Clemson University TigerPrints All eses eses 5-2015 Determination of Chaos in Different Dynamical Systems Sherli Koshy-Chenthiayil Clemson University Follow this and additional works at: hps://tigerprints.clemson.edu/all_theses is esis is brought to you for free and open access by the eses at TigerPrints. It has been accepted for inclusion in All eses by an authorized administrator of TigerPrints. For more information, please contact [email protected]. Recommended Citation Koshy-Chenthiayil, Sherli, "Determination of Chaos in Different Dynamical Systems" (2015). All eses. 2115. hps://tigerprints.clemson.edu/all_theses/2115
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Clemson UniversityTigerPrints
All Theses Theses
5-2015
Determination of Chaos in Different DynamicalSystemsSherli Koshy-ChenthittayilClemson University
Follow this and additional works at: https://tigerprints.clemson.edu/all_theses
This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorizedadministrator of TigerPrints. For more information, please contact [email protected].
Recommended CitationKoshy-Chenthittayil, Sherli, "Determination of Chaos in Different Dynamical Systems" (2015). All Theses. 2115.https://tigerprints.clemson.edu/all_theses/2115
3.1 Values of parameters for microbial model presented in Kot, et.al. [7] 173.2 Table of values for model equations (3.11) - (3.14) used in the numerical
3.1 Manifold plot of forced model in [7] when ω = 5π6
and ε = 0.6 . . . . . 193.2 3-D plot depicting chaos and non-chaos with changes in ε and ω . . . 203.3 3-D plot depicting chaos and non-chaos with changes in D and Si . . 213.4 Time series plot of the solutions to the system in [6] . . . . . . . . . . 243.5 3-D plot of the Lyapunov exponent when X and Z were varied. (The
purple denotes the z-plane at 0) . . . . . . . . . . . . . . . . . . . . . 253.6 3-D plot of the Lyapunov exponent when KSX and Z were varied . . 263.7 3-D plot depicting chaos and non-chaos with changes in D and N . . 303.8 3-D plot depicting chaos and non-chaos with changes in D and P . . 31
5.1 Parallel Coordinates Plot of ε, ω, initial values of the variables x, y, zof the Forced System in [7] and the Maximum Lyapunov Exponent . 35
5.2 Parallel Coordinates Plot of ρ, β, σ, initial values of the variables x, y, zof the Lorenz system and the Maximum Lyapunov Exponent . . . . . 36
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Chapter 1
Introduction
It is an indisputable fact that chaos exists not just in theory. The objective
of this thesis is to find the parameter values for a system that determines chaos via
Lyapunov exponents. Before we delve into chaos, let us go through the background
needed for it.
1.1 Background information
• Dynamical systems A dynamical system consists of a set of possible states,
together with a rule determining the present state based on the previous state
[3]. For example consider a simple dynamical system given by xn+1 = 2xn. Here
the variable n stands for time and xn denotes the population at time n.
• Deterministic Dynamical Systems A deterministic dynamical system is one
in which the present state is determined uniquely from the past states. In our
previous example, the present population is completely determined by the pre-
vious one.
If randomness occurs in the prediction of the new state, then the system is no
1
longer deterministic but a random or stochastic process. An example of such a
process is flipping a fair coin to determine if it will rain or not. A coin has no
predictive power over rain.
Types of Dynamical Systems
• Discrete-time Dynamical Systems : If the rule is applied at discrete times, the
system is called a discrete-time dynamical system. Our example is a discrete
system.
• Continuous-time Dynamical Systems : It is essentially the limit of discrete sys-
tem with smaller and smaller updating times. In this case, the governing rule
will become a set of differential equations. Instead of expressing the current
state as a function of the previous state, the differential equation expresses the
rate of change of the current state as a function of the previous state [3].
We will be considering continuous dynamical systems with ordinary differential
equations.
As we all know, an ordinary differential equation is one in which the solutions are
functions of an independent variable. In our case the independent variable will be
time denoted by t. Such equations come in two types:
• An autonomous differential equation is one in which t does not appear explicitly.
An example for this would be the equation of pendulum given by:
dx
dt= − sinx.
2
• A nonautonomous differential equation is one where t explicitly appears. The
equation of the forced damped pendulum:
(1 + c)dx
dt= − sinx+ ρ sin t
is an example for such an equation.
Any nonautonomous system can be transformed into an autonomous system
by introducing a new variable y and setting it to be equal to t. This conversion
requires an additional differential equation. For the above example the autonomous
version would be:
(1 + c)dx
dt= − sinx+ ρ sin y
dy
dt= 1
1.2 Behavior of the dynamical systems
We shall describe the behavior of the dynamical systems in terms of equilib-
rium solutions, limit cycles and chaos.
• Equilibrium solutions: A constant solution of the autonomous differential
equationdx
dt= f(x) is called an equilibrium of the equation[3]. In other words,
it is a solution which satisfies f(x) = 0. The solutions either converge to the
equilibrium or diverge away from it.
3
• Periodic orbits: If there exists a T > 0 such that F (t + T, v0) = F (t, v0),∀t
and if v0 is not an equilibrium, then the solution F (t, v0) is called a periodic
orbit or cycle. Here F (t, v0) denotes the value of the solution at time t with
initial value v0. Also the periodic orbit traces out a simple closed curve.
• Chaotic orbit: An orbit that exhibits an unstable behavior that is not itself
fixed or periodic is called a chaotic orbit. At any point in such an orbit, there
are points arbitrarily near that will move away from the point during further
iteration. In terms of solutions, it means they are very sensitive to small per-
turbations in the initial conditions and almost all of them do not appear to be
either periodic or converge to equilibrium solutions.
For autonomous differential equations on the real line, bounded solutions must con-
verge to an equilibrium. For planar autonomous systems, solutions that are bounded
may instead converge to periodic orbits or cycles. In this case solutions cannot be
chaotic. There is no such restriction in three-dimensional cases. These results follow
from the Poincare-Bendixson Theorem. A classic three-dimensional system which
displays stable equilibria and chaotic behavior for different values of a parameter is
the Lorenz model given below:
x = σ(y − x)
y = x(ρ− z)− y
z = xy − βz.
4
For σ = 10, β = 8/3, Lorenz found that the system behaved chaotically for ρ ≥ 24.74.
The chaotic attractor is shown below:
Figure 1.1: Lorenz attractor
This figure depicts the orbit of a single set of initial conditions. This is a
numerically observed attractor since the choice of almost any initial condition in a
neighborhood of the chosen set results in a similar figure [3].
A chaotic attractor can be dissipative (volume-decreasing), locally unstable (orbits
do not settle down to stationary, periodic, or quasiperiodic motion) or stable at large
scale(i.e. they get trapped in a strange attractor).
In the next chapter, we discuss ways to measure chaos using the Lyapunov spec-
trum. In the third chapter, we go on to evaluate the spectrum for three continuous-
time dynamical systems based on biological populations. The final chapter considers
5
as future work choosing appropriate sampling algorithms to better understand the
parameter space for which we may obtain chaos.
6
Chapter 2
Evaluation of Lyapunov Spectrum
2.1 Definitions
A usual measure of chaos is finding the Lyapunov spectrum of the system. If
at least one of the Lyapunov exponents is positive then the bounded aperiodic orbit
is said to be chaotic [3]. As the systems investigated in this thesis are continuous, the
definition of the exponent will be given in terms of such a dynamical system.[2]
Consider a continuous dynamical system in an n-dimensional phase space. We are
observing the long term behavior of an infinitesimal n-sphere (i.e. sphere of very
small radius) of initial conditions. Due to the locally deforming nature of the flow,
the sphere eventually becomes an n-ellipsoid. The Lyapunov exponent is calculated
for each dimension and it is dependent on the length of the principal axis of the
ellipsoid. It is given by:
λi = limt→∞
1
tlog2
pi(t)
pi(0)(2.1)
where pi(t) denotes the length of the ellipsoidal principal axis at time t and pi(0)
7
denotes its length at time t = 0.
The exponents are generally given in decreasing order, i.e. λ1 > λ2 > · · · > λn.
The exponents give us an idea of whether a specific direction in the phase space is
contracting or expanding. An expanding direction signifies a positive exponent and
contracting a negative one. As the orientation of the ellipsoid is varying continuously,
we cannot speak of a definite direction with respect to the exponent. For a dissipative
dynamical system, we will have at least one negative Lyapunov exponent. If the
exponent is positive, we wouldn’t expect a bounded attractor unless some folding
of widely separated trajectories takes place. So for that particular direction, the
system goes through a repeated stretching and folding processes. As a result of this,
we cannot predict the long-term behavior of the system given the initial conditions
which is the very definition of chaos.
For a one-dimensional system, the Lyapunov spectrum clearly consists of one
value. For a discrete dynamical system, it is positive for a chaotic regime, zero
for a marginally stable orbit and negative for a periodic orbit [2]. For a continuous
one-dimensional dynamical system, the Lyapunov exponent will always be negative.
For a continuous three-dimensional system which is dissipative (i.e volume decreas-
ing), the possible spectra are as follows:
(+, 0,−) denotes a strange attractor, (0, 0,−) denotes a two-torus, (0,−,−) for a
limit cycle and finally (−,−,−) for a fixed point.
This can be extended to n-dimensions. The magnitude of the Lyapunov exponent
computes the attractor’s dynamics;i.e it tells us the number of orbits after which we
cannot predict the future behavior of the initial condition [2].
8
2.2 Procedure for calculation of Lyapunov Expo-
nents
The definition of Lyapunov exponents requires us to define principal axes with
initial conditions. These axes need to evolve with the equations of the system. The
issue is we cannot guarantee the condition of small separations for times on the
order of hundreds of orbital periods needed for convergence in a chaotic system. To
overcome this, the authors of [2] use a phase space together with a tangent space
approach. A fiducial trajectory (center of the sphere) is obtained by the action of
the non-linear system on some initial conditions. Now to obtain the trajectories of
points on the surface of the sphere, we consider the action of the linearized system
on points very close to the fiducial trajectory. In fact, the principal axes are defined
by the evolution via the linearized equations of an initially orthonormal vector frame
anchored to the fiducial trajectory. [2].
To define the trajectories on the points of the sphere we need the concept of a
linearized system or variational equations. Consider a dynamical system of the form
[1] Lutz Becks, Frank M.Hilker, Horst Malchow, Kalus Jurgens, Hartmut Arndt.Experimental demonstration of chaos in a microbial food web. Nature03627, 435,2005.
[2] Alan Wolf, Jack B.Swift, Harry L.Swinney, John A.Vastano. Determining lya-punov exponents from a time series. Physica, Volume 16D:Pg.285–317, 1985.
[3] Kathleen T.Alligood, Tim D.Sauer, James A.Yorke. Chaos: An Introduction toDynamical Systems. Springer-Verlag New York Inc, 1997.
[4] Siddhartha Chib and Edward Greenberg. Understanding the metropolis-hastingsalgorithm. The American Statistician, 1995.
[5] Liz Bradley , Dept.of Computer Science , Uty . of Colorado. The variationalequation notes for a course.
[6] Nikolay S.Strigul, Lev V.Kravchenko. Mathematical modeling of PGPR inocula-tion into the rhizosphere. Environmental Modeling and Software, 21:1158–1171,2006.
[7] Mark Kot, Gary S.Sayler, Terry W.Schultz. Complex dynamics in a model mi-crobial system. Bulletin of Mathematical Biology, Vol No.54(No.4):Pg.619–648,1992.