the following result:
Theorem (Li and Yorke)
Let f : [a,b]→ R be a continuous function which has a period-threepoint. Then f has a periodic point with minimal period m, for allm ∈ N \ {0}.
Let f : [a,b]→ [a,b] be a continuous function which has a periodicpoint with minimal period n. If n � m in the Sharkovsky ordering, then falso has a periodic point with mimimal period m.
As a corollary, we have the following result:
Theorem (Li and Yorke)
Let f : [a,b]→ R be a continuous function which has a period-threepoint. Then f has a periodic point with minimal period m, for allm ∈ N \ {0}.
Let f : [a,b]→ [a,b] be a continuous function which has a periodicpoint with minimal period n. If n � m in the Sharkovsky ordering, then falso has a periodic point with mimimal period m.
As a corollary, we have the following result:
Theorem (Li and Yorke)
Let f : [a,b]→ R be a continuous function which has a period-threepoint. Then f has a periodic point with minimal period m, for allm ∈ N \ {0}.
Let f : [a,b]→ [a,b] be a continuous function which has a periodicpoint with minimal period n. If n � m in the Sharkovsky ordering, then falso has a periodic point with mimimal period m.
As a corollary, we have the following result:
Theorem (Li and Yorke)
Let f : [a,b]→ R be a continuous function which has a period-threepoint. Then f has a periodic point with minimal period m, for allm ∈ N \ {0}.
If a continuous function f over the closed interval [a,b] has a period-5cycle, then it has cycles of all periods with the possible exception ofperiod-3.
Notice that the possibility of a period-3 is not ruled out.
If f has no period-2 orbits, then there do not exist higher-order periodicorbits, including chaos.
The Sharkovsky theorem, and to some extent the Li-Yorke theorem,show that even systems that exhibit chaotic behavior still have astructure.
If a continuous function f over the closed interval [a,b] has a period-5cycle, then it has cycles of all periods with the possible exception ofperiod-3.
Notice that the possibility of a period-3 is not ruled out.
If f has no period-2 orbits, then there do not exist higher-order periodicorbits, including chaos.
The Sharkovsky theorem, and to some extent the Li-Yorke theorem,show that even systems that exhibit chaotic behavior still have astructure.
If a continuous function f over the closed interval [a,b] has a period-5cycle, then it has cycles of all periods with the possible exception ofperiod-3.
Notice that the possibility of a period-3 is not ruled out.
If f has no period-2 orbits, then there do not exist higher-order periodicorbits, including chaos.
The Sharkovsky theorem, and to some extent the Li-Yorke theorem,show that even systems that exhibit chaotic behavior still have astructure.
If a continuous function f over the closed interval [a,b] has a period-5cycle, then it has cycles of all periods with the possible exception ofperiod-3.
Notice that the possibility of a period-3 is not ruled out.
If f has no period-2 orbits, then there do not exist higher-order periodicorbits, including chaos.
The Sharkovsky theorem, and to some extent the Li-Yorke theorem,show that even systems that exhibit chaotic behavior still have astructure.
From a mathematical viewpoint, a period-doubling bifurcation for amap g(x ;µ) at the fixed point (x∗, µ∗) is characterized by∂g∂x (x∗, µ∗) = −1 and other conditions on higher-order derivatives.
The logistic map displays also a transcritical bifurcation at(x∗, µ∗) = (0,1), where x∗ = 0 loses stability in favor of x∗ = 1− 1
From a mathematical viewpoint, a period-doubling bifurcation for amap g(x ;µ) at the fixed point (x∗, µ∗) is characterized by∂g∂x (x∗, µ∗) = −1 and other conditions on higher-order derivatives.
The logistic map displays also a transcritical bifurcation at(x∗, µ∗) = (0,1), where x∗ = 0 loses stability in favor of x∗ = 1− 1
In general, when a saddle-node bifurcation for a map g(x ;µ) at thefixed point (x∗, µ∗) occurs, a stable (the node) and an unstable (thesaddle) fixed points arise.
Saddle-node bifurcation for a map g(x ;µ) at the fixed point (x∗, µ∗)
With the triple saddle-node (tangent, or fold) bifurcation of the thirditerate of the logistic map, a stable and an unstable period-three cyclesemerge.
The graph of the logistic map fµ for µ in a neighborhood of1 +√
– Elaydi SN (2007) Discrete Chaos, Second Edition: With Applicationsin Science and Engineering. CRC Press, Taylor & Francis Group,Boca Raton, Florida. Chapters 1-2, Paragraphs 1.2, 1.4–1.9, 2.5, 2.6
– Shone R (2002) Economic Dynamics. Phase Diagrams and TheirEconomic Application, second ed. Cambridge University Press,Cambridge. Chapter 3, Paragraphs 3.1–3.5
Introduction to Heterogeneous Agents Models (HAMs)
We will deal with HAMs presenting the real and the financial sides ofthe economy, in order to study the interactions, mainly in terms ofstability/instability, of the two sectors.
We will build our models by blocks of increasing complexity:
1) We will consider just the financial sector.
2) We will consider just the real sector.
3) We will jointly consider the two sectors via the interaction degreeapproach.
4) We will show how to add further elements, such as the switchingmechanism, endogenous beliefs about the fundamental, etc.
For 3) and 4) we will need to analyze 2D and 3D systems.
Introduction to Heterogeneous Agents Models (HAMs)
We will deal with HAMs presenting the real and the financial sides ofthe economy, in order to study the interactions, mainly in terms ofstability/instability, of the two sectors.
We will build our models by blocks of increasing complexity:
1) We will consider just the financial sector.
2) We will consider just the real sector.
3) We will jointly consider the two sectors via the interaction degreeapproach.
4) We will show how to add further elements, such as the switchingmechanism, endogenous beliefs about the fundamental, etc.
For 3) and 4) we will need to analyze 2D and 3D systems.
Introduction to Heterogeneous Agents Models (HAMs)
We will deal with HAMs presenting the real and the financial sides ofthe economy, in order to study the interactions, mainly in terms ofstability/instability, of the two sectors.
We will build our models by blocks of increasing complexity:
1) We will consider just the financial sector.
2) We will consider just the real sector.
3) We will jointly consider the two sectors via the interaction degreeapproach.
4) We will show how to add further elements, such as the switchingmechanism, endogenous beliefs about the fundamental, etc.
For 3) and 4) we will need to analyze 2D and 3D systems.
Introduction to Heterogeneous Agents Models (HAMs)
We will deal with HAMs presenting the real and the financial sides ofthe economy, in order to study the interactions, mainly in terms ofstability/instability, of the two sectors.
We will build our models by blocks of increasing complexity:
1) We will consider just the financial sector.
2) We will consider just the real sector.
3) We will jointly consider the two sectors via the interaction degreeapproach.
4) We will show how to add further elements, such as the switchingmechanism, endogenous beliefs about the fundamental, etc.
For 3) and 4) we will need to analyze 2D and 3D systems.
Introduction to Heterogeneous Agents Models (HAMs)
We will deal with HAMs presenting the real and the financial sides ofthe economy, in order to study the interactions, mainly in terms ofstability/instability, of the two sectors.
We will build our models by blocks of increasing complexity:
1) We will consider just the financial sector.
2) We will consider just the real sector.
3) We will jointly consider the two sectors via the interaction degreeapproach.
4) We will show how to add further elements, such as the switchingmechanism, endogenous beliefs about the fundamental, etc.
For 3) and 4) we will need to analyze 2D and 3D systems.
Introduction to Heterogeneous Agents Models (HAMs)
We will deal with HAMs presenting the real and the financial sides ofthe economy, in order to study the interactions, mainly in terms ofstability/instability, of the two sectors.
We will build our models by blocks of increasing complexity:
1) We will consider just the financial sector.
2) We will consider just the real sector.
3) We will jointly consider the two sectors via the interaction degreeapproach.
4) We will show how to add further elements, such as the switchingmechanism, endogenous beliefs about the fundamental, etc.
For 3) and 4) we will need to analyze 2D and 3D systems.
Introduction to Heterogeneous Agents Models (HAMs)
We will deal with HAMs presenting the real and the financial sides ofthe economy, in order to study the interactions, mainly in terms ofstability/instability, of the two sectors.
We will build our models by blocks of increasing complexity:
1) We will consider just the financial sector.
2) We will consider just the real sector.
3) We will jointly consider the two sectors via the interaction degreeapproach.
4) We will show how to add further elements, such as the switchingmechanism, endogenous beliefs about the fundamental, etc.
For 3) and 4) we will need to analyze 2D and 3D systems.
At first, we assume that the market is populated just byfundamentalists (see Day and Huang, 1990).
Believing that stock prices will return to their fundamental value, theybuy stocks in undervalued markets and sell stocks in overvaluedmarkets.
The market maker determines excess demand and adjusts the stockprice for the next period: if aggregate excess demand is positive(negative), price increases (decreases).
Pt+1 − Pt = γg(Dt ),
where γ > 0 is the market maker reactivity, Dt = F − Pt reflects theorders placed by fundamentalists, and g is a function increasing in Dtand vanishing for Dt = 0.
At first, we assume that the market is populated just byfundamentalists (see Day and Huang, 1990).
Believing that stock prices will return to their fundamental value, theybuy stocks in undervalued markets and sell stocks in overvaluedmarkets.
The market maker determines excess demand and adjusts the stockprice for the next period: if aggregate excess demand is positive(negative), price increases (decreases).
Pt+1 − Pt = γg(Dt ),
where γ > 0 is the market maker reactivity, Dt = F − Pt reflects theorders placed by fundamentalists, and g is a function increasing in Dtand vanishing for Dt = 0.
At first, we assume that the market is populated just byfundamentalists (see Day and Huang, 1990).
Believing that stock prices will return to their fundamental value, theybuy stocks in undervalued markets and sell stocks in overvaluedmarkets.
The market maker determines excess demand and adjusts the stockprice for the next period: if aggregate excess demand is positive(negative), price increases (decreases).
Pt+1 − Pt = γg(Dt ),
where γ > 0 is the market maker reactivity, Dt = F − Pt reflects theorders placed by fundamentalists, and g is a function increasing in Dtand vanishing for Dt = 0.
At first, we assume that the market is populated just byfundamentalists (see Day and Huang, 1990).
Believing that stock prices will return to their fundamental value, theybuy stocks in undervalued markets and sell stocks in overvaluedmarkets.
The market maker determines excess demand and adjusts the stockprice for the next period: if aggregate excess demand is positive(negative), price increases (decreases).
Pt+1 − Pt = γg(Dt ),
where γ > 0 is the market maker reactivity, Dt = F − Pt reflects theorders placed by fundamentalists, and g is a function increasing in Dtand vanishing for Dt = 0.
At first, we assume that the market is populated just byfundamentalists (see Day and Huang, 1990).
Believing that stock prices will return to their fundamental value, theybuy stocks in undervalued markets and sell stocks in overvaluedmarkets.
The market maker determines excess demand and adjusts the stockprice for the next period: if aggregate excess demand is positive(negative), price increases (decreases).
Pt+1 − Pt = γg(Dt ),
where γ > 0 is the market maker reactivity, Dt = F − Pt reflects theorders placed by fundamentalists, and g is a function increasing in Dtand vanishing for Dt = 0.
At first, we assume that the market is populated just byfundamentalists (see Day and Huang, 1990).
Believing that stock prices will return to their fundamental value, theybuy stocks in undervalued markets and sell stocks in overvaluedmarkets.
The market maker determines excess demand and adjusts the stockprice for the next period: if aggregate excess demand is positive(negative), price increases (decreases).
Pt+1 − Pt = γg(Dt ),
where γ > 0 is the market maker reactivity, Dt = F − Pt reflects theorders placed by fundamentalists, and g is a function increasing in Dtand vanishing for Dt = 0.
What happens when introducing a nonlinear price adjustmentmechanism which determines a bounded price variation in every timeperiod, as done in Naimzada and Pireddu (2015a)?
Pt+1 − Pt = γg(Dt ) = γa2
(a1 + a2
a1 exp(−Dt ) + a2− 1),
with a1,a2 positive parameters.
With this choice, g is increasing in Dt and it vanishes when Dt = 0.
Moreover, g is bounded from below by −a2 and from above by a1.
What happens when introducing a nonlinear price adjustmentmechanism which determines a bounded price variation in every timeperiod, as done in Naimzada and Pireddu (2015a)?
Pt+1 − Pt = γg(Dt ) = γa2
(a1 + a2
a1 exp(−Dt ) + a2− 1),
with a1,a2 positive parameters.
With this choice, g is increasing in Dt and it vanishes when Dt = 0.
Moreover, g is bounded from below by −a2 and from above by a1.
What happens when introducing a nonlinear price adjustmentmechanism which determines a bounded price variation in every timeperiod, as done in Naimzada and Pireddu (2015a)?
Pt+1 − Pt = γg(Dt ) = γa2
(a1 + a2
a1 exp(−Dt ) + a2− 1),
with a1,a2 positive parameters.
With this choice, g is increasing in Dt and it vanishes when Dt = 0.
Moreover, g is bounded from below by −a2 and from above by a1.
What happens when introducing a nonlinear price adjustmentmechanism which determines a bounded price variation in every timeperiod, as done in Naimzada and Pireddu (2015a)?
Pt+1 − Pt = γg(Dt ) = γa2
(a1 + a2
a1 exp(−Dt ) + a2− 1),
with a1,a2 positive parameters.
With this choice, g is increasing in Dt and it vanishes when Dt = 0.
Moreover, g is bounded from below by −a2 and from above by a1.
Hence, the price variations are gradual and the presence of the twohorizontal asymptotes prevents the dynamics of the stock market fromdiverging and helps avoiding negativity issues.
The above adjustment mechanism may be implemented assuming thatthe market maker is forced by a central authority to behave in adifferent manner according to the excess demand value.
In order to avoid overreaction phenomena, he/she has to be morecautious in adjusting prices when excess demand is large, whilehe/she has more freedom when excess demand is small, i.e., whenthe system is close to an equilibrium.
Since we allow a1 and a2 to be possibly different, the market makercan react in a different manner to a positive or to a negative excessdemand.
Hence, the price variations are gradual and the presence of the twohorizontal asymptotes prevents the dynamics of the stock market fromdiverging and helps avoiding negativity issues.
The above adjustment mechanism may be implemented assuming thatthe market maker is forced by a central authority to behave in adifferent manner according to the excess demand value.
In order to avoid overreaction phenomena, he/she has to be morecautious in adjusting prices when excess demand is large, whilehe/she has more freedom when excess demand is small, i.e., whenthe system is close to an equilibrium.
Since we allow a1 and a2 to be possibly different, the market makercan react in a different manner to a positive or to a negative excessdemand.
Hence, the price variations are gradual and the presence of the twohorizontal asymptotes prevents the dynamics of the stock market fromdiverging and helps avoiding negativity issues.
The above adjustment mechanism may be implemented assuming thatthe market maker is forced by a central authority to behave in adifferent manner according to the excess demand value.
In order to avoid overreaction phenomena, he/she has to be morecautious in adjusting prices when excess demand is large, whilehe/she has more freedom when excess demand is small, i.e., whenthe system is close to an equilibrium.
Since we allow a1 and a2 to be possibly different, the market makercan react in a different manner to a positive or to a negative excessdemand.
Hence, the price variations are gradual and the presence of the twohorizontal asymptotes prevents the dynamics of the stock market fromdiverging and helps avoiding negativity issues.
The above adjustment mechanism may be implemented assuming thatthe market maker is forced by a central authority to behave in adifferent manner according to the excess demand value.
In order to avoid overreaction phenomena, he/she has to be morecautious in adjusting prices when excess demand is large, whilehe/she has more freedom when excess demand is small, i.e., whenthe system is close to an equilibrium.
Since we allow a1 and a2 to be possibly different, the market makercan react in a different manner to a positive or to a negative excessdemand.
For any given value of γ and η, either smaller or larger than 1, it ispossible to find a1 and a2 sufficiently small, so that our stabilitycondition is satisfied, even for those values of γ and η that make thenonzero steady states in Tramontana et al. (2009) unstable.
When a1 or a2 are sufficiently small, the map ψ is strictly increasing.
This prevents the existence of interesting dynamics.
For any given value of γ and η, either smaller or larger than 1, it ispossible to find a1 and a2 sufficiently small, so that our stabilitycondition is satisfied, even for those values of γ and η that make thenonzero steady states in Tramontana et al. (2009) unstable.
When a1 or a2 are sufficiently small, the map ψ is strictly increasing.
This prevents the existence of interesting dynamics.
For any given value of γ and η, either smaller or larger than 1, it ispossible to find a1 and a2 sufficiently small, so that our stabilitycondition is satisfied, even for those values of γ and η that make thenonzero steady states in Tramontana et al. (2009) unstable.
When a1 or a2 are sufficiently small, the map ψ is strictly increasing.
This prevents the existence of interesting dynamics.
For any given value of γ and η, either smaller or larger than 1, it ispossible to find a1 and a2 sufficiently small, so that our stabilitycondition is satisfied, even for those values of γ and η that make thenonzero steady states in Tramontana et al. (2009) unstable.
When a1 or a2 are sufficiently small, the map ψ is strictly increasing.
This prevents the existence of interesting dynamics.
For any given value of γ and η, either smaller or larger than 1, it ispossible to find a1 and a2 sufficiently small, so that our stabilitycondition is satisfied, even for those values of γ and η that make thenonzero steady states in Tramontana et al. (2009) unstable.
When a1 or a2 are sufficiently small, the map ψ is strictly increasing.
This prevents the existence of interesting dynamics.
For any given value of γ and η, either smaller or larger than 1, it ispossible to find a1 and a2 sufficiently small, so that our stabilitycondition is satisfied, even for those values of γ and η that make thenonzero steady states in Tramontana et al. (2009) unstable.
When a1 or a2 are sufficiently small, the map ψ is strictly increasing.
This prevents the existence of interesting dynamics.
However, for intermediate values of a1 or a2, it is possible that,although X ∗1 and X ∗3 are locally asymptotically stable, they coexist withperiodic or chaotic attractors.
We consider a model with a Keynesian good market of a closedeconomy with public intervention.
The Keynesian equilibrium condition is given by
Y = C + I + G
with
I = I, G = G, C = C + cY ,
where Y is aggregate income, C is aggregate consumption, I isaggregate investment and G is government expenditure.Investment and government expenditures are exogenous and equal toI and G, respectively.In the consumption function, C is autonomous consumption andc ∈ (0,1) is the marginal propensity to consume.
We consider a model with a Keynesian good market of a closedeconomy with public intervention.
The Keynesian equilibrium condition is given by
Y = C + I + G
with
I = I, G = G, C = C + cY ,
where Y is aggregate income, C is aggregate consumption, I isaggregate investment and G is government expenditure.Investment and government expenditures are exogenous and equal toI and G, respectively.In the consumption function, C is autonomous consumption andc ∈ (0,1) is the marginal propensity to consume.
We consider a model with a Keynesian good market of a closedeconomy with public intervention.
The Keynesian equilibrium condition is given by
Y = C + I + G
with
I = I, G = G, C = C + cY ,
where Y is aggregate income, C is aggregate consumption, I isaggregate investment and G is government expenditure.Investment and government expenditures are exogenous and equal toI and G, respectively.In the consumption function, C is autonomous consumption andc ∈ (0,1) is the marginal propensity to consume.
We consider a model with a Keynesian good market of a closedeconomy with public intervention.
The Keynesian equilibrium condition is given by
Y = C + I + G
with
I = I, G = G, C = C + cY ,
where Y is aggregate income, C is aggregate consumption, I isaggregate investment and G is government expenditure.Investment and government expenditures are exogenous and equal toI and G, respectively.In the consumption function, C is autonomous consumption andc ∈ (0,1) is the marginal propensity to consume.
We consider a model with a Keynesian good market of a closedeconomy with public intervention.
The Keynesian equilibrium condition is given by
Y = C + I + G
with
I = I, G = G, C = C + cY ,
where Y is aggregate income, C is aggregate consumption, I isaggregate investment and G is government expenditure.Investment and government expenditures are exogenous and equal toI and G, respectively.In the consumption function, C is autonomous consumption andc ∈ (0,1) is the marginal propensity to consume.
We notice that µ = 21−c < µ when a1 and a2 are small enough.
Indeed, when reducing a1 and a2, we decrease the current variation ofoutput, enlarging the stability region.
With the introduction of the sigmoidal adjustment mechanism, in theinstability regime we have the emergence of an absorbing interval, i.e.,an invariant interval which eventually captures all forward trajectories.
We notice that µ = 21−c < µ when a1 and a2 are small enough.
Indeed, when reducing a1 and a2, we decrease the current variation ofoutput, enlarging the stability region.
With the introduction of the sigmoidal adjustment mechanism, in theinstability regime we have the emergence of an absorbing interval, i.e.,an invariant interval which eventually captures all forward trajectories.
We notice that µ = 21−c < µ when a1 and a2 are small enough.
Indeed, when reducing a1 and a2, we decrease the current variation ofoutput, enlarging the stability region.
With the introduction of the sigmoidal adjustment mechanism, in theinstability regime we have the emergence of an absorbing interval, i.e.,an invariant interval which eventually captures all forward trajectories.
We notice that µ = 21−c < µ when a1 and a2 are small enough.
Indeed, when reducing a1 and a2, we decrease the current variation ofoutput, enlarging the stability region.
With the introduction of the sigmoidal adjustment mechanism, in theinstability regime we have the emergence of an absorbing interval, i.e.,an invariant interval which eventually captures all forward trajectories.
3) Real and financial sectors: the interaction degreeapproach
We start presenting the model in Westerhoff (2012).
The real sector coincides con the linear framework presented above,but now private expenditure also increases with the stock price P.
Hence:I = I, G = G, Ct = C + cYt + αPt ,where c ∈ (0,1) is the marginal propensity to consume and invest fromcurrent income and α ∈ (0,1) is the marginal propensity to consumeand invest from current stock market wealth.
Imposing a linear adjustment mechanism and setting µ = 1,Westerhoff (2012) obtains
3) Real and financial sectors: the interaction degreeapproach
We start presenting the model in Westerhoff (2012).
The real sector coincides con the linear framework presented above,but now private expenditure also increases with the stock price P.
Hence:I = I, G = G, Ct = C + cYt + αPt ,where c ∈ (0,1) is the marginal propensity to consume and invest fromcurrent income and α ∈ (0,1) is the marginal propensity to consumeand invest from current stock market wealth.
Imposing a linear adjustment mechanism and setting µ = 1,Westerhoff (2012) obtains
3) Real and financial sectors: the interaction degreeapproach
We start presenting the model in Westerhoff (2012).
The real sector coincides con the linear framework presented above,but now private expenditure also increases with the stock price P.
Hence:I = I, G = G, Ct = C + cYt + αPt ,where c ∈ (0,1) is the marginal propensity to consume and invest fromcurrent income and α ∈ (0,1) is the marginal propensity to consumeand invest from current stock market wealth.
Imposing a linear adjustment mechanism and setting µ = 1,Westerhoff (2012) obtains
3) Real and financial sectors: the interaction degreeapproach
We start presenting the model in Westerhoff (2012).
The real sector coincides con the linear framework presented above,but now private expenditure also increases with the stock price P.
Hence:I = I, G = G, Ct = C + cYt + αPt ,where c ∈ (0,1) is the marginal propensity to consume and invest fromcurrent income and α ∈ (0,1) is the marginal propensity to consumeand invest from current stock market wealth.
Imposing a linear adjustment mechanism and setting µ = 1,Westerhoff (2012) obtains
3) Real and financial sectors: the interaction degreeapproach
We start presenting the model in Westerhoff (2012).
The real sector coincides con the linear framework presented above,but now private expenditure also increases with the stock price P.
Hence:I = I, G = G, Ct = C + cYt + αPt ,where c ∈ (0,1) is the marginal propensity to consume and invest fromcurrent income and α ∈ (0,1) is the marginal propensity to consumeand invest from current stock market wealth.
Imposing a linear adjustment mechanism and setting µ = 1,Westerhoff (2012) obtains
Suppose first that Pt = P. National income is then driven by theone-dimensional linear map Yt+1 = A + cYt + αP. Its unique steadystate Y ∗ = (A + αP)/(1− c) is positive and globally asymptoticallystable.Suppose now that Yt = Y . The stock price is then determined by theone-dimensional nonlinear map Pt+1 = Pt + η(Pt − dY ) +σ(dY −Pt )
3.There are three coexisting steady states P∗1 = dY and
P∗2,3 = P∗1 ±√
ησ . Steady state P∗1 is positive, yet unstable. Steady
Suppose first that Pt = P. National income is then driven by theone-dimensional linear map Yt+1 = A + cYt + αP. Its unique steadystate Y ∗ = (A + αP)/(1− c) is positive and globally asymptoticallystable.Suppose now that Yt = Y . The stock price is then determined by theone-dimensional nonlinear map Pt+1 = Pt + η(Pt − dY ) +σ(dY −Pt )
3.There are three coexisting steady states P∗1 = dY and
P∗2,3 = P∗1 ±√
ησ . Steady state P∗1 is positive, yet unstable. Steady
For the parameter values considered in Westerhoff (2012), raising theinterconnection between markets destabilizes Y ∗ and leads toincreasing income oscillations.
Since η > 1 all equilibria of the isolated stock market are unstable.Raising the interconnection between markets slightly increases themodulus of oscillations.
For the parameter values considered in Westerhoff (2012), raising theinterconnection between markets destabilizes Y ∗ and leads toincreasing income oscillations.
Since η > 1 all equilibria of the isolated stock market are unstable.Raising the interconnection between markets slightly increases themodulus of oscillations.
For the financial sector, we consider the framework with chartists andfundamentalists in Tramontana et al. (2009) and in Westerhoff (2012),but with a linear demand for fundamentalists, too.
Pt+1 = Pt + γ(DCt + DF
t ) = Pt + γ(η(Pt − Ft ) + σ(Ft − Pt )).
Similarly to Westerhoff (2012), we suppose that speculators perceivethe following relation between the fundamental value and a proxy Yt ofnational income
Ft = dYt ,
where d > 0 is a parameter capturing the above described directrelationship.
For the financial sector, we consider the framework with chartists andfundamentalists in Tramontana et al. (2009) and in Westerhoff (2012),but with a linear demand for fundamentalists, too.
Pt+1 = Pt + γ(DCt + DF
t ) = Pt + γ(η(Pt − Ft ) + σ(Ft − Pt )).
Similarly to Westerhoff (2012), we suppose that speculators perceivethe following relation between the fundamental value and a proxy Yt ofnational income
Ft = dYt ,
where d > 0 is a parameter capturing the above described directrelationship.
For the financial sector, we consider the framework with chartists andfundamentalists in Tramontana et al. (2009) and in Westerhoff (2012),but with a linear demand for fundamentalists, too.
Pt+1 = Pt + γ(DCt + DF
t ) = Pt + γ(η(Pt − Ft ) + σ(Ft − Pt )).
Similarly to Westerhoff (2012), we suppose that speculators perceivethe following relation between the fundamental value and a proxy Yt ofnational income
Ft = dYt ,
where d > 0 is a parameter capturing the above described directrelationship.
For the financial sector, we consider the framework with chartists andfundamentalists in Tramontana et al. (2009) and in Westerhoff (2012),but with a linear demand for fundamentalists, too.
Pt+1 = Pt + γ(DCt + DF
t ) = Pt + γ(η(Pt − Ft ) + σ(Ft − Pt )).
Similarly to Westerhoff (2012), we suppose that speculators perceivethe following relation between the fundamental value and a proxy Yt ofnational income
Ft = dYt ,
where d > 0 is a parameter capturing the above described directrelationship.
Since the functioning of financial markets is such that their priceadjustment mechanism is much faster than the mechanism ofadjustment of good market prices, we assume that γ → +∞.
Since the functioning of financial markets is such that their priceadjustment mechanism is much faster than the mechanism ofadjustment of good market prices, we assume that γ → +∞.
Since the functioning of financial markets is such that their priceadjustment mechanism is much faster than the mechanism ofadjustment of good market prices, we assume that γ → +∞.
Since the functioning of financial markets is such that their priceadjustment mechanism is much faster than the mechanism ofadjustment of good market prices, we assume that γ → +∞.
The bifurcation diagram w.r.t. ω for the map ξ with A = 5, c = 0.2,α = 1, d = 0.55, a1 = 3, a2 = 2, µ = 6, Y = P = 1
We highlight a multistability phenomenon characterized by thecoexistence of chaotic and periodic attractors (in red) with aperiod-eight orbit (in blue).
The bifurcation diagram w.r.t. ω for the map ξ with A = 5, c = 0.2,α = 1, d = 0.55, a1 = 3, a2 = 2, µ = 6, Y = P = 1
We highlight a multistability phenomenon characterized by thecoexistence of chaotic and periodic attractors (in red) with aperiod-eight orbit (in blue).
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