The Melnikov method and subharmonic orbits in a piecewise smooth system

In this work we consider a two-dimensional piecewise smooth system, defined in two domains separated by the switching manifold x = 0. We assume that there exists a piecewise-defined continuous Hamiltonian that is a first integral of the system. We also suppose that the system possesses an invisible fold-fold at the origin and two heteroclinic orbits connecting two hyperbolic critical points on either side of x = 0. Finally, we assume that the region closed by these heteroclinic connections is fully covered by periodic orbits surrounding the origin, whose periods monotonically increase as they approach the heteroclinic connection. When considering a non-autonomous (T-periodic) Hamiltonian perturbation of amplitude ", using an impact map, we rigorously prove that, for every n and m relatively prime and " > 0 small enough, there exists a nT-periodic orbit impacting 2m times with the switching manifold at every period if a modified subharmonic Melnikov function possesses a simple zero. We also prove that, if the orbits are discontinuous when they cross x = 0, then all these orbits exist if the relative size of " > 0 with respect to the magnitude of this jump is large enough. We also obtain similar conditions for the splitting of the heteroclinic connections.Preprin

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

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

The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks

We consider a non-autonomous dynamical system formed by coupling two piecewise-smooth systems in \RR^2 through a non-autonomous periodic perturbation. We study the dynamics around one of the heteroclinic orbits of one of the piecewise-smooth systems. In the unperturbed case, the system possesses two $C^0$ normally hyperbolic invariant manifolds of dimension two with a couple of three dimensional heteroclinic manifolds between them. These heteroclinic manifolds are foliated by heteroclinic connections between $C^0$ tori located at the same energy levels. By means of the {\em impact map} we prove the persistence of these objects under perturbation. In addition, we provide sufficient conditions of the existence of transversal heteroclinic intersections through the existence of simple zeros of Melnikov-like functions. The heteroclinic manifolds allow us to define the {\em scattering map}, which links asymptotic dynamics in the invariant manifolds through heteroclinic connections. First order properties of this map provide sufficient conditions for the asymptotic dynamics to be located in different energy levels in the perturbed invariant manifolds. Hence we have an essential tool for the construction of a heteroclinic skeleton which, when followed, can lead to the existence of Arnol'd diffusion: trajectories that, on large time scales, destabilize the system by further accumulating energy. We validate all the theoretical results with detailed numerical computations of a mechanical system with impacts, formed by the linkage of two rocking blocks with a spring

An Application of the Melnikov Method to a Piecewise Oscillator

In this paper we present a new application of the Melnikov method to a class of periodically perturbed Duffing equations where the nonlinearity is non-smooth as otherwise required in the classical applications. Extensions of the Melnikov method to these situations is a topic with growing interests from the researchers in the past decade. Our model, motivated by the study of mechanical vibrations for systems with “stops”, considers a case of a nonlinear equation with piecewise linear components. This allows us to provide a precise analytical representation of the homoclinic orbit for the associated autonomous planar system and thus obtain simply computable conditions for the zeros of the associated Melnikov function

Study of periodic orbits in periodic perturbations of planar reversible Filippov systems having a two-fold cycle

We study the existence of periodic solutions in a class of planar Filippov systems obtained from non-autonomous periodic perturbations of reversible piecewise smooth differential systems. It is assumed that the unperturbed system presents a simple two-fold cycle, which is characterized by a closed trajectory connecting a visible two-fold singularity to itself. It is shown that under certain generic conditions the perturbed system has sliding and crossing periodic solutions. In order to get our results, Melnikov's ideas were applied together with tools from the geometric singular perturbation theory. Finally, a study of a perturbed piecewise Hamiltonian model is performed

Periodic solutions and chaotic dynamics in a Duffing equation model of charged particles

The emergence of chaotic behavior in many physical systems has triggered the curiosity of scientists for a long time. Their study has been concentrated in understanding which are the underlying laws that govern such dynamics and eventually aim to suppress such (often) undesired behavior. In layman terms, a system is defined chaotic when two orbits that initially are very near to each other will diverge in exponential time. Clearly, this translates to the fact that a chaotic system can hardly have regular behavior, a property that is also often required even for human-made systems. An example is that of particle accelerators used a lot in the study of experimental physics. The main principle is that of forcing a large number of particles to move periodically in a toroidal space in order to collide with each other. Another example is that of the tokamak, a particular accelerator built to generate plasma, one of the states of the matter. In both cases, it is crucial for the sake of the accelerating process, to have regular periodic behavior of the particles instead of a chaotic one. In this dissertation, we have studied the question of chaos in mathematical models for the motion of magnetically charged particles inside the tokamak in the presence or absence of plasma. We start by a model introduced by Cambon et al., which describes in general mathematical terms, also known as the Duffing modes, the formalism of the above problem. The central core of this work reviews the necessary mathematical tools to tackle this problem, such as the theorem of the Linked Twisted maps and the variational Hamiltonian equations which describe the evolutionary dynamics of the system under consideration. Extensive analytical and numerical tools are required in this thesis work to investigate the presence of chaos, known as chaos indicator. The main ones we have used here are the Poincar \u301e Map, the Maximum Lyapunov Exponent (MLE), and the SALI and GALI methods. Using the techniques mentioned above, we have studied our problem analytically and validated our results numerically for the particular case of the Duffing equation, which applies to the motion of charged particles in the tokamak. In detail, we first discuss the presence of chaotic dynamics of charged particles inside an idealized magnetic field, sug- gested by a tokamak type configuration. Our model is based on a periodically perturbed Hamiltonian system in a half-plane r \ubf 0. We propose a simple mechanism producing complex dynamics, based on the theory of Linked Twist Maps jointly with the method of stretching along the paths. A key step in our argument relies on the monotonicity of the period map associated with the unperturbed planar system. In the second part of our results, we give an analytical proof of the presence of complex dynamics for a model of charged particles in a magnetic field. Our method is based on the theory of topological horseshoes and applied to a periodically perturbed Duffing equation. The existence of chaos is proved for sufficiently large, but explicitly computable, periods. In the latter part, we study the generalized forementioned Duffing equations and study the chaoticity using the Melnikov topological method and verify the results numerically for the models of Wang & You and the tokamak one