Скачать или смотреть Analysis of Nonlinear Systems, Part 4 (Trapping Regions, Singular Perturbations, Poincare-Bendixson)

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(0:07) This video will include a concept near and dear to me for my research: trapping regions for systems of ordinary differential equations. (0:29) Competing species model (dx/dt = 2x(5/2 - x) - x*y, dy/dt = y(6-y) - 3x*y. Factor to find nullclines and equilibrium points. Draw the phase portrait in the phase plane. (7:55) We could linearize with the Jacobian matrix, but I won't take the time to do that in this video. Instead, we'll just sketch it as fast as possible without linearization. There are some separatrix solutions as well (separatrices). Think about the real life meaning for the competing species. One species or the other could die off. (13:44) Is it possible to rigorously prove that solutions entering the triangular regions must approach the equilibrium point in that region? Yes. Topological arguments can help. The upper triangle is an example of a trapping region. Solutions with initial conditions along the boundary have to stay in the region for all positive times. In fact, there must also be an invariant set which consists of points on solutions which stay inside the region for all time (positive and negative). In two dimensions, the limiting invariant set of a solution curve must be an equilibrium point, a periodic solution, or a separatrix cycle. (19:42) A more significant application of a trapping region: a modified version of the Van der Pol system (add a small parameter as well). We want to prove there is a periodic solution when the parameter is positive and small. (21:20) The geometric idea of the proof is based on the idea of a singular perturbation. The singular system is easy to analyze. It has infinitely many equilibrium points along a cubic curve. When the parametric is a small positive number, there exists a period solution that has a fast-slow nature (it move "fast" sometimes and "slow" other times). (29:23) Once the trapping region is made, then the Poincare-Bendixson theorem can be used to prove there is a periodic solution inside it (as long as there are no equilibrium points inside it). For small values of the parameter, the trapping region is not too hard to make (though it is a bit tricky). (34:47) This is related to my research with the Conley index for singularly perturbed systems. I came up with techniques to isolate the periodic solution with a simpler region. Technically it's not a trapping region, even when epsilon is small. But the periodic solution is still inside of it. Instead, the region is what is called a singular isolating neighborhood. It contains an isolated invariant set whose Conley index has certain properties.

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