1. Chp 2. Flows on the Line

2. Geometric Thinking x’ = sin x. dt= dx/sin x. t=-ln |csc x + cot x| + C If x=x o at t=0 then C= ln|csc x o + cot x o | t = ln |csc x o + cot x o / csc x + cot x| Concepts: Flow, fixed points, stable and unstable.

3. Fixed Points and Stability Based on a view of x’=f(x) Phase Point Trajectory Phase Portrait Equilibrium points (stable and unstable) – Idea of locally stable and infinity

4. Population Growth One model is N’= rN(1- N/K) – r = growth rate & r > 0. K = carrying capacity.

5. Linear Stability Analysis Looking for rate of decay to a stable fixed point. n(t) = x(t) –x*, thus n’ = d/dt (x – x*) = x’ Becomes f(x*+n), by Taylor’s expansion – f(x*+n) = f(x*) n * f’ (x*) + O(n 2 ) – x* is a fixed point. n’ = nf’(x*) +O(n 2 ) – n’ ~= nf’(x*) if f’(x*) != 0

6. More on n’=nf(x*) If f’(x*) >0, grows, f’(x*) < 0 decays If f’(x*) = 0, test by case Example x’ = sin x. f’(x*) = cos (k*pi) – +1, k is even, -1 if k odd.

7. Oscillations First order systems can not deal with oscillations. – Flows never reverse in direction. – Can do oscillators if in a overdamped system.

8. Potentials f(x) = -dV/dx is a potential. Figure out V(x(t)) by chain rule – dV/dt = (dV/dx) (dx/dt) – By x’ = -dV/dx we get dV/dt = -(dV/dx) 2 <=0 – Means decreases along trajectories toward stable points.

9. Computer Solving Euler’s method is x n+1 = x n + f(x n ) delta t. Good, but some errors. Can do better. Use fourth-order Runge-Kutta method. – K 1 = f(x n ) delta t – K 2 = f(x n + (1/2) K 1 ) delta t – K 3 = f(x n + (1/2) K 2 ) delta t – K 4 = f(x n + K 3 ) delta t – x n+1 = x n + 1/6( K 1 + 2K 2 + 2K 3 + K 4 )

10. Problems with Computer Round off error from small steps.