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

Impulsive perturbations to differential equations: stable/unstable pseudo-manifolds, heteroclinic connections, and flux

State-dependent time-impulsive perturbations to a two-dimensional autonomous flow with stable and unstable manifolds are analysed by posing in terms of an integral equation which is valid in both forwards- and backwards-time. The impulses destroy the smooth invariant manifolds, necessitating new definitions for stable and unstable pseudo-manifolds. Their time-evolution is characterised by solving a Volterra integral equation of the second kind with discontinuous inhomogeniety. A criteria for heteroclinic trajectory persistence in this impulsive context is developed, as is a quantification of an instantaneous flux across broken heteroclinic manifolds. Several examples, including a kicked Duffing oscillator and an underwater explosion in the vicinity of an eddy, are used to illustrate the theory

Some contributions to the analysis of piecewise linear systems.

This thesis consists of two parts, with contributions to the analysis of dynamical systems in continuous time and in discrete time, respectively. In the first part, we study several models of memristor oscillators of dimension three and four, providing for the first time rigorous mathematical results regarding the rich dynamics of such memristor oscillators, both in the case of piecewise linear models and polynomial models. Thus, for some families of discontinuous 3D piecewise linear memristor oscillators, we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible so to justify the periodic behavior exhibited by such three dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems. By using the first-order Melnikov theory, we derive the bifurcation set for a three-parametric family of Bogdanov-Takens systems with symmetry and deformation. As an applications of these results, we study a family of 3D memristor oscillators where the characteristic function of the memristor is a cubic polynomial. In this family we also show the existence of an infinity number of invariant manifolds. Also, we clarify some misconceptions that arise from the numerical simulations of these systems, emphasizing the important role of invariant manifolds in these models. In a similar way than for the 3D case, we study some discontinuous 4D piecewise linear memristor oscillators, and we show that the dynamics in each stratum is topologically equivalent to a continuous 3D piecewise linear dynamical system. Some previous results on bifurcations in such reduced systems, allow us to detect rigorously for the first time a multiple focus-center-cycle bifurcation in a three-parameter space, leading to the appearance of a topological sphere in the original model, completely foliated by stable periodic orbits. In the second part of this thesis, we show that the two-dimensional stroboscopic map defined by a second order system with a relay based control and a linear switching surface is topologically equivalent to a canonical form for discontinuous piecewise linear systems. Studying the main properties of the stroboscopic map defined by such a canonical form, the orbits of period two are completely characterized. At last, we give a conjecture about the occurrence of the big bang bifurcation in the previous map

Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators

Energy harvesting systems based on oscillators aim to capture energy from mechanical oscillations and convert it into electrical energy. Widely extended are those based on piezoelectric materials, whose dynamics are Hamiltonian submitted to different sources of dissipation: damping and coupling. These dissipations bring the system to low energy regimes, which is not desired in long term as it diminishes the absorbed energy. To avoid or to minimize such situations, we propose that the coupling of two oscillators could benefit from theory of Arnold diffusion. Such phenomenon studies $O(1)$ energy variations in Hamiltonian systems and hence could be very useful in energy harvesting applications. This article is a first step towards this goal. We consider two piezoelectric beams submitted to a small forcing and coupled through an electric circuit. By considering the coupling, damping and forcing as perturbations, we prove that the unperturbed system possesses a $4$-dimensional Normally Hyperbolic Invariant Manifold with $5$ and $4$-dimensional stable and unstable manifolds, respectively. These are locally unique after the perturbation. By means of the parameterization method, we numerically compute parameterizations of the perturbed manifold, its stable and unstable manifolds and study its inner dynamics. We show evidence of homoclinic connections when the perturbation is switched on

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

Nonlinear Dynamics Of Coupled Capillary-Surface Oscillators

The nonlinear dynamics of coupled liquid droplets and bridges are examined. By restricting droplet and bridge shapes to equilibrium states, the quasi-static dynamics of such systems may be studied using ordinary differential equations, and the techniques of nonlinear dynamics may be applied. For example, liquid droplets are restricted to spherical-caps, whose shapes may be deduced solely from their volume. Networks of liquid droplets are first considered. Static solutions are grouped into families, each with some p droplets large and some q = n [-] p small. The twodroplet system is modeled as a conservative second-order oscillator and fixed points undergo a pitchfork bifurcation as the total volume is increased; furthermore, when subjected to periodic forcing, chaotic dynamics are possible. Bounds for chaotic dynamics are investigated by using Melnikov's method and calculating Lyapunov exponents. Results are compared qualitatively with experimental results, thereby confirming the existence of chaotic motions. The two-droplet model is then extended to a n-droplet frictionless Sn symmetric model that consists of n [-] 1 second-order differential equations. Symmetry of the system is fundamental. In particular, independent of the equations, fixedpoints may be grouped into families by the number of small and large droplets. Within the families, stability is invariant and hence significantly reduces the number of equilibria to be considered. All equilibria, and their associated stability, are calculated analytically for an arbitrary number of droplets. For three droplets, the system is fourth order and thus trajectories are (in general) quasi-periodic or chaotic. Because the equations are S3 symmetric, trajectories may also possess S3 , or one of the three flip symmetries. Since there is no dissipation there are no asymptotically stable attractors. As such, trajectories of interest are away from equilibrium. In particular, trajectories with no initial velocity are analyzed to ascertain their symmetry as well as their dynamic nature. Both these determinations may be done in an automated fashion through the use of symmetry detectives and Lyapunov exponents, respectively. For this system, the results of these two methods reflect a strong correlation between symmetry and nonlinear dynamics; chaotic trajectories are S3 symmetric while quasi-periodic trajectories possess one of the three flip symmetries. Next, a non-smooth switching bridge-droplet system is considered. The system has two states: droplet-droplet and bridge-droplet. The switching system can be obtained from the two droplet system by introducing a planar substrate below one of the droplets. As the system oscillates, it may transition between states if the droplet impacts the wall or the liquid bridge breaks. The two transitions occur at different places in state space which results in a region for which the system is multiply defined. In addition, transitions are assumed to be instantaneous with no loss of velocity. The two states are first analyzed separately. The bridgedroplet state undergoes a cusp bifurcation in a two parameter expansion. Boundary equilibrium bifurcations also occur when an equilibrium point collides with a nonsmooth boundary. If the bridge-droplet and droplet-droplet states are combined, a two parameter bifurcation diagram for the switching system is realized. Switching trajectories are of particular interest because each switching cycle dampens the system until it no longer switches. These trajectories are mapped into a semi- infinite cylindrical space in which long-term behavior can be described solely by the dynamics in the multiply defined region. In the final chapter models for pull-off adhesive failure are considered. Recognizing engineering applications (i.e. a capillary adhesion device) as well as a phenomenon found in nature (i.e. defense mechanism of palm beetle), models for pull-off adhesive failure are developed for different loading conditions and compared with available observations. In particular, array geometry and the relationship of adhesive failure to the instabilities of a single liquid bridge are emphasized