Love Dynamics, Love Triangles, & Chaos DONT TRY THIS AT HOME!
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Love Dynamics, Love Triangles,
& ChaosDON’T TRY THIS AT HOME!
Overview The Numerical Method:
Runge-Kutta-Fehlberg Modeling Stability Simulations Special Solutions
Runge-Kutta-Fehlberg Assumption: f(t, y) is smooth enough
IVP: y’ = f(t, y) a ≤ t ≤ b
y(a) = y0
Taylor Polynomial about ti
y(ti+1) = y(ti) + h y’(ti) +…+ hn/n! y(n)(ti) + O(hn+i) y’ = f(t, y), y’’ = f’(t, y),…, y(k) = f(k-1)(t, y)
Runge-Kutta-Fehlberg
Then,
y(ti+1) = y(ti) + h f(ti, y(ti)) +…
+ hn/n! f(n-1)(ti, y(ti)) + O(hn+i)
OR
yi+1 = yi + h f(ti, yi) +…
+ hn/n! f(n-1)(ti, yi) + O(hn+i)
Runge-Kutta-Fehlberg
Taylor’s Method, Order 2
w0 = y0
wi+1 = wi + h f(ti, wi) + h2/2 f’(ti, wi)
Good: Truncation Error is O(hn) Bad: Computation of Derivatives
(complicated and time consuming)
Runge-Kutta-Fehlberg
Runge Kutta Methods: Truncation Error is O(hn) No Computation of Derivatives
Runge-Kutta-FehlbergIllustration:
Taylor’s Order 2 Needs (1)
f(t, y) + h/2 f’(t, y)
= f(t, y) + h/2 [ ft(t, y) + fy(t, y) y’]
= f(t, y) + h/2 ft(t, y) + h/2 fy(t, y) f(t, y) Taylor’s (again!)
c f(t+a, y+b) = c f(t,y) + a c ft(t,y)
+ b c fy(t,y) + c R(*)
Runge-Kutta-Fehlberg
Matching Coefficients: c = 1 c a = h/2 c b = h/2
which gives c = 1 a = h/2 b = h/2 f(t, y)
Runge-Kutta-Fehlberg
Then (1) can be written:
f( t + h/2, y + h/2 f(t, y) )
RK Order 2:w0 = y0
k1 = h f(ti, wi)
k2 = h f(ti + h/2, wi + k1/2)
wi+1 = wi + k2
Runge-Kutta-Fehlberg
RK Order 4:w0 = y0
k1 = h f(ti, wi)
k2 = h f(ti + h/2, wi + k1/2)
k3 = h f(ti + h/2, wi + k2/2)
k4 = h f(ti+1, wi + k3)
wi+1 = wi + 1/6 ( k1 + 2 k2 + 2 k3 + k4 )
Runge-Kutta-Fehlberg
Further Improvement: Control the Error (predefined tolerance)
Minimize the Number of Mesh Points
Runge-Kutta-Fehlberg
Runge-Kutta-Fehlberg Compute RK Order 4 approximation, wi+1
Compute RK Order 5 approximation, ŵi+1
τi+1 (q h) = q4/h (ŵi+1 - wi+1) ≤ TOL
Take q ≤ ( h TOL / | ŵi+1 - wi+1 |) ¼
Result: ODE45 Command in MATLAB
Modeling Linear Systems:
ů = A*u Solution: u(t) = u(0)exp(At)
Predetermined No Chaos Well Documented
Non-Linear Systems: ů = f(u, λ)
Stability
Stability Linear
Re(λ) < 0 implies Asymptotic Stability Non-Linear
Linearize Local Stability
Some Models R’ = aR + bJ
J’ = cR + dJ Romantic Styles
Eager Beaver: a > 0, b > 0 Narcissistic Nerd: a > 0, b < 0 Cautious Lover: a < 0, b > 0 Hermit: a < 0, b < 0
Some Models R’ = aR + bJ
J’ = cR + dJ Simple Linear Model
Out of Touch with One’s Own Feelings: a = d = 0 Fire and Ice: c = -b, d = -a Peas in a Pod: c = b, a = d
Some Models: Love Triangles Rj’ = aRj + b(J-G)
J’ = cRj + dJ
Rg’ = aRg + b(G-J)
G’ = eRg+ fG
Some Models: Nonlinear R’ = aR + bJ(1-|J|)
J’ = cR(1-|R|) + dJ Simple Nonlinear Model
Eager Beaver: c = d = 1 Hermit: a = b = -2
Some Models: Nonlinear Love Triangles
Rj’ = aRj + b(J - G)(1 - |J – G|)
J’ = cRj(1 - |Rj|)+ dJ
Rg’ = aRg + b(G - J )(1 - |G – J|)
G’ = eRg(1 - |Rg|)+ fG Love Triangle Nonlinear Model
Cautious Lovers: a = -3, b = 4; e = 2, f = -1 Narcissistic Nerd: c = -7, d = 2
Special Solutions Chaos
Nonlinear Unpredictable Non-stable
Periodic Orbits Out of Touch: Nerd plus Lover Fire and Ice: Nerd plus Lover (|a| < |b|)
Special Solutions Strange Attractors – Nonlinear Love Triangle:
Romeo: Lover Juliet: Nerd Guinevere: Lover
Stability (cont.) Hyperbolic Equilibrium Point:
An equilibrium point is hyperbolic if the Jacobian has no eigenvalues with the real part equal to zero (stability is based on the real part)
Stability (cont.) Hartman-Grobman Theorem
Let ů=A*u be the linearization of ů=f(u). If A is hyperbolic, then both systems are equivalent around the equilibrium point.
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