Download presentation

Presentation is loading. Please wait.

Published byCarol Ezell Modified over 4 years ago

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.

Similar presentations

OK

2.12.1 Complex Zeros and the Fundamental Theorem of Algebra.

2.12.1 Complex Zeros and the Fundamental Theorem of Algebra.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Oral cavity anatomy and physiology ppt on cells Ppt on types of forest in india Ppt on role of youth in indian politics Ppt on cloud computing pdf Ppt on sound navigation and ranging system sensor Ppt on job evaluation and merit rating Ppt on statistics and probability for dummies Ppt on common man made disasters Ppt on water softening techniques to improve Ppt on 14 principles of henri fayol principles