Related papers: Dynamics of phase slips in systems with time-perio…
Modulation instability in a nonlinear optical waveguide array with alternating positive and negative refractive indices is investigated analytically. Particular solutions of a system of coupled nonlinear equations are found. These solutions…
We develop a complete stability theory for two-dimensional periodic traveling waves of reaction-diffusion systems. More precisely, we identify a diffusive spectral stability assumption, prove that it implies nonlinear stability and provide…
We describe a method for analyzing the phase space structures of Hamiltonian systems. This method is based on a time-frequency decomposition of a trajectory using wavelets. The ridges of the time-frequency landscape of a trajectory, also…
The May-Leonard model for three competing species, symmetric with respect to cyclic permutation of the variables and extended by diffusive terms, is considered. Exact time-periodic solutions of the system have been found, and their…
In the presence of wave dissipation, phase-space structures emerge in nonlinear Vlasov dynamics. Their dynamics can lead to a nonlinear continuous shifting of the wave frequency (chirping). This report summarizes my personal contribution to…
The presence of a dispersed phase substantially modifies small-scale turbulence. However, there has not been a comprehensive mechanistically-based understanding to predict turbulence modulation. Based on the energy flux balance, we propose…
In weakly nonlinear systems, the main effect of cubic nonlinearity on wave propagation is an amplitude-dependent correction of the dispersion relation. This phenomenon can manifest either as a frequency shift or as a wavenumber shift…
We study the Floquet edge states in arrays of periodically curved optical waveguides described by the modulated Su-Schrieffer-Heeger model. Beyond the bulk-edge correspondence, our study explores the interplay between band topology and…
Triadic interactions are the fundamental mechanism of energy transfer in fluid flows. This work introduces bispectral mode decomposition as a direct means of educing flow structures that are associated with triadic interactions from…
In nonlinear dynamical systems with highly nonorthogonal linear eigenvectors, linear non-modal analysis is more useful than normal mode analysis in predicting turbulent properties. However, the non-trivial time evolution of non-modal…
A large variety of rhythms are observed in nature. Rhythms such as electroencephalogram signals in the brain can often be regarded as interacting. In this study, we investigate the dynamical properties of rhythmic systems in two populations…
We consider the Ginzburg-Landau equation, $ \partial_t u= \partial_x^2 u + u - u|u|^2 $, with complex amplitude $u(x,t)$. We first analyze the phenomenon of phase slips as a consequence of the {\it local} shape of $u$. We next prove a {\it…
Stability is a key property of dynamical systems. In some cases, we want to change unstable system into stable one to achieve certain goals in engineering. Here, we present an example of a $3$ dimensional switched system that alternates…
Nonequilibrium statistical mechanics exhibit a variety of complex phenomena far from equilibrium. It inherits challenges of equilibrium, including accurately describing the joint distribution of a large number of configurations, and also…
This paper introduces the novel class of modulated cyclostationary processes, a class of non-stationary processes exhibiting frequency coupling, and proposes a method of their estimation from repeated trials. Cyclostationary processes also…
In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized…
Nonlinear dynamics emerge through either nonlinear interactions between the variables or through nonlinearities imposed on their linear interactions. Their interactions can be conceptualized as modulations of input-output (I/O) functions,…
The concept of a common modulated oscillation spanning multiple time series is formalized, a method for the recovery of such a signal from potentially noisy observations is proposed, and the time-varying bias properties of the recovery…
A coupled-phase oscillator model where each oscillator has an angular velocity that varies due to the interaction with other oscillators is studied. This model is proposed to deepen the understanding of the relationship between the…
The nonexponential relaxation ocurring in complex dynamics manifested in a wide variety of systems is analyzed through a simple model of diffusion in phase space. It is found that the inability of the system to find its equilibrium state in…