Dynamic transitions through scattors in dissipative systems

Collisions of localized patterns in a nonvariational Swift-Hohenberg equation

The cubic-quintic Swift-Hohenberg equation (SH35) has been proposed as an order parameter description of several convective systems with reflection symmetry in the layer midplane, including binary fluid convection. We use numerical continuation, together with extensive direct numerical simulations (DNSs), to study SH35 with an additional nonvariational quadratic term to model the effects of breaking the midplane reflection symmetry. The nonvariational structure of the model leads to the propagation of asymmetric spatially localized structures (LSs). An asymptotic prediction for the drift velocity of such structures, derived in the limit of weak symmetry breaking, is validated numerically. Next, we present an extensive study of possible collision scenarios between identical and nonidentical traveling structures, varying a temperaturelike control parameter. These collisions are inelastic and result in stationary or traveling structures. Depending on system parameters and the types of structures colliding, the final state may be a simple bound state of the initial LSs, but it can also be longer or shorter than the sum of the two initial states as a result of nonlinear interactions. The Maxwell point of the variational system, where the free energy of the global pattern state equals that of the trivial state, is shown to have no bearing on which of these scenarios is realized. Instead, we argue that the stability properties of bound states are key. While individual LSs lie on a modified snakes-and-ladders structure in the nonvariational SH35, the multipulse bound states resulting from collisions lie on isolas in parameter space, disconnected from the trivial solution. In the gradient SH35, such isolas are always of figure-eight shape, but in the present nongradient case they are generically more complex, although the figure-eight shape is preserved in a small subset of cases. Some of these complex isolas are shown to terminate in T-point bifurcations. A reduced model is proposed to describe the interactions between the tails of the LSs. The model consists of two coupled ordinary differential equations (ODEs) capturing the oscillatory nature of SH35 profiles at the linear level. It contains three parameters: two interaction amplitudes and a phase, whose values are deduced from high-resolution DNSs using gradient descent optimization. For collisions leading to the formation of simple bound states, the reduced model reproduces the trajectories of LSs with high quantitative accuracy. When nonlinear interactions lead to the creation or deletion of wavelengths, the model performs less well. Finally, we propose an effective signature of a given interaction in terms of net attraction or repulsion relative to free propagation. It is found that interactions can be attractive or repulsive in the net, irrespective of whether the two closest interacting extrema are of the same or opposite signs. Our findings highlight the rich temporal dynamics described by this bistable nonvariational SH35, and show that the interactions in this system can be quantitatively captured, to a significant extent, by a highly reduced ODE model.

Non-reciprocity induces resonances in a two-field Cahn-Hilliard model

We consider a non-reciprocally coupled two-field Cahn-Hilliard system that has been shown to allow for oscillatory behaviour and suppression of coarsening. After introducing the model, we first review the linear stability of steady uniform states and show that all instability thresholds are identical to the ones for a corresponding two-species reaction-diffusion system. Next, we consider a specific interaction of linear modes-a 'Hopf-Turing' resonance-and derive the corresponding amplitude equations using a weakly nonlinear approach. We discuss the weakly nonlinear results and finally compare them with fully nonlinear simulations for a specific conserved amended FitzHugh-Nagumo system. We conclude with a discussion of the limitations of the employed weakly nonlinear approach. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Phase-field-crystal description of active crystallites: Elastic and inelastic collisions

The active Phase-Field-Crystal (aPFC) model combines elements of the Toner-Tu theory for self-propelled particles and the classical Phase-Field-Crystal (PFC) model that describes the transition between liquid and crystalline phases. In the liquid-crystal coexistence region of the PFC model, crystalline clusters exist in the form of localized states that coexist with a homogeneous background. At sufficiently strong activity (related to self-propulsion strength), they start to travel. We employ numerical path continuation and direct time simulations to first investigate the existence regions of different types of localized states in one spatial dimension. The results are summarized in morphological phase diagrams in the parameter plane spanned by activity and mean density. Then we focus on the interaction of traveling localized states, studying their collision behavior. As a result, we distinguish "elastic" and "inelastic" collisions. In the former, localized states recover their properties after a collision, while in the latter, they may completely or partially annihilate, forming resting bound states or various traveling states.

Why a Large-Scale Mode Can Be Essential for Understanding Intracellular Actin Waves

During the last decade, intracellular actin waves have attracted much attention due to their essential role in various cellular functions, ranging from motility to cytokinesis. Experimental methods have advanced significantly and can capture the dynamics of actin waves over a large range of spatio-temporal scales. However, the corresponding coarse-grained theory mostly avoids the full complexity of this multi-scale phenomenon. In this perspective, we focus on a minimal continuum model of activator-inhibitor type and highlight the qualitative role of mass conservation, which is typically overlooked. Specifically, our interest is to connect between the mathematical mechanisms of pattern formation in the presence of a large-scale mode, due to mass conservation, and distinct behaviors of actin waves.

Excitable solitons: Annihilation, crossover, and nucleation of pulses in mass-conserving activator-inhibitor media

Excitable pulses are among the most widespread dynamical patterns that occur in many different systems, ranging from biological cells to chemical reactions and ecological populations. Traditionally, the mutual annihilation of two colliding pulses is regarded as their prototypical signature. Here we show that colliding excitable pulses may exhibit solitonlike crossover and pulse nucleation if the system obeys a mass conservation constraint. In contrast to previous observations in systems without mass conservation, these alternative collision scenarios are robustly observed over a wide range of parameters. We demonstrate our findings using a model of intracellular actin waves since, on time scales of wave propagations over the cell scale, cells obey conservation of actin monomers. The results provide a key concept to understand the ubiquitous occurrence of actin waves in cells, suggesting why they are so common, and why their dynamics is robust and long-lived.

Origin of finite pulse trains: Homoclinic snaking in excitable media

Many physical, chemical, and biological systems exhibit traveling waves as a result of either an oscillatory instability or excitability. In the latter case a large multiplicity of stable spatially localized wavetrains consisting of different numbers of traveling pulses may be present. The existence of these states is related here to the presence of homoclinic snaking in the vicinity of a subcritical, finite wavenumber Hopf bifurcation. The pulses are organized in a slanted snaking structure resulting from the presence of a heteroclinic cycle between small and large amplitude traveling waves. Connections of this type require a multivalued dispersion relation. This dispersion relation is computed numerically and used to interpret the profile of the pulse group. The different spatially localized pulse trains can be accessed by appropriately customized initial stimuli, thereby blurring the traditional distinction between oscillatory and excitable systems. The results reveal a new class of phenomena relevant to spatiotemporal dynamics of excitable media, particularly in chemical and biological systems with multiple activators and inhibitors.

Delayed feedback induces motion of localized spots in reaction-diffusion systems

We study the formation of localized structures, often called localized spots, in reaction-diffusion systems subject to time delayed feedback control. We focus on the regime close to a second-order critical point marking the onset of a hysteresis loop. We show that the space-time dynamics of the FitzHugh-Nagumo model in the vicinity of that critical point could be described by the delayed Swift-Hohenberg equation. We show that the delayed feedback induces a spontaneous motion of localized spots. We characterize this motion by computing analytically the velocity and the threshold above which localized structures start to move in an arbitrary direction. Numerical solutions of the governing equation are in close agreement with those obtained from the delayed Swift-Hohenberg equation.

Mathematical model for contemplative amoeboid locomotion

It has recently been reported that even single-celled organisms appear to be "indecisive" or "contemplative" when confronted with an obstacle. When the amoeboid organism Physarum plasmodium encounters the chemical repellent quinine during migration along a narrow agar lane, it stops for a period of time (typically several hours) and then suddenly begins to move again. When movement resumes, three distinct types of behavior are observed: The plasmodium continues forward, turns back, or migrates in both directions simultaneously. Here, we develop a continuum mathematical model of the cell dynamics of contemplative amoeboid movement. Our model incorporates the dynamics of the mass flow of the protoplasmic sol, in relation to the generation of pressure based on the autocatalytic kinetics of pseudopod formation and retraction (mainly, sol-gel conversion accompanying actin-myosin dynamics). The biological justification of the model is tested by comparing with experimentally measured spatiotemporal profiles of the cell thickness. The experimentally observed types of behavior are reproduced in simulations based on our model, and the core logic of the modeled behavior is clarified by means of nonlinear dynamics. An on-off transition between the refractory and activated states of the chemical reactivity that takes place at the leading edge of the plasmodium plays a key role in the emergence of contemplative behavior.

Stationary periodical structure emitting an infinite number of traveling impulses in a model of a one-dimensional infinite excitable reaction-diffusion system

A two-variable model of a one-dimensional infinite excitable reaction-diffusion system describing an expanding stationary periodical structure emitting traveling impulses is presented. The model is based on two coupled catalytic (enzymatic) reactions. The chemical scheme consists of mono- and bimolecular reactions.

Onset of unidirectional pulse propagation in an excitable medium with asymmetric heterogeneity

Heterogeneity is one of the most important and ubiquitous types of external perturbations in dissipative systems. To know the behaviors of pulse waves in such media is closely related to studying the collision process between the pulse and the heterogeneity-induced-ordered pattern. In particular, we focus on unidirectional propagation of pulses in a medium with an asymmetric bump heterogeneity. This topic has attracted much interest recently with respect to potential computational aspects of chemical pulse propagation as well as with respect to pulse propagation in biological signal processing. We employ a three-component reaction-diffusion system with one activator and two inhibitor species to illustrate these issues. The reduced dynamics near a drift bifurcation describes the phenomena in the full partial differential equations by ordinary differential equations. Such a reduced dynamics is able to capture unidirectional propagation properties of pulses near an asymmetric heterogeneity in a qualitatively correct way. A remarkable feature is that such unidirectional behavior is linked to the imperfection of global bifurcation structure and the resulting asymmetric locations of critical points.

Self-organized pacemakers and bistability of pulses in an excitable medium

Pattern formation in an excitable medium described by a three-component reaction-diffusion system is investigated. Our focus is on stable self-organized pacemakers which give rise to spatially extended target patterns. Bistability of pulse solutions in the excitable regime is also reported, and interactions of the different pulses with each other and the pacemaker are studied. Self-organized pacemakers are created by a suitable perturbation from the steady state or through interaction of pulses. Bound states of one-dimensional pacemakers and phase flips are also observed.

Dynamics of traveling pulses in heterogeneous media

One of the fundamental issues of pulse dynamics in dissipative systems is clarifying how the heterogeneity in the media influences the propagating manner. Heterogeneity is the most important and ubiquitous type of external perturbation. We focus on a class of one-dimensional traveling pulses, the associated parameters of which are close to drift and/or saddle-node bifurcations. The advantage in studying the dynamics in such a class is twofold: First, it gives us a perfect microcosm for the variety of outputs in a general setting when pulses encounter heterogeneities. Second, it allows us to reduce the original partial differential equation dynamics to a tractable finite-dimensional system. Such pulses are sensitive when they run into heterogeneities and show rich responses such as annihilation, pinning, splitting, rebound, as well as penetration. The reduced ordinary differential equations (ODEs) explain all these dynamics and the underlying bifurcational structure controlling the transitions among different dynamic regimes. It turns out that there are hidden ordered patterns associated with the critical points of ODEs that play a pivotal role in understanding the responses of the pulse; in fact, the depinning of pulses can be explained in terms of global bifurcations among those critical points. We focus mainly on a bump and periodic types of heterogeneity, however our approach is also applicable to general cases. It should be noted that there appears to be spatio-temporal chaos for a periodic type of heterogeneity when its period becomes comparable with the size of the pulse.

Heterogeneity-induced defect bifurcation and pulse dynamics for a three-component reaction-diffusion system

We consider the dynamics when traveling pulses encounter heterogeneities in a three-component reaction diffusion system of one-activator-two-inhibitor type, which typically arises as a qualitative model of a gas-discharge system. We focused on the case where one of the kinetic coefficients changes similar to a smoothed step function, which is basic for more general heterogeneity as in periodic or random media. Since the heterogeneity is introduced to the kinetic part in an additive way, it causes the system to produce various types of localized structures smoothing the jump heterogeneity called the defects at the jump point, which makes a sharp contrast with the multiplicative heterogeneous case for the Gray-Scott model. The main issue is to study the collision dynamics between traveling pulses and defects, and show that their global bifurcation structure plays a key role in clarifying the underlying mechanism. Five outputs are observed after collisions including annihilation, rebound, and pinning. Unstable steady states are identified as separators between two different dynamic regimes: penetration and rebound, the role of which is very close to that of scattors arising in collision process. An organizing center producing the traveling pulses, defects, and scattors via unfolding with respect to the parameters is also presented.

Periodical survival or decay of traveling impulse in a model of a one-dimensional reaction-diffusion system

A two-variable model of a one-dimensional (1D), open, excitable reaction-diffusion system describing space-time evolutions of traveling impulses is investigated. It is shown that depending on the size of the system, the traveling impulse can survive or decay. Continuous increase of the size of the system causes periodical repetitions of surviving and decay of the impulse. The qualitative properties of the model, which allow us to expect the phenomenon, are described. Numerical solutions confirm this expectation. The chemical reaction scheme is realistic and may be a stimulus for seeking the phenomenon in experiments.

Chaotically oscillating interfaces in a parametrically forced system

Structures and motions of a single interface exhibiting chaotic behavior are studied in the one-dimensional parametrically forced complex Ginzburg-Landau equation. There exist two kinds of chaotic interfaces whose differences are characterized by their chiral symmetry and the diffusivity of their motion. The transition between these behaviors is also investigated from the viewpoint of singularities of several dynamical variables, such as the diffusion constant, the resident time to each state, and the maximum trapping time to the unstable solution.

Complex patterns in reactive microemulsions: self-organized nanostructures?

In a reverse microemulsion consisting of water, oil (octane), an anionic surfactant [aerosol OT (AOT)], and the reactants of the oscillating Belousov-Zhabotinsky (BZ) reaction, a variety of complex spatiotemporal patterns appear. These include traveling and standing waves, spirals that move either toward or away from their centers, spatiotemporal chaos, Turing patterns, segmented waves, and localized structures, both stationary and oscillatory. The system consists of nanometer-sized droplets of water containing the BZ reactants surrounded by a monolayer of AOT, swimming in a sea of oil, through which nonpolar BZ intermediates can diffuse rapidly. We present experimental and computational results on this fascinating system and comment on possible future directions for research.

Scattering of traveling spots in dissipative systems

One of the fundamental questions for self-organization in pattern formation is how spatial periodic structure is spontaneously formed starting from a localized fluctuation. It is known in dissipative systems that splitting dynamics is one of the driving forces to create many particle-like patterns from a single seed. On the way to final state there occur many collisions among them and its scattering manner is crucial to predict whether periodic structure is realized or not. We focus on the colliding dynamics of traveling spots arising in a three-component system and study how the transition of scattering dynamics is brought about. It has been clarified that hidden unstable patterns called "scattors" and their stable and unstable manifolds direct the traffic flow of orbits before and after collisions. The collision process in general can be decomposed into several steps and each step is controlled by such a scattor, in other words, a network among scattors forms the backbone for scattering dynamics. A variety of input-output relations comes from the complexity of the network as well as high Morse indices of the scattor. The change of transition manners is caused by the switching of the network from one structure to another, and such a change is caused by the singularities of scattors. We illustrate a typical example of the change of transition caused by the destabilization of the scattor. A new instability of the scattor brings a new destination for the orbit resulting in a new input-output relation, for instance, Hopf instability for the scattor of peanut type brings an annihilation.

On the back-firing instability

The onset of the back-firing instability is studied in a one-dimensional spatially extended and dissipative system, where propagating localized solutions become unstable. It corresponds to the emission in the tail of a solitary wave of a new wave propagating in the opposite direction. The transition is illustrated, in geometrical terms, using a model normal form equation. (c) 2004 American Institute of Physics.

Phase-dependent output of scattering process for traveling breathers

Scattering process between one-dimensional traveling breathers (TBs), i.e., oscillatory traveling pulses, for the complex Ginzburg-Landau equation (CGLE) with external forcing and a three-component activator-substrate-inhibitor model are studied. The input-output relation depends in general on the phase of two TBs at collision point, which makes a contrast to the case for the steady traveling pulses. A hidden unstable solution called the scattor plays a crucial role to understand the scattering dynamics. Stable and unstable manifolds of the scattor direct the traffic flows of the scattering process. A transition point of the input-output relation in a parameter space such as from preservation to annihilation corresponds to when the orbit crosses the stable manifold of the scattor. The phase dependency of input-output relation comes from the fact that the profiles at collision point make a loop parametrized by the phase and it traverses the stable manifold of the scattor. A global bifurcation viewpoint is quite useful not only to understand how TBs emerge but also to detect scattors. It turns out that the scattor for the CGLE (respectively the three-component system) becomes an unstable time-periodic (respectively stationary) solution.