Localized Patterns in Periodically Forced Systems: II. Patterns with Non-Zero Wavenumber

In pattern-forming systems, localized patterns are readily found when stable patterns exist at the same parameter values as the stable unpatterned state. Oscillons are spatially localized, time-periodic structures, which have been found experimentally in systems that are driven by a time-periodic force, for example, in the Faraday wave experiment. This paper examines the existence of oscillatory localized states in a PDE model with single-frequency time-dependent forcing, introduced in [A. M. Rucklidge and M. Silber, SIAM J. Appl. Dyn. Syst., 8 (2009), pp. 298--347] as a phenomenological model of the Faraday wave experiment. We choose parameters so that patterns set in with non-zero wavenumber (in contrast to [A. S. Alnahdi, J. Niesen, and A. M. Rucklidge, SIAM J. Appl. Dyn. Syst., 13 (2014), pp. 1311--1327]). In the limit of weak damping, weak detuning, weak forcing, small group velocity, and small amplitude, we reduce the model PDE to the coupled forced complex Ginzburg--Landau equations. We find localized solutions and snaking behavior in the coupled forced complex Ginzburg--Landau equations and relate these to oscillons that we find in the model PDE. Close to onset, the agreement is excellent. The periodic forcing for the PDE and the explicit derivation of the amplitude equations make our work relevant to the experimentally observed oscillons

Pattern Formation on Networks:from Localised Activity to Turing Patterns

Systems of dynamical interactions between competing species can be used to model many complex systems, and can be mathematically described by {\em random} networks. Understanding how patterns of activity arise in such systems is important for understanding many natural phenomena. The emergence of patterns of activity on complex networks with reaction-diffusion dynamics on the nodes is studied here. The connection between solutions with a single activated node, which can bifurcate from an undifferentiated state, and the fully developed system-scale patterns are investigated computationally. The different coexisting patterns of activity the network can exhibit are shown to be connected via a snaking type bifurcation structure, similar to those responsible for organising localised pattern formation in regular lattices. These results reveal the origin of the multistable patterns found in systems organised on complex networks. A key role is found to be played by nodes with so called {\em optimal degree}, on which the interaction between the reaction kinetics and the network structure organise the behaviour of the system. A statistical representation of the density of solutions over the parameter space is used as a means to answer important questions about the number of accessible states that can be exhibited in systems with such a high degree of complexity

Stationary peaks in a multivariable reaction--diffusion system: Foliated snaking due to subcritical Turing instability

An activator-inhibitor-substrate model of side-branching used in the context of pulmonary vascular and lung development is considered on the supposition that spatially localized concentrations of the activator trigger local side-branching. The model consists of four coupled reaction-diffusion equations and its steady localized solutions therefore obey an eight-dimensional spatial dynamical system in one dimension (1D). Stationary localized structures within the model are found to be associated with a subcritical Turing instability and organized within a distinct type of foliated snaking bifurcation structure. This behavior is in turn associated with the presence of an exchange point in parameter space at which the complex leading spatial eigenvalues of the uniform concentration state are overtaken by a pair of real eigenvalues; this point plays the role of a Belyakov-Devaney point in this system. The primary foliated snaking structure consists of periodic spike or peak trains with $N$ identical equidistant peaks, $N=1,2,\dots \,$, together with cross-links consisting of nonidentical, nonequidistant peaks. The structure is complicated by a multitude of multipulse states, some of which are also computed, and spans the parameter range from the primary Turing bifurcation all the way to the fold of the $N=1$ state. These states form a complex template from which localized physical structures develop in the transverse direction in 2D.Comment: 30 pages, 14 figure

Oscillons: localized patterns in a periodically forced system

Spatially localized, time-periodic structures, known as oscillons, are common in patternforming systems, appearing in fluid mechanics, chemical reactions, optics and granular media. This thesis examines the existence of oscillatory localized states in a PDE model with single frequency time dependent forcing, introduced in [70] as phenomenological model of the Faraday wave experiment. Firstly in the case where the prefered wavenumber at onset is zero, we reduce the PDE model to the forced complex Ginzburg– Landau equation in the limit of weak forcing and weak damping. This allows us to use the known localized solutions found in [15]. We reduce the forced complex Ginzburg– Landau equation to the Allen–Cahn equation near onset, obtaining an asymptotically exact expression for localized solutions. In the strong forcing case, we get the Allen–Cahn equation directly. Throughout, we use continuation techniques to compute numerical solutions of the PDE model and the reduced amplitude equation. We do quantitative comparison of localized solutions and bifurcation diagrams between the PDE model, the forced complex Ginzburg–Landau equation, and the Allen–Cahn equation. The second aspect in this work concerns the investigation of the existence of localized oscillons that arise with non-zero preferred wavenumber. In the limit of weak damping, weak detuning, weak forcing, small group velocity, and small amplitude, asymptotic reduction of the model PDE to the coupled forced complex Ginzburg–Landau equations is done. In the further limit of being very close to onset, we reduce the coupled forced complex Ginzburg–Landau equations to the real Ginzburg–Landau equation. We have qualitative prediction of finding exact localized solutions from the real Ginzburg–Landau equation limited by computational constraints of domain size. Finally, we examine the existence of localized oscillons in the PDE model with cubic–quintic nonlinearity in the strong damping, strong forcing and large amplitude case. We find two snaking branches in the bistability region between stable periodic patterns and the stable trivial state in one spatial dimension in a manner similar to systems without time dependent forcing. We present numerical examples of localized oscillatory spots and rings in two spatial dimensions

Localized patterns in periodically forced systems

Spatially localized, time-periodic structures are common in pattern-forming systems, appearing in fluid mechanics, chemical reactions, and granular media. We examine the existence of oscillatory localized states in a PDE model with single frequency time dependent forcing, introduced in [20] as phenomenological model of the Faraday wave experiment. In this study, we reduce the PDE model to the forced complex Ginzburg-Landau equation in the limit of weak forcing and weak damping. This allows us to use the known localized solutions found in [7]. We reduce the forced complex Ginzburg-Landau equation to the Allen-Cahn equation near onset, obtaining an asymptotically exact expression for localized solutions. We also extend this analysis to the strong forcing case recovering Allen-Cahn equation directly without the intermediate step. We find excellent agreement between numerical localized solutions of the PDE, localized solutions of the forced complex Ginzburg-Landau equation, and the Allen-Cahn equation. This is the first time that a PDE with time dependent forcing has been reduced to the Allen-Cahn equation, and its localized oscillatory solutions quantitatively studied. This paper is dedicated to the memory of Thomas Wagenknecht