Pattern formation and competition in photorefractive oscillators

We introduce a general model of pattern formation in optical systems made of a cavity with an active medium as a photorefractive crystal fed by a pump. The model is based on the interplay of a diffractive equation for the optical field and a diffusive equation for the medium refractivity. The aim of the model is to describe a series of experiments which have shown mode competition (periodic or chaotic alternation) for low Fresnel numbers (F) and mode coexistence, leading to short range space correlations, for high F. For low F, a linear stability analysis provides the set of modes above threshold as a function of the transverse wave number. Due to the interplay of the optical and the diffusive interactions, different behaviors result depending on the thickness of the medium as compared to the optical absorption length and diffusion length. Including the leading nonlinearities compatible with the symmetry constraints, we introduce normal form equations which describe the time-dependent mode competition. In the case of a large number of modes (high F), nonlinear mode-mode interaction is equivalent to a self-induced noise. In this limit, the relevant feature to be compared with the experiment is the power spectrum.     

Clustering of arrays of chaotic chemical oscillators by feedback and forcing

Feedback and external forcing are applied to an array of chaotic electrochemical oscillators through variations in the applied potential. We see transitions from intermittent clusters to stable chaotic clusters to stable periodic clusters to synchronized states as the feedback gain and forcing amplitude, respectively, are varied. With forcing up to four clusters are observed in stable states. The transition to synchronization with feedback occurs by the increase in the size of one cluster at the expense of the others.     

Localized excitations in arrays of synchronized laser oscillators

We investigate the spatiotemporal dynamics of a large array of laser oscillators. The oscillators are locally coupled and their natural frequencies are randomly detuned. We show that synchronization of the array elements results in localized excitations wandering along well-defined trajectories.     

Synchronization in arrays of coupled self-induced friction oscillators

We investigate synchronization phenomena in systems of self-induced dry friction oscillators with kinematic excitation coupled by linear springs. Friction force is modelled according to exponential model. Initially, a single degree of freedom mass-spring system on a moving belt is considered to check the type of motion of the system (periodic, non-periodic). Then the system is coupled in chain of identical oscillators starting from two, up to four oscillators. A reference probe of two coupled oscillators is applied in order to detect synchronization thresholds for both periodic and non-periodic motion of the system. The master stability function is applied to predict the synchronization thresholds for longer chains of oscillators basing on two oscillator probe. It is shown that synchronization is possible both for three and four coupled oscillators under certain circumstances. Our results confirmed that this technique can be also applied for the systems with discontinuities.     

Synchronization from disordered driving forces in arrays of coupled oscillators

The effects of disorder in external forces on the dynamical behavior of coupled nonlinear oscillator networks are studied. When driven synchronously, i.e., all driving forces have the same phase, the networks display chaotic dynamics. We show that random phases in the driving forces result in regular, periodic network behavior. Intermediate phase disorder can produce network synchrony. Specifically, there is an optimal amount of phase disorder, which can induce the highest level of synchrony. These results demonstrate that the spatiotemporal structure of external influences can control chaos and lead to synchronization in nonlinear systems.     

Frequency down-conversion using cascading arrays of coupled nonlinear oscillators

A novel coupling scheme using M≥2 arrays of coupled nonlinear elements arranged in a specific configuration can produce multifrequency patterns or a frequency down-converting effect on an external (input) signal. In such a configuration, each array contains N≥3 nonlinear elements with similar dynamics and each element is coupled unidirectionally within the array. The subsequent arrays in the cascade are coupled in a similar fashion except that the coupling direction is arranged in the opposite direction with respect to that of the preceding array. Previous theoretical work and numerical results have already been reported in [P. Longhini, A. Palacios, V. In, J. Neff, A. Kho, A. Bulsara, Exploiting dynamical symmetry in coupled nonlinear elements for efficient frequency down-conversion, Phys. Rev. E 76 (2007) 026201]. This paper is focused on results of experiments implemented on two distinct systems: the first system is fabricated using discrete component circuits to approximate an overdamped bistable Duffing oscillator described by a quartic potential system, and the second system is built in a microcircuit, where the nonlinearity is described by a hyperbolic tangent function, with the option of applying an external signal to investigate resonant effects. In particular, the circuit implementations for each case use M=2 arrays, but their voltage oscillations already demonstrate that the frequency relations between each of the successive arrays decrease by a rational factor, conforming to earlier theoretical and numerical results for the general case containing M arrays. This behavior is important for efficient frequency down-converting applications which are essential in many communication systems where heterodyning is typically used and it involves multi-step processes with complicated circuitry.     

Experimental observation of multifrequency patterns in arrays of coupled nonlinear oscillators

Frequency-related oscillations in coupled oscillator systems, in which one or more oscillators oscillate at different frequencies than the other oscillators, have been studied using group theoretical methods by Armbruster and Chossat [Phys. Lett. A 254, 269 (1999)] and more recently by Golubitsky and Stewart [in Geometry, Mechanics, and Dynamics, edited by P. Newton, P. Holmes, and A. Weinstein (Springer, New York, 2002), p. 243]. We demonstrate, experimentally, via electronic circuits, the existence of frequency-related oscillations in a network of two arrays of N oscillators, per array, coupled to one another. Under certain conditions, one of the arrays can be induced to oscillate at N times the frequency of the other array. This type of behavior is different from the one observed in a driven system because it is dictated mainly by the symmetry of the coupled system.     

Nonadiabatic pattern formation in optical parametric oscillators

A nonadiabatic mechanism for pattern formation in parametrically forced systems subjected to a slow periodic modulation of the excitation frequency is proposed and discussed in detail for the case of an optical parametric oscillator. It is demonstrated that nonautonomous dynamics may induce nonadiabatic off-axis emission of down-converted photons even when the signal field is blueshifted from the nearby cavity resonance.     

Entrainment of chemical oscillators and resonance dissipation

Nonlinear chemical oscillators can only exist far from equilibrium and therefore dissipate chemical energy. It has been found earlier that this dissipation can be reduced when the oscillator is driven by a periodic input, provided the driving frequency is near resonance with the autonomous oscillation. This observation, which was based on numerical experimentation is now confirmed analytically within the framework of reductive perturbation theory, i.e. for “weakly nonlinear” oscillations. The analysis explains all qualitative features of the numerical dissipation spectra, notably the enhancement of dissipation at the transition between periodic and quasiperiodic response behavior.     

On the stability of coupled chemical oscillators

The coupling of chemical oscillators is investigated in the case of the Brusselator model. The stable steady states obtained by coupling two, three and more Brusselators in parallel, in a diffusion like manner are discussed. Results are given for identical, identical with perturbation (i.e. almost identical), and completely dissimilar oscillators.Parameter domains in which stability and multistability can be found are analyzed. These domains usually increase with the number of cells - thus a bigger system of oscillators has a greater chance to be stabilized. The symmetry patterns of the stable domains are discussed.     

Anti-phase synchronization and ergodicity in arrays of oscillators coupled by an elastic force

We have proposed a mechanism of interaction between two non-linear dissipative oscillators, leading to exact and robust anti-phase and in-phase synchronization. The system we have analyzed is a model for the Huygens’s two pendulum clocks system, as well as a model for synchronization mediated by an elastic media. Here, we extend these results to arrays, finite or infinite, of conservative pendula coupled by linear elastic forces. We show that, for two interacting pendula, this mechanism leads always to synchronized anti-phase small amplitude oscillations, and it is robust upon variation of the parameters. For three or more interacting pendula, this mechanism leads always to ergodic non-synchronized oscillations. In the continuum limit, the pattern of synchronization is described by a quasi-periodic longitudinal wave.     

Discrete X-wave formation in nonlinear waveguide arrays

We present experimental evidence for the spontaneous formation of discrete X waves in AlGaAs waveguide arrays. This new family of optical waves has been excited, for the first time, by using the interplay between discrete diffraction and normal temporal dispersion, in the presence of Kerr nonlinearity. Our experimental results are in good agreement with theoretical predictions.     

Anomalous band formation in arrays of terahertz nanoresonators

We investigate band formation in one-dimensional periodic arrays of rectangular holes which have a nanoscale width but a length of 100 μm. These holes are tailored to work as resonators in the terahertz frequency regime. We study the evolution of the electromagnetic response with the period of the array, showing that this dependence is not monotonic due to both the oscillating behavior of the coupling between holes and its long-range character.     

Pattern recognition minimizes entropy production in a neural network of electrical oscillators

We investigate the physical principle driving pattern recognition in a previously introduced Hopfield-like neural network circuit (Hölzel and Krischer, 2011 [13]). Effectively, this system is a network of Kuramoto oscillators with a coupling matrix defined by the Hebbian rule. We calculate the average entropy production 〈dS/dt〉$〈\mathrm{d}S/\mathrm{d}t〉$ of all neurons in the network for an arbitrary network state and show that the obtained expression for 〈dS/dt〉$〈\mathrm{d}S/\mathrm{d}t〉$ is a potential function for the dynamics of the network. Therefore, pattern recognition in a Hebbian network of Kuramoto oscillators is equivalent to the minimization of entropy production for the implementation at hand. Moreover, it is likely that all Hopfield-like networks implemented as open systems follow this mechanism.     

Pattern formation in active fluids

We discuss pattern formation in active fluids in which active stress is regulated by diffusing molecular components. Nonhomogeneous active stress profiles create patterns of flow which transport stress regulators by advection. Our work is motivated by the dynamics of the actomyosin cell cortex in which biochemical pathways regulate active stress. We present a mechanism in which a single diffusing species up regulates active stress, resulting in steady flow and concentration patterns. We also discuss general pattern-formation behaviors of reaction-diffusion systems placed in active fluids.     

Pattern formation in drying drops

Ring formation in an evaporating sessile drop is a hydrodynamic process in which solids dispersed in the drop are advected to the contact line. After all the liquid evaporates, a ring-shaped deposit is left on the substrate that contains almost all the solute. Here I show that the drop itself can generate one of the essential conditions for ring formation to occur: contact line pinning. Furthermore, I show that when self-induced pinning is the only source of pinning an array of patterns-that include cellular and lamellar structures, sawtooth patterns, and Sierpinski gaskets-arises from the competition between dewetting and contact line pinning.     

Pattern formation in Ferroelastic Transitions

We study pattern formation in ferroelastic materials using the Ginzburg–Landau approach. Since ferroelastic transitions are driven by strain, the nonlinear elastic free energy is expressed as an expansion in the appropriate (i.e., order parameter) strain variables. However, the displacement fields are the real independent variables, whereas the components of the strain tensor are related to each other through elastic compatibility relations. These constraints manifest as an anisotropic long-range interaction which drastically influences the underlying microstructure. The evolution of the microstructure is demonstrated for (i) a hexagonal-to-orthorhombic transition using a strain-based approach with explicit long-range interactions; and (ii) a cubic-to-tetragonal transition by solving the force-balance equations for the displacement fields.