set5 - Geometrical Analysis of 1-D Dynamical Systems...

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1 Geometrical Analysis of 1-D Dynamical Systems Logistic equation: Equilibria or fixed points : initial conditions n * where you start and stay without evolving for all time. They correspond to zeros of the velocity function: Phase diagram The length of the arrows magnitude of the velocity (function) at that point. velocity function n rn(1 n)   n * =0 n * =1 n f(n)
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2 limit set of a point (initial condition) : It is defined as the set of limit points of the trajectory started at , for t → - . Thus, limit set of a point is the set Existence of a potential function: Consider (Gradient Dynamical System) Let there be a function V(n) Example: for the Logistic equation i.e., 00 t (n ) n lim (n ,t) n        n f(n) such that f(n) V/ n   0 n 0 n t (n ) n lim (n ,t) n  f(n) rn(1 n)  23 V(n) rn /2 rn /3   0 n
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3 Then, note that the equilibrium points for the system (a Gradient Dynamical System) are at the local extrema of the potential function. This is where the similarity with mechanical systems with potential energy functions ends!! Considering the Logistic equation: the plot of the potential function, and the equilibrium points are as follows: 23 rn rn V(n)  V(n) n n*=0 n*=1
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4 Observations: Oscillatory behavior is not possible in 1-D autonomous systems Trajectories approach the equilibrium point n * =1, but never reach it in finite time. Invariant subspaces are regions in phase space where if then for all negative and positive flow times (- < t < ). For the Logistic Equation, the invariant subspaces are: Thus, the state space is decomposed into: limit sets of any initial condition 0 n I, 0 (t,n ) I  1 2 3 I { ,0} , I {0} , } , I {0,1   45 I {1} , I {1, }  1 2 3 4 5 I I I I I and  0 n: I
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5 If limit set is 0 limit set is If limit sets are the same If Stability of Equilibria/Fixed Points An equilibrium point of say x=x * , is stable if for any initial condition x 0 , with Otherwise, it is unstable.  0 n I, the    02 n I , then and   0 3 0 n I , then (n ) {0}, and 0 (n ) {1} so on.  x f(x), 0, ( ) 0 such that      ** 00 (x x ) , (t,x ) x for all t, 0 t .        
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6 This definition of stability is very difficult to use directly to deduce stability of an equilibrium point. One needs to a priori know the solution for every given initial condition starting inside the region of size δ. Thus, one really needs to find other criteria that can be used to characterize stability without solving the differential equation . If in addition, is an asymptotically stable equilibrium.
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set5 - Geometrical Analysis of 1-D Dynamical Systems...

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