Love Dynamics, Love Triangles, & Chaos DONT TRY THIS AT HOME!

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Love Dynamics, Love Triangles, & Chaos DON’T TRY THIS AT HOME!

Transcript of Love Dynamics, Love Triangles, & Chaos DONT TRY THIS AT HOME!

Page 1: Love Dynamics, Love Triangles, & Chaos DONT TRY THIS AT HOME!

Love Dynamics, Love Triangles,

& ChaosDON’T TRY THIS AT HOME!

Page 2: Love Dynamics, Love Triangles, & Chaos DONT TRY THIS AT HOME!

Overview The Numerical Method:

Runge-Kutta-Fehlberg Modeling Stability Simulations Special Solutions

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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)

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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)

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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)

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Runge-Kutta-Fehlberg

Runge Kutta Methods: Truncation Error is O(hn) No Computation of Derivatives

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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(*)

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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)

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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

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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 )

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Runge-Kutta-Fehlberg

Further Improvement: Control the Error (predefined tolerance)

Minimize the Number of Mesh Points

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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

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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)

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Stability

Stability Linear

Re(λ) < 0 implies Asymptotic Stability Non-Linear

Linearize Local Stability

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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

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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

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Some Models: Love Triangles Rj’ = aRj + b(J-G)

J’ = cRj + dJ

Rg’ = aRg + b(G-J)

G’ = eRg+ fG

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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

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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

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Special Solutions Chaos

Nonlinear Unpredictable Non-stable

Periodic Orbits Out of Touch: Nerd plus Lover Fire and Ice: Nerd plus Lover (|a| < |b|)

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Special Solutions Strange Attractors – Nonlinear Love Triangle:

Romeo: Lover Juliet: Nerd Guinevere: Lover

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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)

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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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