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

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Love Dynamics, Love Triangles, & Chaos DONT TRY THIS AT HOME! Slide 2 Overview The Numerical Method: Runge-Kutta-Fehlberg Modeling Stability Simulations Special Solutions Slide 3 Runge-Kutta-Fehlberg Assumption: f(t, y) is smooth enough IVP: y = f(t, y)a t b y(a) = y 0 Taylor Polynomial about t i y(t i+1 ) = y(t i ) + h y(t i ) ++ h n /n! y (n) (t i ) + O(h n+i ) y = f(t, y), y = f(t, y),, y (k) = f (k-1) (t, y) Slide 4 Runge-Kutta-Fehlberg Then, y(t i+1 ) = y(t i ) + h f(t i, y(t i )) + + h n /n! f (n-1) (t i, y(t i )) + O(h n+i ) OR y i+1 = y i + h f(t i, y i ) + + h n /n! f (n-1) (t i, y i ) + O(h n+i ) Slide 5 Runge-Kutta-Fehlberg Taylors Method, Order 2 w 0 = y 0 w i+1 = w i + h f(t i, w i ) + h 2 /2 f(t i, w i ) Good: Truncation Error is O(h n ) Bad: Computation of Derivatives (complicated and time consuming) Slide 6 Runge-Kutta-Fehlberg Runge Kutta Methods: Truncation Error is O(h n ) No Computation of Derivatives Slide 7 Runge-Kutta-Fehlberg Illustration: Taylors Order 2 Needs (1) f(t, y) + h/2 f(t, y) = f(t, y) + h/2 [ f t (t, y) + f y (t, y) y] = f(t, y) + h/2 f t (t, y) + h/2 f y (t, y) f(t, y) Taylors (again!) c f(t+a, y+b) = c f(t,y) + a c f t (t,y) + b c f y (t,y) + c R(*) Slide 8 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) Slide 9 Runge-Kutta-Fehlberg Then (1) can be written: f( t + h/2, y + h/2 f(t, y) ) RK Order 2: w 0 = y 0 k 1 = h f(t i, w i ) k 2 = h f(t i + h/2, w i + k 1 /2) w i+1 = w i + k 2 Slide 10 Runge-Kutta-Fehlberg RK Order 4: w 0 = y 0 k 1 = h f(t i, w i ) k 2 = h f(t i + h/2, w i + k 1 /2) k 3 = h f(t i + h/2, w i + k 2 /2) k 4 = h f(t i+1, w i + k 3 ) w i+1 = w i + 1/6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) Slide 11 Runge-Kutta-Fehlberg Further Improvement: Control the Error (predefined tolerance) Minimize the Number of Mesh Points Slide 12 Runge-Kutta-Fehlberg Compute RK Order 4 approximation, w i+1 Compute RK Order 5 approximation, i+1 i+1 (q h) = q 4 /h ( i+1 - w i+1 ) TOL Take q ( h TOL / | i+1 - w i+1 |) Result: ODE45 Command in MATLAB Slide 13 Modeling Linear Systems: = A*u Solution: u(t) = u(0)exp(At) Predetermined No Chaos Well Documented Non-Linear Systems: = f(u, ) Slide 14 Stability Linear Re() < 0 implies Asymptotic Stability Non-Linear Linearize Local Stability Slide 15 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 Hermit: a < 0, b < 0 Slide 16 Some Models R = aR + bJ J = cR + dJ Simple Linear Model Out of Touch with Ones Own Feelings: a = d = 0 Fire and Ice: c = -b, d = -a Peas in a Pod: c = b, a = d Slide 17 Slide 18 Some Models: Love Triangles R j = aR j + b(J-G) J = cR j + dJ R g = aR g + b(G-J) G = eR g + fG Slide 19 Slide 20 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 Slide 21 Slide 22 Slide 23 Some Models: Nonlinear Love Triangles R j = aR j + b(J - G)(1 - |J G|) J = cR j (1 - |R j |)+ dJ R g = aR g + b(G - J )(1 - |G J|) G = eR g (1 - |R g |)+ fG Love Triangle Nonlinear Model Cautious Lovers: a = -3, b = 4; e = 2, f = -1 Narcissistic Nerd: c = -7, d = 2 Slide 24 Slide 25 Slide 26 Special Solutions Chaos Nonlinear Unpredictable Non-stable Periodic Orbits Out of Touch: Nerd plus Lover Fire and Ice: Nerd plus Lover (|a| < |b|) Slide 27 Special Solutions Strange Attractors Nonlinear Love Triangle: Romeo: Lover Juliet: Nerd Guinevere: Lover Slide 28 Slide 29 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) Slide 30 Stability (cont.) Hartman-Grobman Theorem Let =A*u be the linearization of =f(u). If A is hyperbolic, then both systems are equivalent around the equilibrium point. Slide 31 Slide 32 Slide 33 Slide 34