Abstract
In this paper, the focus is on a bifurcation of period-mathcal{K} orbit that can occur in a class of Filippov-type four-dimensional homogenous linear switched systems. We introduce a theoretical framework for analyzing the generalized Poincaré map corresponding to switching manifold. This provides an approach to capturing the possible results concerning the existence of a period-mathcal{K} orbit, stability, a number of invariant cones, and related bifurcation phenomena. Moreover, the analysis identifies criteria for the existence of multi-sliding bifurcation depending on the sensitivity of the system behavior with respect to changes in parameters. Our results show that a period-two orbit involves multi-sliding bifurcation from a period-one orbit. Further, the existence of invariant torus, crossing-sliding, and grazing-sliding bifurcation is investigated. Numerical simulations are carried out to illustrate the results.
Highlights
Higher dimensional systems (n > 3) are of great significance for applications as modeling problems often require higher dimensions
The existence of invariant cones is equivalent to the existence of positive real eigenvalues of the return Poincaré map. We extend this approach to investigate the existence of the period-K orbit, a number of invariant cones, and associated phenomena
4 Conclusion In this paper, we have discussed the bifurcation of period-K orbit of a class of fourdimensional homogeneous linear SDSs in two possible behaviors: transversal crossing and attractive sliding mode
Summary
Higher dimensional systems (n > 3) are of great significance for applications as modeling problems often require higher dimensions. The existing results for period-K orbit (K > 1) are only associated with the problems involving nonlinear or non-homogeneous operators [5, 22], we point out in this paper that the homogeneous linear four-dimensional SDS can exhibit period-K orbits with/without sliding mode, which is quite different from what is known for three-dimensional SDSs with single discontinuity surface. In such cases we observe a sudden transition through the discontinuity manifold
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