Periodic orbits for perturbations of piecewise linear systems

Carmona, Victoriano; Fernández-García, Soledad; Freire, Emilio

doi:10.1016/j.jde.2010.10.025

Cited by 11 publications

(3 citation statements)

References 29 publications

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“…Concerning smooth systems, the existence of invariant manifold were addressed in [17,18,19] among others. In the piecewise linear context, several authors study the existence, the number and the stability of invariant cones and limit cycles in piecewise linear systems, see [4,5,9,10] and references therein. For the case of nonlinear piecewise smooth systems similar results are obtained in [12,21,22] where the considered systems are approximated by linear ones.…”

Section: Introductionmentioning

confidence: 99%

Invariant manifolds for piecewise smooth differential systems in $\mathbb{R}^{3}$

Buzzi¹,

Euzébio²,

Mereu³

2020

Preprint

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In this paper some piecewise smooth perturbations of a three-dimensional differential system are considered. The existence of invariant manifolds filled by periodic orbits is obtained after suitable small perturbations of the original differential system. These manifolds emerge from a continuum of cylinders of R 3 which does exist for the piecewise smooth differential systems after a rotation of some planar algebraic polynomial curves. The main tool used in order to obtain the results is the averaging theory for piecewise smooth differential systems.

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Section: Introductionmentioning

confidence: 99%

Invariant manifolds for piecewise smooth differential systems in $\mathbb{R}^{3}$

Buzzi¹,

Euzébio²,

Mereu³

2020

Preprint

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“…However, due to the difficulty in applying these last ideas in higher dimensional systems, it has been considered, in general, for planar systems (see, for instance, [2,20,22,23]). As far as we know there are only a few works dealing this problem in higher dimensional nonsmooth systems (see, for instance, [4]). Therefore, in this paper, our interest lies in studying the persistence of periodic orbits for n-dimensional piecewise smooth vector fields having a periodannulus of periodic solutions contained in an invariant hyperplane.…”

Section: Introductionmentioning

confidence: 99%

Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

Gouveia

Llibre

Novaes

et al. 2016

Journal of Differential Equations

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Abstract. We consider a n-dimensional piecewise smooth vector fields with two zones separated by a hyperplane Σ which admits an invariant hyperplane Ω transversal to Σ containing a period-annulus A fulfilled by crossing periodic solutions. For small discontinuous perturbations of these systems we develop a Melnikov-like function to control the persistence of periodic solutions contained in A. When n = 3 we provide normal forms in the piecewise linear case. Finally we apply the Melnikov-like function to study discontinuous perturbations of the normal forms established.

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“…After writing the system in actionangle variables, these ideas were applied in [KKY97] to a different system to prove the existence of such tori. The use of perturbation methods for the existence of periodic orbits of some specific linear systems can be found in [TA07,CFGF11]. Other works have also been applied in [DLZ08,LH10,DL12] to general autonomous systems for the persistence of periodic orbits.…”

Section: Extension Of Classical Melnikov Methodsmentioning

confidence: 99%

Local and global phenomena in piecewise-defined systems: from big bang bifurcations to splitting of heteroclinic manifolds

Granados Corsellas

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In the first part, we formally study the phenomenon of the so-called big bang bifurcations, both for one and two-dimensional piecewise-smooth maps with a single switching boundary. These are a special type of organizing centers consisting on points in parameter space with co-dimension higher than one from which an infinite number of bifurcation curves emerge. These separate existence regions of periodic orbits with arbitrarily large periods. We show how a mechanism for their occurrence in piecewise-defined maps is the simultaneous collision of fixed (or periodic) points with the switching boundary. For the one-dimensional case, the sign of the eigenvalues associated with the colliding fixed points determines the possible bifurcation scenarios. When they are attracting, we show how the two typical bifurcation structures, so-called period incrementing and period adding, occur if they have different sign or both are positive, respectively. Providing rigorous arguments, we also conjecture sufficient conditions for their occurrence in two-dimensional piecewise-defined maps. In addition, we also apply these results to first and second order systems controlled with relays, systems in slide-mode control. In the second part of this thesis, we discuss global aspects of piecewise-defined Hamiltonian systems. These are piecewise-defined systems such that, when restricted to each domain given in its definition, the system is Hamiltonian. We first extend classical Melnikov theory for the case of one degree of freedom under periodic non-autonomous perturbations. We hence provide sufficient conditions for the persistence of subharmonic orbits and for the existence of transversal heteroclinic/homoclinic intersections. The crucial tool to achieve this is the so-called impact map, a regular map for which classical theory of dynamical systems can be applied. We also extend these sufficient conditions to the case when the trajectories are forced to be discontinuous by means of restitution coefficient simulating a loss of energy at the impacts. As an example, we apply our results to a system modeling the dynamical behaviour of a rocking block. Finally, we also consider the coupling of two of the previous systems under a periodic perturbation: a two and a half degrees of freedom piecewise-defined Hamiltonian system. By means of a similar technique, we also provide sufficient conditions for the existence of transversal intersections between stable and unstable manifolds of certain invariant manifolds when the perturbation is considered. In terms of the rocking blocks, these are associated with the mode of movement given by small amplitude rocking for one block while the other one follows large oscillations of small frequency. This heteroclinic intersections allow us to define the so-called scattering map, which links asymptotic dynamics in the invariant manifolds through heteroclinic connections. It is the essential tool in order to construct a heteroclinic skeleton which, when followed, can lead to the existence of Arnold diffusion: trajectories that, in large time scale destabilize the system by further accumulating energy.

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Periodic orbits for perturbations of piecewise linear systems

Cited by 11 publications

References 29 publications

Invariant manifolds for piecewise smooth differential systems in $\mathbb{R}^{3}$

Invariant manifolds for piecewise smooth differential systems in $\mathbb{R}^{3}$

Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

Local and global phenomena in piecewise-defined systems: from big bang bifurcations to splitting of heteroclinic manifolds

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