Pernament dynamics of particle-like solutions in out-of-equilibrium systems

Abstract

This dissertation aims to understand how nonvariational effects affect the dynamics of particle-type solutions in spatially extended systems. In the first part of our research, we study nonvariational perturbations on the dynamics of fronts and localized structures that can be understood as particle-like solutions associating continuous variables such as position, velocity, and size. In Chapter 1, we explain some preliminary concepts useful to understand the results of this thesis. Chapter 2 is dedicated to understanding how nonvariational effects can affect the dynamics of fronts connecting a stable state with an unstable one. From an analytical and numerical point of view, we show how the nonvariational effect can induce a pulled-pushed transition in this kind of front and show experimental evidence of our results in a LCLV setup with optical feedback. Chapter 3 reveals how nonvariational effects induce propagation in normal fronts, obtaining in such a way a new propagation mechanism caused only by the shape of the front connecting the stable states that the nonvariational parameters can control. In addition, we verify our theoretical findings by studying the LCLV setup with optical feedback in the regime where the system is multistable. Chapter 4 shows a transition from motionless to traveling localized states due to a spontaneous symmetry breaking in the nonvariational Turing-Swift-Hohenberg equation as an archetypical model of pattern-forming systems. A nonvariational effect drives such transition, and it is characterized numerically and analytically. Moreover, we generalize such kinds of states in higher dimensions. Chapter 5 and 6 explores a new kind of chimera states in continuous media, namely, traveling and wandering chimera states, respectively. Chimera states are characterized by the coexistence of coherent and incoherent spatiotemporal dynamics. Traveling chimeras propagates in a specific direction with a well-defined average speed; meanwhile, wandering chimeras exhibit an erratic motion resembling a random walk. We characterize its statistical and dynamical properties and give numerical insights about its bifurcation diagram. In chapter 7, we study the dynamics of fronts but consider another kind of nonvariational perturbation, namely, deterministic fluctuations englobing chaotical and spatiotemporal chaotical dynamics. We show that despite these fluctuations, a front may remain pinned, in contrast to fronts in systems with Gaussian white noise fluctuations, and explore the pinning-depinning transition. We describe this transition by deriving an equation for the front position, which takes the form of an overdamped system with a ratchet potential and chaotic forcing; this equation can, in turn, be transformed into a linear parametrically driven oscillator with a chaotically oscillating frequency. Finally, in chapter 8, we extend our study of deterministic perturbations to localized structures. Mobility properties of spatially localized structures arising from chaotic but deterministic forcing of the bistable Swift-Hohenberg equation are compared with the corresponding results when the chaotic forcing is replaced by white noise. For the family of LS studied, we found that shorter structures are more fragile than longer ones, and their stability region can be displaced outside the pinning region for constant forcing. In addition, we discuss the nature of randomness and how it can emerge from deterministic dynamics.