Saddle point in trapping region

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The discussion centers on saddle points within a trapping region in a 2D nonlinear system, referencing Steven Strogatz's work on index theory. It is established that the index of the trapping region is +1, indicating that the sum of the indices of fixed points inside must also equal +1. The conversation highlights the possibility of having two unstable fixed points and one saddle point within this region, with the implication that limit cycles may exist around the unstable points. The user seeks clarification on whether all trajectories converge to the saddle point's stable manifold.

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  • Understanding of saddle points in dynamical systems
  • Familiarity with index theory in nonlinear systems
  • Knowledge of stable and unstable fixed points
  • Basic concepts of limit cycles in dynamical systems
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  • Study Strogatz's "Nonlinear Dynamics and Chaos" for deeper insights into index theory
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Students and researchers in mathematics, physics, and engineering focusing on nonlinear dynamics, particularly those interested in the behavior of trajectories in trapping regions and the implications of saddle points.

Appaloosa
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hi all,

i have a question regarding saddle points. I'm looking at a 2D system which has a trapping region, all trajectories on the boundary point in. from strogatz's book on nonlinear systems I've read about index theory and so as i understand it the index of this region is +1 and the sum of the indices of any fixed points inside that region should also = +1. i understand that stable and unstable fixed points have index +1 whilst saddle points have index -1.

presumably this mean that it is possible for two unstable fixed points and a saddle to exist within this trapping region. i can't really visualize this topology however and I'm trying to work out whether limit cycles would have to exist about the two unstable fixed points or if all trajectories some how end up on the stable manifold of the saddle and are attracted to the saddle point? I'm pretty new to nonlinear systems analysis so I've probably misunderstood something somewhere which would explain which of these two scenarios actually occurs. hopefully someone can point me in the right direction?
 
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Those examples (of vector fields) can always be visualized by some drawings. I'm not sure what index in this context is, but it seems pretty obvious that saddle points relate to unstable equilibria.
 

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