Related papers: Analyzing Turing's Systems via Dynamic Bifurcation…
In this work, we focus on an autocatalytic reaction-diffusion model and carry out multiple scale weakly nonlinear analysis. A cubic and a quadratic autocatalytic reaction system is analysed. We develop a framework to identify the critical…
A free-energy minimization approach is used to address the secular & dynamical instabilities & the bifurcations along sequences of rotating, self-gravitating fluid and stellar systems. Our approach stems from the Landau-Ginzburg theory of…
We numerically investigate nucleation processes in the transient dynamics of the two-dimensional complex Ginzburg-Landau equation towards its "frozen" state with quasi-stationary spiral structures. We study the transition kinetics from…
We propose a kind of reaction-diffusion equations for cell differentiation, which exhibits the Turing instability. If the diffusivity of some variables is set to be infinity, we get coupled competitive reaction-diffusion equations with a…
This paper investigates the stability and bifurcation of the two-dimensional viscous primitive equations with full diffusion under thermal forcing. The system governs perturbations about a motionless basic state with a linear temperature…
In this work we study the effect of density dependent nonlinear diffusion on pattern formation in the Lengyel--Epstein system. Via the linear stability analysis we determine both the Turing and the Hopf instability boundaries and we show…
Turing theory of pattern formation is among the most popular theoretical means to account for the variety of spatio-temporal structures observed in Nature and, for this reason, finds applications in many different fields. While Turing…
The Turing-Hopf type spatiotemporal patterns in a diffusive Holling-Tanner model with discrete time delay is considered. A global Turing bifurcation theorem for $\tau=0$ and a local Turing bifurcation theorem for $\tau>0$ are given by the…
We investigate Turing pattern formation in a stochastic and spatially discretized version of a reaction diffusion advection (RDA) equation, which was previously introduced to model synaptogenesis in \textit{C. elegans}. The model describes…
Linear stability of solitary waves near transcritical bifurcations is analyzed for the generalized nonlinear Schroedinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions. Bifurcation of…
We introduce a two-state non-conserving driven-diffusive system in one-dimension under a discrete-time updating scheme. We show that the steady-state of the system can be obtained using a matrix product approach. On the other hand, the…
Using various techniques from dynamical systems theory, we rigorously study an experimentally validated model by [Barkley et al., Nature, 526:550-553, 2015], which describes the rise of turbulent pipe flow via a PDE system of reduced…
This paper is a preliminary work to address the problem of dynamical systems with parameters varying in time. An idea to predict their behaviour is proposed. These systems are called \emph{transient systems}, and are distinguished from…
A novel method for stability and instability study of autonomous dynamical systems using the flow and divergence of the vector field is proposed. A relation between the method of Lyapunov functions and the proposed method is established.…
We hypothesize that dynamical systems concepts used to study the transition to turbulence in shear flows are applicable to other transition phenomena in fluid mechanics. In this paper, we consider a finite air bubble that propagates within…
We consider the one-dimensional Swift-Hohenberg equation coupled to a conservation law. As a parameter increases the system undergoes a Turing bifurcation. We study the dynamics near this bifurcation. First, we show that stationary,…
We analyze the emergence of diffractive focusing in the transition from discrete to continuous space-time variables. Three types of dynamical equations are studied in a top-to-bottom approach, starting with the most general system. First we…
Pattern formation mechanisms of a reaction-diffusion-advection system, with one diffusivity, differential advection, and (Robin) boundary conditions of Danckwerts type, are being studied. Pattern selection requires mapping the domains of…
Two-dimensional (2D) turbulence, despite being an idealization of real flows, is of fundamental interest as a model of the spontaneous emergence of order from chaotic flows. The emergence of order often displays critical behavior, whose…
In this paper, I prove necessary and sufficient conditions for the existence of Turing instabilities in a general system with three interacting species. Turing instabilities describe situations when a stable steady state of a reaction…