Periodically Forced Nonlinear Oscillators With Hysteretic Damping

We perform a detailed study of the dynamics of a nonlinear, one-dimensional oscillator driven by a periodic force under hysteretic damping, whose linear version was originally proposed and analyzed by Bishop in [1]. We first add a small quadratic stiffness term in the constitutive equation and construct the periodic solution of the problem by a systematic perturbation method, neglecting transient terms as $t\rightarrow \infty$. We then repeat the analysis replacing the quadratic by a cubic term, which does not allow the solutions to escape to infinity. In both cases, we examine the dependence of the amplitude of the periodic solution on the different parameters of the model and discuss the differences with the linear model. We point out certain undesirable features of the solutions, which have also been alluded to in the literature for the linear Bishop's model, but persist in the nonlinear case as well. Finally, we discuss an alternative hysteretic damping oscillator model first proposed by Reid [2], which appears to be free from these difficulties and exhibits remarkably rich dynamical properties when extended in the nonlinear regime.Comment: Accepted for publication in the Journal of Computational and Nonlinear Dynamic

Chimera States in a Two-Population Network of Coupled Pendulum-Like Elements

More than a decade ago, a surprising coexistence of synchronous and asynchronous behavior called the chimera state was discovered in networks of nonlocally coupled identical phase oscillators. In later years, chimeras were found to occur in a variety of theoretical and experimental studies of chemical and optical systems, as well as models of neuron dynamics. In this work, we study two coupled populations of pendulum-like elements represented by phase oscillators with a second derivative term multiplied by a mass parameter $m$ and treat the first order derivative terms as dissipation with parameter $\epsilon>0$. We first present numerical evidence showing that chimeras do exist in this system for small mass values $0<m<<1$. We then proceed to explain these states by reducing the coherent population to a single damped pendulum equation driven parametrically by oscillating averaged quantities related to the incoherent population

Stability Properties of 1-Dimensional Hamiltonian Lattices with Non-analytic Potentials

We investigate the local and global dynamics of two 1-Dimensional (1D) Hamiltonian lattices whose inter-particle forces are derived from non-analytic potentials. In particular, we study the dynamics of a model governed by a "graphene-type" force law and one inspired by Hollomon's law describing "work-hardening" effects in certain elastic materials. Our main aim is to show that, although similarities with the analytic case exist, some of the local and global stability properties of non-analytic potentials are very different than those encountered in systems with polynomial interactions, as in the case of 1D Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Our approach is to study the motion in the neighborhood of simple periodic orbits representing continuations of normal modes of the corresponding linear system, as the number of particles $N$ and the total energy $E$ are increased. We find that the graphene-type model is remarkably stable up to escape energy levels where breakdown is expected, while the Hollomon lattice never breaks, yet is unstable at low energies and only attains stability at energies where the harmonic force becomes dominant. We suggest that, since our results hold for large $N$, it would be interesting to study analogous phenomena in the continuum limit where 1D lattices become strings.Comment: Accepted for publication in the International Journal of Bifurcation and Chao

Spatial Control of Localized Oscillations in Arrays of Coupled Laser Dimers

Arrays of coupled semiconductor lasers are systems possessing radically complex dynamics that makes them useful for numerous applications in beam forming and beam shaping. In this work, we investigate the spatial controllability of oscillation amplitudes in an array of coupled photonic dimers, each consisting of two semiconductor lasers driven by differential pumping rates. We consider parameter values for which each dimer's stable phase-locked state has become unstable through a Hopf bifurcation and we show that, by assigning appropriate pumping rate values to each dimer, large-amplitude oscillations coexist with negligibly small amplitude oscillations. The spatial profile? of the amplitude of oscillations across the array can be dynamically controlled by appropriate pumping rate values in each dimer. This feature is shown to be quite robust, even for random detuning between the lasers, and suggests a mechanism for dynamically reconfigurable production of a large diversity of spatial profiles of laser amplitude oscillations.Comment: 8 pages, 10 figure

Cauchy distributions for the integrable standard map

We consider the integrable (zero perturbation) two--dimensional standard map, in light of current developments on ergodic sums of irrational rotations, and recent numerical evidence that it might possess non-trivial q-Gaussian statistics. Using both classical and recent results, we show that the phase average of the sum of centered positions of an orbit, for long times and after normalization, obeys the Cauchy distribution (a q-Gaussian with q=2), while for almost all individual orbits such a sum does not obey any distribution at all. We discuss the question of existence of distributions for KAM tori.Comment: 6 pages, 2 figure

Antiresonances and Ultrafast Resonances in a Twin Photonic Oscillator

We consider the properties of the small-signal modulation response of symmetry-breaking phase-locked states of twin coupled semiconductor lasers. The extended stability and the varying asymmetry of these modes allows for the introduction of a rich set of interesting modulation response features, such as sharp resonances and anti-resonance as well as efficient modulation at very high frequencies exceeding the free running relaxation frequencies by orders of magnitude.Comment: 6 pages, 5 figure

Energy transmission in Hamiltonian systems of globally interacting particles with Klein-Gordon on-site potentials

We consider a family of 1-dimensional Hamiltonian systems consisting of a large number of particles with on-site potentials and global (long range) interactions. The particles are initially at rest at the equilibrium position, and are perturbed sinusoidally at one end using Dirichlet data, while at the other end we place an absorbing boundary to simulate a semi-infinite medium. Using such a lattice with quadratic particle interactions and Klein-Gordon type on-site potential, we use a parameter 0 ≤ α < ∞as a measure of the “length” of interactions, and show that there is a sharp threshold above which energy is transmitted in the form of large amplitude nonlinear modes, as long as driving frequencies Ω lie in the forbidden band-gap of the system. This process is called nonlinear supratransmission and is investigated here numerically to show that it occurs at higher amplitudes the longer the range of interactions, reaching a maximum at a value α = αmax . 1.5 that depends on Ω. Below this αmax supratransmission thresholds decrease sharply to values lower than the nearest neighbor α = ∞ limit. We give a plausible argument for this phenomenon and conjecture that similar results are present in related systems such as the sine-Gordon, the nonlinear Klein-Gordon and the double sine-Gordon type

COMPLEX DYNAMICS AND STATISTICS OF 1-D HAMILTONIAN LATTICES: LONG RANGE INTERACTIONS AND SUPRATRANSMISSION

In this paper, I review a number of results that my co-workers and I have obtained in the field of 1–Dimensional (1D) Hamiltonian lattices. This field has grown in recent years, due to its importance in revealing many phenomena that concern the occurrence of chaotic behavior in conservative physical systems with a high number of degrees of freedom. After the establishment of the Kolomogorov–Arnol’d–Moser (KAM) theory in the 1960s, a wealth of results were obtained about such systems as small perturbations of completely integrable N degree-of-freedom Hamiltonians, where ordered motion is dominant in the form of invariant tori. Since the 1980s, however, and particularly in the last two decades, there has been great progress in understanding the properties of Hamiltonian 1D lattices far from the KAM regime, where "weak" and "strong" forms of chaos begin to play an increasingly significant role. It is the purpose of this review to address and highlight some of these advances, in which the author has made several contributions concerning the dynamics and statistics of these lattice