5_qual_analysis - 5 `Qualitative Analysis of Nonlinear Systems The Center Manifold Theorem Key points Qualitative analysis is an extension of

5_qual_analysis - 5 `Qualitative Analysis of Nonlinear...

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5 ` Qualitative Analysis of Nonlinear Systems: The Center Manifold Theorem Definitions A bifurcation is a qualitative change in the nature of the solution trajectories due to a parameter change. A catastrophe is a sudden jump between equilibrium surfaces. Key points Qualitative analysis is an extension of phase-plane analysis to higher-order systems. The center manifold theorem is an extension of Lyapunov’s first method to evaluate stability of systems with eigenvalues that have zero real part. References: Sastry chapter 7, Guckenheimer and Holmes chapters 1 and 3
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First-Order Example (due to Carr) [ ] ) , ( 2 1 3 c x f c x c x x = + + = & = 2 1 c c c For this example, a potential function Φ exists, obtained from x f Φ = x c x c x fdx 2 2 1 4 2 4 + + = = Φ A potential function is an “energy-type” function (this problem resembles a “marble on a hill”). Stable equilibrium points are minima of Φ . Let c 1 be fixed and c 1 = -3, and let’s vary c 2 .
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Notes: We have “jump” conditions for |c 2 | = 2. The system has one equilibrium point for |c 2 | > 2. The system has three equilibrium points for |c 2 | < 2. The system displays hysteresis (In the “marble on a hill” analogy, where the marble goes depends on which way you are coming, or on whether the marble starts on the left or the right). A bifurcation surface is a surface at which the equilibrium surfaces separate (or bifurcate) from each other. This surface is defined by: ) ( 0 ) , ( ) ( 0 ) , ( 2 2 jump x c x x f m equilibriu c x f = Φ = = For our example: 0 1 3 0 2 2 1 3 = + = + + c x c x c x By eliminating x in the two above equations, one can get the equation for the bifurcation surface: 0 4 1 27 1 2 2 3 1 = + c c The parameter c 1 is called the splitting factor. If c 1 < 0 a catastrophe can occur. If c 1 > 0 no catastrophes can occur.
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It is possible to plot 3D surfaces showing c 1 , c 2 and x eq (results in interesting geometries). Notes: It becomes difficult to apply these techniques for higher order systems because of their dimensionality. The center manifold theorem provides a technique to analyze a bifurcation/catastrophe. It is a generalization of known results for linear systems, which we will review in the following section. Linear Systems Results We consider linear systems results dealing with flows and invariant subspaces. Ax x = & 0 0 ) , ( x e t x x At = We can think of e At as a mapping of x 0 onto x(t), or as a flow generated by the vector field, Ax . Suppose A has linearly independent eigenvectors v j , j=1,…,n, that is: n j j j A I v A I λ λ λ λ ,... 0 | | 0 ] [ 1 = = Then, j t n j v e t x j λ α = 1 ) ( (where the α j depend on the initial conditions). Certain solutions are invariant under e At , that is, solutions starting on eigenvectors stay on eigenvectors.
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  • Spring '08
  • Nonlinear system, Stability theory, Dynamical systems, stable manifold, Center manifold, Center Manifold Theorem

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