Reduction to invariant cones for non-smooth systems

Küpper, Tassilo; Hosham, Hany A.

doi:10.1016/j.matcom.2010.10.004

Cited by 21 publications

(14 citation statements)

References 12 publications

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“…In fact, as exhibited in [16,17], we also identify an invariant manifold in our model. Morevoer, the trajectories converge to it in a very unusual way.…”

Section: Introductionmentioning

confidence: 93%

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Attractivity, Degeneracy and Codimension of a Typical Singularity in 3D Piecewise Smooth Vector Fields

Carvalho

Teixeira

2019

Milan J. Math.

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We address the problem of understanding the dynamics around typical singular points of 3D piecewise smooth vector fields. A model Z0 in 3D presenting a T-singularity is considered and a complete picture of its dynamics is obtained in the following way: (i) Z0 has an invariant plane π0 filled up with periodic orbits (this means that the restriction Z0|π 0 is a center around the singularity), (ii) All trajectories of Z0 converge to the surface π0, and such attraction occurs in a very non-usual and amazing way, (iii) given an arbitrary integer k ≥ 0 then Z0 can be approximated by π0-invariant piecewise smooth vector fields Zε such that the restriction Zε|π 0 has exactly k-hyperbolic limit cycles, (iv) the origin can be chosen as an asymptotic stable equilibrium of Zε when k = 0, and finally, (v) Z0 has infinite codimension in the set of all 3D piecewise smooth vector fields.2010 Mathematics Subject Classification. Primary 34A36, 34C23, 37G10.

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“…In fact, as exhibited in [16,17], we also identify an invariant manifold in our model. Morevoer, the trajectories converge to it in a very unusual way.…”

Section: Introductionmentioning

confidence: 93%

“…Moreover when we restrict the flow of Z ρ ε to π 0 , by Propositions 12 and 4, the fixed points of the first return map ϕ Z ρ ε = ϕ Y ρ ε • ϕ X occurs when t = t 3 = 4x 0 . So, take p 0 = (x 0 , −x 0 , 0) and we get (16) ϕ…”

Section: Proposition 12 Given An Arbitrary Pointmentioning

confidence: 99%

Attractivity, Degeneracy and Codimension of a Typical Singularity in 3D Piecewise Smooth Vector Fields

Carvalho

Teixeira

2019

Milan J. Math.

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“…Further, when μ = 0, we find M s := {φ}, which means that there is no sliding motion area. Furthermore, the intersection times (hit times) are constants on rays in all regions due to the homogeneity of system (3) (see [16]). Last, if ξ ∈ M 0 , we find e T 1 Aξ = e T 1 A + ξ = 0 in (4), and therefore the Filippov system has a singular point.…”

Section: • the Sliding Motion Is Governed By Linear Equations If λmentioning

confidence: 99%

“…Our approach (in collaboration with Küpper and Weiss, see [12,15,16,18,25,26]) is linked with the existence of invariant surfaces in the phase space which is separated by a discontinuity manifold. It has been shown that the existence of invariant cones for SDS plays a central role in understanding the often complicated dynamical behavior near fixed points.…”

Section: Introductionmentioning

confidence: 99%

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Bifurcations in four-dimensional switched systems

Hosham¹

2018

Adv Differ Equ

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In this paper, the focus is on a bifurcation of period-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-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.

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