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

2. Overview The Numerical Method: Runge-Kutta-Fehlberg Modeling Stability Simulations Special Solutions

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)

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 )

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)

6. Runge-Kutta-Fehlberg Runge Kutta Methods: Truncation Error is O(h n ) No Computation of Derivatives

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(*)

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)

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

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 )

11. Runge-Kutta-Fehlberg Further Improvement: Control the Error (predefined tolerance) Minimize the Number of Mesh Points

12. Runge-Kutta-Fehlberg Compute RK Order 4 approximation, w i+1 Compute RK Order 5 approximation, ŵ i+1 τ i+1 (q h) = q 4 /h (ŵ i+1 - w i+1 ) TOL Take q ( h TOL / | ŵ i+1 - w i+1 |) ¼ Result: ODE45 Command in MATLAB

13. 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)

14. Stability Linear Re(λ) < 0 implies Asymptotic Stability Non-Linear Linearize Local Stability

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

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

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

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

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

26. Special Solutions Chaos Nonlinear Unpredictable Non-stable Periodic Orbits Out of Touch: Nerd plus Lover Fire and Ice: Nerd plus Lover (|a| < |b|)

27. Special Solutions Strange Attractors – Nonlinear Love Triangle: Romeo: Lover Juliet: Nerd Guinevere: Lover

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)

30. 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.