oscillators. The only known realization of similar process in the short homogenous chain with more then two elements refers to the three-well quantum system with the same coupling between the wells providing the special condition on the equations describing the system. However, this condition does not appear to hold in classical homogenous cou- pled systems, that makes the full periodic energy exchange in the systems of three classi- cal oscillators questionable. Here we prove that fully reciprocal periodic energy transport between the ends of a short homogenous chain of nonlinear oscillators is possible in the conservative classical system with the ‘soft’ nonlinearity. The analogy with the quantum system of the coherent (harmonic) quantum Rabi oscillations in a superposition of three quantum states helps to reveal the special condition on the effective on-site potentials, which does not exist in the classical linear system or system of more than two oscillators with ‘hard’ nonlinearity. We study the periodic energy transport and its localization with use of the regular asymptotic analysis in the reduced phase space. The reported effects can be significant for many fundamental and applied areas of sciences where the coherent energy transport is important.
Stochastic Resonance (SR) is a well-known noise-induced phenomenon widely reported in dynamical systems with a threshold, while Inverse Stochastic Resonance (ISR) is an opposing phenomenon observed in the dynamical systems which exhibit bistability between a stable node and a stable limit cycle. This study shows a co-occurrence of SR and ISR, in a minimal circuit of synaptically coupled spiking neurons that is designed to show bistability between quiescence and a persistent firing mode. We identify noise, synaptic and intrinsic parameters ranges that allow for ISR. The minimal computational model, is investigated for a range of parameters, and our simulations indicate that the main features of SR, are the direct results of dynamical properties which lead to ISR.
We introduce a nonlinear Schrödinger equation (NLSE) which combines the pseudo-stimulated-Raman-scattering (pseudo-SRS) term, i.e., a non-conservative cubic one with the first spatial derivative, and an external potential, which helps to stabilize solitons against the pseudo-SRS effect. Dynamics of solitons is addressed by means of analytical and numerical methods. The quasi-particle approximation (QPA) for the solitons demonstrates that the SRS-induced downshift of the soliton's wavenumber may be compensated by a potential force, producing a stable stationary soliton. Three physically relevant potentials are considered: a harmonic-oscillator (HO) trap, a spatially periodic cosinusoidal potential, and the HO trap subjected to periodic temporal modulation. Both equilibrium positions of trapped pulses (solitons) and their regimes of motion with trapped and free trajectories are accurately predicted by the QPA and corroborated by direct simulations of the underlying NLSE. In the case of the time-modulated HO trap, a parametric resonance is demonstrated, in the form of the motion of the driven soliton with an exponentially growing amplitudes of oscillations.
Gamma rhythm (20-100 Hz) plays a key role in numerous cognitive tasks: working memory, sensory processing and in routing of information across neural circuits. In comparison with lower frequency oscillations in the brain, gamma-rhythm associated firing of the individual neurons is sparse and the activity is locally distributed in the cortex. Such “weak” gamma rhythm results from synchronous firing of pyramidal neurons in an interplay with the local inhibitory interneurons in a "pyramidal-interneuron gamma" or PING. Experimental evidence shows that individual pyramidal neurons during such oscillations tend to fire at rates below gamma, with the population showing clear gamma oscillations and synchrony. One possible way to describe such features is that this gamma oscillation is generated within local synchronous neuronal clusters. The number of such synchronous clusters defines the overall coherence of the rhythm and its spatial structure. The number of clusters in turn depends on the properties of the synaptic coupling and the intrinsic properties of the constituent neurons. We previously showed that a slow spike frequency adaptation current in the pyramidal neurons can effectively control cluster numbers. These slow adaptation currents are modulated by endogenous brain neuromodulators such as dopamine, whose level is in turn related to cognitive task requirements. Hence we postulate that dopaminergic modulation can effectively control the clustering of weak gamma and its coherence. In this paper we study how dopaminergic modulation of the network and cell properties impacts the cluster formation process in a PING network model.
The issue of a recurrence of the modulationally unstable water wave trains within the framework of the fully nonlinear potential Euler equations is addressed. It is examined, in particular, if a modulation which appears from nowhere (i.e., is infinitesimal initially) and generates a rogue wave which then disappears with no trace. If so, this wave solution would be a breather solution of the primitive hydrodynamic equations. It is shown with the help of the fully nonlinear numerical simulation that when a rogue wave occurs from a uniform Stokes wave train, it excites other waves which have different lengths, what prevents the complete recurrence and, eventually, results in a quasi-periodic breathing of the wave envelope. Meanwhile the discovered effects are rather small in magnitude, and the period of the modulation breathing may be thousands of the dominant wave periods. Thus, the obtained solution may be called a quasi-breather of the Euler equations.
The objective of this paper is to establish that algorithms, which reconstruct the coupling between solar proxies based on the properties of the Kuramoto equations, and algorithms, based on the van der Pol equations, might produce similar estimates. To this end, the inverse problem is formulated as follows: reconstruct the coupling based on the solutions of the corresponding equations. For either system of the equations we construct an algorithm solving the inverse problem and establish that there exists a range of moderate values of the correlation such that the algorithms produce practically identical coupling within the established range. The lower boundary of this range is dependent on the half-difference of the oscillators' frequencies. Then, we apply the two reconstruction algorithms to solar index ISSN and the geomagnetic index aa, which are proxies to the toroidal and poloidal magnetic fields of the Sun respectively. Their correlation belongs within the range that yields the proximity of the coupling reconstructed with all solar cycles from 11 till 23 except 20 and, possibly, 21. Our finding relate the reconstruction of characteristics of solar activity inferred by Blanter et al [Sol. Phys. 2014, 289, 4309; Sol. Phys. 2016, 291, 1003] from the Kuramoto model to the state of the art solar dynamo theory based on the magnetohydrodynamic equations.
The process of rogue wave formation on deep water is considered. A wave of extreme amplitude is born against the background of uniform waves (Gerstner waves) under the action of external pressure on free surface. The pressure distribution has a form of a quasi-stationary “pit”. The fluid motion is supposed to be a vortex one and is described by an exact solution of equations of 2D hydrodynamics for an ideal fluid in Lagrangian coordinates. Liquid particles are moving around circumferences of different radii in the absence of drift flow. Values of amplitude and wave steepness optimal for rogue wave formation are found numerically. The influence of vorticity distribution and pressure drop on parameters of the fluid is investigated.
Solitons and cavitons (the latter are localized solutions with singularities) for the nonlocal Whitham equations are studied. The fourth order differential equation for traveling waves with a parameter in front of the fourth derivative is reduced to a reversible Hamiltonian system defined on a two-sheeted four-dimensional space. Solutions of the system which stay on one sheet represent smooth solutions of the equation but those which perform transitions through the branching plane represent solutions with jumps. They correspond to solutions with singularities of the fourth order differential equation – breaks of the first and third derivatives but continuous even derivatives. The Hamiltonian system can have two types of equilibria on different sheets, they can be saddle-centers or saddle-foci. Using analytic and numerical methods we found many types of homoclinic orbits to these equilibria both with a monotone asymptotics and oscillating ones. They correspond to solitons and cavitons of the initial equation. When we deal with homoclinic orbits to a saddle-center, the values of the second parameter (physical wave speed) are discrete but for the case of a saddle-focus they are continuous. The presence of multiplicity of such solutions displays the very complicated dynamics of the system.
Compactons are studied in the framework of the Korteweg–de Vries (KdV) equation with the sublinear nonlinearity. Compactons represent localized bell-shaped waves of either polarity which propagate to the same direction as waves of the linear KdV equation. Their amplitude and width are inverse proportional to their speed. The energetic stability of compactons with respect to symmetric compact perturbations with the same support is proven analytically. Dynamics of compactons is studied numerically, including evolution of pulse-like disturbances and interactions of compactons of the same or opposite polarities. Compactons interact inelastically, though almost restore their shapes after collisions. Compactons play a two-fold role of the long-living soliton-like structures and of the small-scale waves which spread the wave energy.
The nonlinear dynamics of a parametrically excited pendulum is addressed. The proposed analytical approach aims at describing the pendulum dynamics beyond the simplified regimes usually considered in literature, where stationary and small amplitude oscilla- tions are assumed. Thus, by combining complexification and Limiting Phase Trajectory (LPT) concepts, both stationary and non-stationary dynamic regimes are considered in the neighborhood of the main parametric resonance, without any restriction on the pendulum oscillation amplitudes. The advantage of the proposed approach lies in the possibility of identifying the strongly modulated regimes for arbitrary initial conditions and high- amplitude excitation, cases in which the conventionally used quasilinear approximation is not valid. The identification of the bifurcations of the stationary states as well as the large-amplitude corrections of the stability thresholds emanating from the main paramet- ric resonance are also provided.
We investigated the phenomenological model of ensemble of two FitzHugh–Nagumo neuron-like elements with symmetric excitatory couplings. The main advantage of proposed model is the new approach to model the coupling which is implemented by smooth function that approximates rectangular function and reflects main important properties of biological synaptic coupling. The proposed coupling depends on three parameters that define: a) the beginning of activation of an element α, b) the duration of the activation δ and c) the strength of the coupling g. We observed a rich diversity of different types of neuron-like activity, including regular in-phase, anti-phase and sequential spiking. In the phase space of the system, these regular regimes correspond to specific asymptotically stable periodic motions (limit cycles). We also observed the canard in-phase solutions and the chaotic anti-phase activity, which corresponds to a strange attractor that appears via the cascade of period doubling bifurcations of limit cycles.
In addition, we investigated an interesting phenomenon when two different chaotic attractive regimes corresponding for two different types of chaotic anti-phase activity merge in a single strange attractor. As a result, a new type of chaotic anti-phase regime appears by explosion from the collision of these two strange attractors.
We also provided the detailed study of bifurcations which lead to the transitions between all these regimes. We detected on the (α, δ) parameter plane regions that correspond to the above-mentioned regimes. We also showed numerically the existence of bistability regions where various non-trivial regimes coexist. For example, in some regions, one can observe either anti-phase or in-phase oscillations depending on initial conditions. We also specified regions corresponding to coexisting various types of sequential activity.
The dynamics of two-component solitons is studied, analytically and numerically, in the framework of a system of coupled extended nonlinear Schrödinger equations, which incorporate the cross-phase modulation, pseudo-stimulated-Raman-scattering (pseudo-SRS), cross-pseudo-SRS, and spatially inhomogeneous second-order dispersion (SOD). The system models co-propagation of electromagnetic waves with orthogonal polarizations in plasmas. It is shown that the soliton's wavenumber downshift, caused by pseudo-SRS, may be compensated by an upshift, induced by the inhomogeneous SOD, to produce stable stationary two-component solitons. The corresponding approximate analytical solutions for stable solitons are found. Analytical results are well confirmed by their numerical counterparts. Further, the evolution of inputs composed of spatially even and odd components is investigated by means of systematic simulations, which reveal three different outcomes: formation of a breather which keeps opposite parities of the components; splitting into a pair of separating vector solitons; and spreading of the weak odd component into a small-amplitude pedestal with an embedded dark soliton.