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1 - L Karp Notes.1 for Dynamics I Basic Ideas of ODE's 1...

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˙ x / dx dt ± f ( x , t ) dx dt / x ± α dx x ± α dt x ± ce α t x ± φ ( t ) or x ± φ ( t ; c ) or x ± φ ( t ; c 0 , t 0 ) L. Karp Notes.1 for Dynamics, September 10, 2001 I. Basic Ideas of ODE's 1) Basic terms of ordinary differential equations (ODE's). 2) Basics of phase plane analysis. 3) Solutions and stability of linear ODE's. 4) Linear approximations of nonlinear ODE's. What is a differential equation? ODE solution to ODE x = N ( t ) family of curves t x ± = f ( x ) autonomous (no dependence on t , or more generally, on independent variable) e.g. x ± = " x ln( x ) = " t + c
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1:2 Suppose I have a boundary condition (or initial condition) x = 7 3 7 = ce Y c = 7 e ± 3 -3 ± B.C. 9 9 x = 7 e x = N ( t ; t , x ) t t 0 0 ± ( t -3) Definitions: (Equilibrium point) steady state x * x ± = 0 = f ( x , t ) * Linear autonomous example, steady state is 0. In general case, can have multiple steady states.
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* x o ² x ± * < δ Y * φ ( t ; t 0 , x 0 ) ² x ± * < ε ° 1:3 Stability : x is stable if , > 0, t $ 0 * 0 * ( , , t ) such that 0 (If I begin close to steady state, I stay close.) Asymptotic Stability: requires x to be stable and N ( t ; t , x ) 6 x as t 6 * * 0 0 (If I begin close to S.S, I approach it.) Global asymptotic stability (GAS). x converges to x regardless of initial condition. * Note: for linear example x ± = " x , x = 0 is GAS iff " < 0. We can see this by looking at solution, * or graph. Consider nonlinear autonomous example with multiple steady states x ± = f ( x ). Some steady states are stable, other are not stable. In the following figure, B & D are asymptotically stable. The stable points B & D are separated by the unstable steady state C. Note: At a steady state, f(x) = 0 and at a stable steady state, f(x) is decreasing. Therefore, if f(x) is continuous, there must be an unstable steady state between any two stable steady states.
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˙ x ± F ( x , y ) ˙ y ± G ( x , y ) x 2 ( t ) ³ y 2 ( t ) 1:4 Analysis of 2 dimensional systems. Suppose we are given a autonomous system. We want to know what happens to x and y over time. (Example: fish stocks that interact, state and control in optimal control problems, capital stock and shadow value of capital in rational expectation models.) See Clark page 169 for following existence theorem: Theorem:. If F&G are continuous, bounded, with continuous first derivatives, then unique solution to system that satisfies the initial condition x (0) = x , y (0) = y . Solution is either defined 0 0 for all t $ 0, or norm approaches 6 4 Norm: ************* Example of analysis using phase space x ± = y y ± = x 2 2 Suppose that I had an initial condition for this system (e,g, the values of x and y at time t=0) and that I solved the system of equations to obtain the value of x and y for arbitrary t. That is, I find the functions x(t) and y(t) that satisfy the system of ODEs and the initial condition. I can then project these functions on to the x,y plane. The resulting curve in x,y space is called a trajectory. (It is more precise to call it the projection of a trajectory onto phase space, but I will often refer to it merely as a A trajectory @ ). For different initial conditions I have different trajectories. The sketch of these different trajectories is called a phase portrait. We will analyze the evolution (over time) of the variable x and y by studying the phase portrait.
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