One-Dimensional Dynamics

Transcription

1 One-Dimensional Dnamics We aim to characterize the dnamics completel describe the phase portrait) of the one-dimensional autonomous differential equation Before proceeding we worr about 1.1) having a solution. x = Fx). 1.1) Definition 1.1 We sa a function f : I R is uniforml Lipschitz on an interval I if a constant M exists so that fx) f) M x for all x, I. Example 1. Consider fx) = x on I = [ 1,1]. Then the above definition applies with M = 1, and so f is Lipschitz but not differentiable) on I. Example 1.3 Suppose f : [a,b] R is continuousl differentiable we sa f C 1 [a,b])). Then f is Lipschitz on [a,b]. Indeed, the mean-value theorem applies here so that fx) fx) = f c)x ) for some c between x and. Since f is continuous, it has a max and min. Thus f c) M for some M. It follows fx) fx) M x and f is Lipschitz. The main existence and uniqueness theorem follows. Its proof is at a MAT 371 level and can be found in man elementar texts on ODEs. Theorem 1.4 Consider the differential equation x = Fx), x0) = x 0. Suppose F is Lipschitz on an interval I with 0 I. Then an interval I 1 exists with 0 I 1 ) so that the ODE has an unique C 1 solution on I 1. Proof. We prove the uniqueness onl. Suppose two solutions, x 1 t), x t) exist. Set wt) = x 1 t) x t). Then w solves w = Fx 1 ) Fx ), w0) = 0. 1

2 We need to show w = 0 and we do this b showing w 0. Multipling this ODE b w and using the properties of F we find ) 1 w w = w = Fx 1 ) Fx ))w Fx 1 ) Fx ) w M w. Let us set = w. Then we have We ma write this as Integrating, we get t M. e Mt ) 0. e Mt ) dt 0, or t) 0). That is, w t) 0 and we see w = 0 or x 1 t) = x t). 0 Even if the function F is nice, solutions ma reach infinit in finite time. We will have to live with this. Example 1.5 Consider B a straightforward integration we find x = x x0) = x 0. xt) = x 0 1 x 0 t, and for positive x 0 the solution blows up at t = 1/x 0. Using the uniqueness of solutions we ma characterize the dnamics of 1.1). Henceforth we assume the conditions of Theorem 1.4 hold. Definition 1.6 We call a point x, satisfing Fx ) = 0, a fixed point. Theorem 1.7 If the initial data for 1.1) is a fixed point, then the solution is xt) = x for all t R. Proof. Note xt) = x is a solution to x = Fx) x0) = x. Since the solution is unique, this is the onl solution.

3 Theorem 1.8 Suppose a solution xt) exists on an interval I and xt 0 ) = x for some t 0 I. Then xt) = x for all t R. Proof. Set zt) = xt+t 0 ). Then z solves z = x t+t 0 ) = Fxt+t 0 )) = Fz), z0) = x on some interval. Theorem 1.7 shows zt) = x for all t R. That is, xt+t 0 ) = x for all t. The result follows. The next result shows trajectories cannot cross fixed points. Corollar 1.9 If x0) x, then xt) x for all t. Proof. This is just the contrapositive of the previous theorem. Corollar 1.10 Solutions to 1.1) are strictl monotone. That is, the either move, for all t, in the positive x direction, or the move, for all t, in the negative direction. Proof. If a solution xt) changes direction, then for some t 1,t we would have x t 1 ) > 0 and x t ) < 0 or visa versa). Since x enjos the intermediate-value propert, there would beatime t c between t 1 and t such that x t c ) = 0. This implies thesolution passed through a fixed point, which cannot happen. Corollar 1.11 Solution of 1.1) cannot oscillate. There are no periodic solutions. Proof. As we have seen, trajectories are monotone. Thus we ma completel characterize the behavior dnamics) of solutions to 1.1): The solutions to x = Fx) either approach a fixed point or diverge to ±. There is a no-think wa of determining the stabilit of a fixed point. 3

4 Theorem 1.1 Poincaré-Lpunov) Consider x = Fx), x0) = x 0. Suppose F M near x 0 so the solution is unique but ma blow up in finite time). Suppose further that Fx ) = 0 and F x ) < 0. Then x is asmptoticall stable. That is, for x 0 x r, with r = F x ) M, we have lim xt) = t x. In particular, solutions starting near x exists for all positive time and converge to x. Proof. We note that x ) = 0. Thus b Talor s theorem x x ) = Fx) = F x )+F ct)) x x ), where ct) is between xt) and x. Set zt) = x x. Then z = F x )z +F ct)) z. We show z 0 as t. Suppose z0) < r. Let t 1 be the largest time so that zt) r on 0 < t t 1. Multipling b z we find ) 1 z = F x )z +F ct))z z F x )z +Mr z. In an attempt to make this more familiar, set = z. Then 1 F x )+ Mr ) = F x )+ F x ) ) = F x ) F x ) ) = F x ). It follows t)e F x )t ) 0. Integrating both sides we find t) 0)e Fx )t r e Fx )t < r or zt) < r on 0 < t t 1. This implies that t 1 = and z 0 as t. 4

5 Some Worked Examples Example.1..1) Find the phase portrait for x = 4x 16. Solution. We first find the fixed points. The solve 4x 16 = 0 or x = ±. To find the stabilit, we compute F x ) = 8x. Thus F ) > 0 and F ) < 0. B the Poincaré-Lpunov theorem x = is unstable and x = is asmptoticall stable. B our characterization of 1D dnamics solutions starting less than converge monotonicall to x = ; solutions starting in,) converge to x = ; solutions starting above converge to + probabl finite time). 5