Suppressing chaos in damped driven systems by non-harmonic excitations: experimental robustness against potential’s mismatches

The robustness of a chaos-suppressing scenario against potential mismatches is experimentally studied through the universal model of a damped, harmonically driven two-well Duffing oscillator subject to non-harmonic chaos-suppressing excitations. We consider a second order analogous electrical circuit having an extremely simple two-well potential that differs from that of the standard two-well Duffing model, and compare the main theoretical predictions regarding the chaos-suppressing scenario from the latter with experimental results from the former. Our experimental results prove the high robustness of the chaos-suppressing scenario against potential mismatches regardless of the (constant) values of the remaining parameters. Specifically, the predictions of an inverse dependence of the regularization area in the control parameter plane on the impulse of the chaos-suppressing excitation as well as of a minimal effective amplitude of the chaos-suppressing excitation when the impulse transmitted is maximum were experimentally confirmed.Ministerio de Ciencia, Innovación y Universidades (MICIU, Spain) and FEDER funds Project No. PID2019-108508GB-100/AEI/10.13039/501100011033Junta de Extremadura (JEx, Spain) and FEDER funds Project No. GR1808

Energy thresholds for the existence of breather solutions and traveling waves on lattices

We discuss the existence of breathers and of energy thresholds for their formation in DNLS lattices with linear and nonlinear impurities. In the case of linear impurities we present some new results concerning important differences between the attractive and repulsive impurity which is interplaying with a power nonlinearity. These differences concern the coexistence or the existence of staggered and unstaggered breather profile patterns. We also distinguish between the excitation threshold (the positive minimum of the power observed when the dimension of the lattice is greater or equal to some critical value) and explicit analytical lower bounds on the power (predicting the smallest value of the power a discrete breather one-parameter family), which are valid for any dimension. Extended numerical studies in one, two and three dimensional lattices justify that the theoretical bounds can be considered as thresholds for the existence of the frequency parametrized families. The discussion reviews and extends the issue of the excitation threshold in lattices with nonlinear impu- rities while lower bounds, with respect to the kinetic energy, are also discussed for traveling waves in FPU periodic lattices

DNLS with Impurities

The past few years have witnessed an explosion of interest in discrete models and intrinsic localized modes (discrete breathers or solitons) that has been summarized in a number of recent reviews [1–3]. This growth has been motivated by numerous applications of nonlinear dynamical lattice models in areas as broad and diverse as the nonlinear optics of waveguide arrays [4], the dynamics of Bose–Einstein condensates in periodic potentials [5, 6], micro-mechanical models of cantilever arrays [7], or even simple models of the complex dynamics of the DNA double strand [8]. Arguably, the most prototypical model among the ones that emerge in these settings is the Discrete Nonlinear Schrödinger (DNLS) equation, the main topic of this book

Impulse-induced localized control of chaos in starlike networks

Locally decreasing the impulse transmitted by periodic pulses is shown to be a reliable method of taming chaos in starlike networks of dissipative nonlinear oscillators, leading to both synchronous periodic states and equilibria (oscillation death). Specifically, the paradigmatic model of damped kicked rotators is studied in which it is assumed that when the rotators are driven synchronously, i.e., all driving pulses transmit the same impulse, the networks display chaotic dynamics. It is found that the taming effect of decreasing the impulse transmitted by the pulses acting on particular nodes strongly depends on their number and degree of connectivity. A theoretical analysis is given explaining the basic physical mechanism as well as the main features of the chaos-control scenario.Ministerio de Economía y Competitividad (MINECO, Spain) Project No. FIS2012-34902 cofinanced by FEDER fundsJunta de Extremadura (JEx, Spain) Project No. GR1514

Experimental and numerical observation of dark and bright breathers in the band gap of a diatomic electrical lattice

We observe dark and bright intrinsic localized modes (ILMs), also known as discrete breathers, experimentally and numerically in a diatomic-like electrical lattice. The experimental generation of dark ILMs by driving a dissipative lattice with spatially homogenous amplitude is, to our knowledge, unprecedented. In addition, the experimental manifestation of bright breathers within the band gap is also novel in this system. In experimental measurements the dark modes appear just below the bottom of the top branch in frequency. As the frequency is then lowered further into the band gap, the dark ILMs persist, until the nonlinear localization pattern reverses and bright ILMs appear on top of the finite background. Deep into the band gap, only a single bright structure survives in a lattice of 32 nodes. The vicinity of the bottom band also features bright and dark self-localized excitations. These results pave the way for a more systematic study of dark breathers and their bifurcations in diatomic-like chains.VI Plan Propio of the University of Seville, Spain (VI PPITUS)AEI/FEDER, UE MAT2016- 79866-

Nonlinear edge modes in a honeycomb electrical lattice near the Dirac points

We examine - both experimentally and numerically - a two-dimensional nonlinear driven electrical lattice with honeycomb structure. Drives are considered over a range of frequencies both outside (below and above) and inside the band of linear modes. We identify a number of discrete breathers both existing in the bulk and also (predominantly) ones arising at the domain boundaries, localized either along the arm-chair or along the zig-zag edges. The types of edge-localized breathers observed and computed emerge in distinct frequency bands near the Dirac-point frequency of the dispersion surface while driving the lattice subharmonically (in a spatially homogeneous manner). These observations/computations can represent a starting point towards the exploration of the interplay of nonlinearity and topology in an experimentally tractable system such as the honeycomb electrical lattice.AEI/FEDER (UE) MAT2016- 79866-RUniversity of Seville (Spain) VI PPITU

Moving discrete breathers in a Klein–Gordon chain with an impurity

We analyse the influence of an impurity in the evolution of moving discrete breathers in a Klein–Gordon chain with non-weak nonlinearity. Three different types of behaviour can be observed when moving breathers interact with the impurity: they pass through the impurity continuing their direction of movement; they are reflected by the impurity; they are trapped by the impurity, giving rise to chaotic breathers, as their Fourier power spectra show. Resonance with a breather centred at the impurity site is conjectured to be a necessary condition for the appearance of the trapping phenomenon. This paper establishes a difference between the resonance condition of the non-weak nonlinearity approach and the resonance condition with the linear impurity mode in the case of weak nonlinearity.European Commission under the RTN project LOCNET, HPRN-CT-1999-0016

Numerical study of two-dimensional disordered Klein-Gordon lattices with cubic soft anharmonicity

Localized oscillations appear both in ordered nonlinear lattices (breathers) and in disordered linear lattices (Anderson modes). Numerical studies on a class of two-dimensional systems of the Klein-Gordon type show that there exist two different types of bifurcation in the path from nonlinearity-order to linearity-disorder: inverse pitchforks, with or without period doubling, and saddle-nodes. This was discovered for a one-dimensional system in a previous work of Archilla, MacKay and Marin. The appearance of a saddle-node bifurcation indicates that nonlinearity and disorder begin to interfere destructively and localization is not possible. In contrast, the appearance of a pitchfork bifurcation indicates that localization persists

Discrete embedded solitary waves and breathers in one-dimensional nonlinear lattices

For a one-dimensional linear lattice, earlier work has shown how to systematically construct a slowly- decaying linear potential bearing a localized eigenmode embedded in the continuous spectrum. Here, we extend this idea in two directions: The first one is in the realm of the discrete nonlinear Schrödinger, where the linear operator of the Schrödinger type is considered in the presence of a Kerr focusing or defocusing nonlinearity and the embedded linear mode is continued into the nonlinear regime as a discrete solitary wave. The second case is the Klein-Gordon setting, where the presence of a cubic nonlinearity leads to the emergence of embedded-in-the-continuum discrete breathers. In both settings, it is seen that the stability of the modes near the linear limit turns into instability as nonlinearity is increased past a critical value, leading to a dynamical delocalization of the solitary wave (or breathing) state. Finally, we suggest a concrete experiment to observe these embedded modes using a bi-inductive electrical lattice.Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía and EU (FEDER program 2014-2020) project P18-RT-3480Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía and EU (FEDER program 2014-2020) project US-1380977MICINN and AEI project PID2019-110430GB-C21MICINN and AEI project PID2020-112620GB-I00Ministerio de Ciencia, Innovación y Universidades (MICIU, Spain) and FEDER funds Project No. PID2019-108508GB-I0

Intersection of crisis loci in a driven nonlinearly damped oscillator

We report on a phenomenon observed in a driven nonlinearly damped oscillator when two control parameters, the frequency of the external excitation and the nonlinear damping coefficient, are varied simultaneously. An interior crisis locus and a boundary crisis locus, corresponding to two different chaotic attractors, intersect in a point of the parameter space. There exists an interchange in the type of crisis that each attractor suffers after crossing the intersection point