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Martins, Ricardo M.; Tonon, Durval J.

The chaotic behaviour of piecewise smooth differential equations on two-dimensional torus and sphere. (English) Zbl 1433.34023

Dyn. Syst. 34, No. 2, 356-373 (2019).
The authors study the global dynamics of piecewise-smooth, autonomous differential equations defined on two-dimensional sphere and torus.
Let \(M\) be either the two-dimensional sphere or the torus. Assume that there exists a smooth curve \(\Sigma\) breaking the manifold \(M\) in two connected components, \(M^+\) and \(M^-\), on which the system is smooth. Moreover the dynamics on \(\Sigma\) is assumed to satisfy the Filippov convention.
In this paper, conditions are given so that “generic” families of piecewise smooth equations on \(M\) admit periodic and dense trajectories. Further, a non-deterministic chaotic behaviour is investigated for such equations and global bifurcations are also classified.
Reviewer: Alessandro Calamai (Ancona)

MSC:

34A36 Discontinuous ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C40 Ordinary differential equations and systems on manifolds
34C28 Complex behavior and chaotic systems of ordinary differential equations

Keywords:

piecewise smooth differential equations; ordinary differential equation on manifolds; structural stability; global dynamics; bifurcations; non-wandering set
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References:

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