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Space-time map showing a Turing mode invading a Hopf background in a system of length 300 during 200 units of time. No-flux BC are applied. The mean velocity of the front is slower than one Turing wavelength for one temporal oscillation and hence the system evolves through intermediate localized oscillations . The initial condition is a stable front obtained for the same values of parameters as in Fig. 2 and B9. The front is set unstable by suddenly decreasing B to 8.8 in order to go outside the pinning domain.
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Spatiotemporal dynamics resulting from the interaction of two instabilities breaking, respectively, spatial and temporal symmetries are studied in the framework of the amplitude equation formalism. The corresponding bifurcation scenarios feature steady-Hopf bistability with corresponding localized structures but also different types of mixed states...
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... locking is one wavelength for one frequency but other ratios are possible as long as there is an integer number of wavelengths per period of oscillation or vice versa. In these situations, the front may progress faster or slower and, in order to satisfy the nonadiabatic constraint, the system then sometimes creates temporary localized subzones Fig. 4 ...
Citations
... We show that it is a consequence of the nonlinear interaction between a supercritical Hopf bifurcation of the trivial state and a nearby subcritical Turing instability, and in particular of the presence of tertiary Hopf bifurcations on both the Turing branches and the associated LS inherited from the primary Hopf bifurcations of the trivial state. As such this behavior appears characteristic of the interaction between the Turing and Hopf instabilities 17,27,[38][39][40][41] and we demonstrate its presence in two different RD systems. ...
... We have revisited the Hopf-Turing interaction that arises in a number of two-species reaction-diffusion systems. 17,27,39,41 In particular, in Ref. 17 the authors considered the Brusselator model with supercritical Hopf and Turing branches in a regime with bistability between the two, finding a large multiplicity of stable Turing states embedded in an oscillating background obtained via DNS. Since no continuation was performed the observed states were not linked to homoclinic snaking, a notion that was developed only subsequently. ...
... We have revisited the Hopf-Turing interaction that arises in a number of two-species reaction-diffusion systems. 17,27,39,41 In particular, in Ref. 17 the authors considered the Brusselator model with supercritical Hopf and Turing branches in a regime with bistability between the two, finding a large multiplicity of stable Turing states embedded in an oscillating background obtained via DNS. Since no continuation was performed the observed states were not linked to homoclinic snaking, a notion that was developed only subsequently. ...
Numerical continuation is used to compute solution branches in a two-component reaction-diffusion model of Leslie–Gower type. Two regimes are studied in detail. In the first, the homogeneous state loses stability to supercritical spatially uniform oscillations, followed by a subcritical steady state bifurcation of Turing type. The latter leads to spatially localized states embedded in an oscillating background that bifurcate from snaking branches of localized steady states. Using two-parameter continuation, we uncover a novel mechanism whereby disconnected segments of oscillatory states zip up into a continuous snaking branch of time-periodic localized states, some of which are stable. In the second, the homogeneous state loses stability to supercritical Turing patterns, but steady spatially localized states embedded either in the homogeneous state or in a small amplitude Turing state are nevertheless present. We show that such behavior is possible when sideband Turing states are strongly subcritical and explain why this is so in the present model. In both cases, the observed behavior differs significantly from that expected on the basis of a supercritical primary bifurcation.
... Over the decades, the Brusselator model has been extensively studied, shaping ideas on oscillations and pattern formation far from thermodynamic equilibrium. It's a wellestablished system in the realm of non-equilibrium instabilities, with numerous research papers dedicated to its exploration, and the model has been subject to extensive investigation, encompassing limit cycles, Turing-Hopf bifurcation [12] and coupled Email addresses: manivelan.saminathan@gmail.com (S. ...
... A similar type of pattern was previously observed in the context of an asymmetric subharmonic Turing mixed mode. 42 Next, we turn our attention to point T 2 (ϕ 0 = 1.4), located closely to the Turing curve as depcited in the bifurcation diagram ( Fig. 1) which exhibits a range of intriguing phenomena resulting from the interaction between periodic photo-illumination and CPEF. The comprehensive diagram presented in Fig. 9 illustrates the system's response to periodic effects. ...
Designing and predicting self-organized pattern formation in out-of-equilibrium chemical and biochemical reactions holds fundamental significance. External perturbations like light and electric fields exert a crucial influence on reaction-diffusion systems involving ionic species. While the separate impacts of light and electric fields have been extensively studied, comprehending their combined effects on spatiotemporal dynamics is paramount for designing versatile spatial orders. Here, we theoretically investigate the spatiotemporal dynamics of chlorine dioxide-iodine-malonic acid reaction-diffusion system under photo-illumination and circularly polarized electric field (CPEF). By applying CPEF at varying intensities and frequencies, we observe the predominant emergence of oscillating hexagonal spot-like patterns from homogeneous stable steady states. Furthermore, our study unveils a spectrum of intriguing spatiotemporal instabilities, encompassing stripe-like patterns, oscillating dumbbell-shaped patterns, spot-like instabilities with square-based symmetry, and irregular chaotic patterns. However, when we introduce periodic photo-illumination to the hexagonal spot-like instabilities induced by CPEF in homogeneous steady states, we observe periodic size fluctuations. Additionally, the stripe-like instabilities undergo alternating transitions between hexagonal spots and stripes. Notably, within the Turing region, the interplay between these two external influences leads to the emergence of distinct superlattice patterns characterized by hexagonal-and square-based symmetry. These patterns include parallel lines of spots, target-like formations, black-eye patterns, and other captivating structures. Remarkably, the simple perturbation of the system through the application of these two external fields offers a versatile tool for generating a wide range of pattern-forming instabilities, thereby opening up exciting possibilities for future experimental validation.
... Note that the presence of additional subdominant neutral modes (e.g., resulting from additional conservation laws) or the simultaneous onset of several distinct instabilities would (possibly in extension of the present work) also result in amplitude equations on a higher level of the "codimension hierarchy" [30,62]. ...
Oscillatory behavior is ubiquitous in out-of-equilibrium systems showing spatiotemporal pattern formation. Starting from a linear large-scale oscillatory instability—a conserved-Hopf instability—that naturally occurs in many active systems with two conservation laws, we derive a corresponding amplitude equation. It belongs to a hierarchy of such universal equations for the eight types of instabilities in homogeneous isotropic systems resulting from the combination of three features: large-scale vs small-scale instability, stationary vs oscillatory instability, and instability without and with conservation law(s). The derived universal equation generalizes a phenomenological model of considerable recent interest, namely, the nonreciprocal Cahn-Hilliard model, and may be of a similar relevance for the classification of pattern forming systems as the complex Ginzburg-Landau equation.
... Spatiotemporal localized solutions characterized by the coexistence of Turing and uniform Hopf states have been observed in the vicinity of a codimension-two TH point, where both bifurcations are supercritical [68][69][70][71] . These dynamical states consist in a static Turing localized pattern embedded in a oscillatory uniform background field, and undergo a kind of snaking structure similar to the standard homoclinic snaking of stationary states, as shown by Tzou et al. 71 . ...
... Furthermore, the locations of the interfaces between the Turing and Hopf regions remain constant in time. This same scenario has been analyzed in other reaction-diffusion systems 68,69,71 , and is morphologically identical to those previously shown in a supercritical Turing scenario 68,69,71 . ...
... Furthermore, the locations of the interfaces between the Turing and Hopf regions remain constant in time. This same scenario has been analyzed in other reaction-diffusion systems 68,69,71 , and is morphologically identical to those previously shown in a supercritical Turing scenario 68,69,71 . ...
Spatially extended patterns and multistability of possible different states are common in many ecosystems, and their combination has an important impact on their dynamical behaviors. One potential combination involves tristability between a patterned state and two different uniform states. Using a simplified version of the Gilad–Meron model for dryland ecosystems, we study the organization, in bifurcation terms, of the localized structures arising in tristable regimes. These states are generally related to the concept of wave front locking and appear in the form of spots and gaps of vegetation. We find that the coexistence of localized spots and gaps, within tristable configurations, yields the appearance of hybrid states. We also study the emergence of spatiotemporal localized states consisting of a portion of a periodic pattern embedded in a uniform Hopf-like oscillatory background in a subcritical Turing–Hopf dynamical regime.
... Note that for systems without conservation laws, resonances are frequently studied. Examples include Hopf-Turing, Turing-Turing and wave-Turing resonances in RD systems or nonlinear optical systems[62][63][64][65][66]. ...
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’.
... Its importance equals that of the complex Ginzburg-Landau equation that describes the universal behavior in the vicinity of a standard Hopf bifurcaton in systems without conservation laws [1,11,21,41]. Note that the occurrence of additional subdominant neutral modes (e.g., resulting from additional conservation laws) or the simultaneous onset of several instabilities result in higher-codimension variants of the eight basic cases [12,55]. Although it is known that the presented linear instability is a phenomenon that is not covered by the complex Ginzburg-Landau equation [38] only very few stud-ies have considered its (weakly) nonlinear behavior by corresponding amplitude equations [18,40], normally, in special cases. ...
... The indicated scalings of the band of unstable wavenumbers and of the maximal growth rate are as in Fig. 1 of the main text. Again we consider the general multi-component model (11), naively use the same ansatz (12) and only discuss the differences for the current case as compared to the analysis done in Section 2 of the Supplementary Material. First, all terms at order ε and ε 2 are unchanged. ...
Oscillatory behavior is ubiquitous in out-of-equilibrium systems showing spatio-temporal pattern formation. Starting from a large-scale linear oscillatory instability - a conserved-Hopf instability - that naturally occurs in systems with two conservation laws, we derive the corresponding amplitude equation. This completes the set of such universal equations for the eight types of instabilities in homogeneous isotropic systems resulting from the combination of three features: large-scale vs. small-scale instability, stationary vs. oscillatory instability, and instability without and with conservation law(s). The derived universal equation generalizes a phenomenological model of considerable recent interest, namely, the nonreciprocal Cahn-Hilliard equation, and plays a similar role in the systematics of pattern formation as the complex Ginzburg-Landau equation.
... Spatiotemporal localized solutions characterized by the coexistence of Turing and uniform Hopf states have been observed in the vicinity of a codimension-two TH point, where both bifurcations are supercritical [58][59][60][61] . These dynamical states consist in a static Turing localized pattern embedded in a oscillatory uniform background field, and undergo a kind of snaking structure similar to the standard homoclinic snaking of stationary states, as shown by Tzou et al. 61 . ...
... Furthermore, the locations of the interfaces between the Turing and Hopf regions remain constant in time. This same scenario has been analyzed in other reaction-diffusion systems 58,59,61 , and is morphologically identical to those previously shown in a supercritical Turing scenario 58,59,61 . ...
... The UV state undergoes a Hopf bifurcation, which, in the absence of diffusion, develops periodic oscillations. When diffusion is considered TI comes into play, leading to complex dynamical scenarios where the Turing and Hopf modes interact 59 . The last part of our work (see Sec. V) has focused on the analysis of how the stability and dynamics of gap biomass LSs modify when entering the TH instability region. ...
Spatially extended patterns and multistability of possible different states is common in many ecosystems, and their combination has an important impact on their dynamical behaviours. One potential combination involves tristability between a patterned state and two different uniform states. Using a simplified version of the Gilad-Meron model for dryland ecosystems, we study the organization, in bifurcation terms, of the localized structures arising in tristable regimes. These states are generally related with the concept of wave front locking, and appear in the form of spots and gaps of vegetation. We find that the coexistence of localized spots and gaps, within tristable configurations, yield the appearance of hybrid states. We also study the emergence of spatiotemporal localized states consisting in a portion of a periodic pattern embedded in a uniform Hopf-like oscillatory background in a subcritical Turing-Hopf dynamical regime.
... For two control parameters, the surfaces will be curves which can terminate at a special value of the parameters giving what is called a codimension-two point. A simple dynamical system capable of showing a wide variety of fixed points is the Brusselator [30,31]. The addition of the reaction diffusion term to the Brusselator has led to the generation of a large number of interesting possibilities over almost three decades [30][31][32][33][34]. ...
... A simple dynamical system capable of showing a wide variety of fixed points is the Brusselator [30,31]. The addition of the reaction diffusion term to the Brusselator has led to the generation of a large number of interesting possibilities over almost three decades [30][31][32][33][34]. ...
... A similar study on the codimensiontwo point in the Brusselator model is available in Refs. [30,31], where the generic structures of the amplitude equations are discussed and their numerical solutions have been obtained. Pure states as well as mixed states that are born out of the interacting Hopf and steady state pattern modes are predicted. ...
We study the linear instabilities and bifurcations in the Selkov model for glycolysis with diffusion. We show that this model has a zero wave-vector, finite frequency Hopf bifurcation, which is a forward or supercritical bifurcation, to a growing oscillatory but spatially homogeneous state and a saddle-node bifurcation, which is a backward or subcritical bifurcation, to a growing inhomogeneous state with a steady pattern characterised by a finite wavevector. We further demonstrate that by tuning the relative diffusivity of the two concentrations, it is possible to make both the instabilities to occur at the same point in the parameter space, leading to an unusual type of codimension-two bifurcation. We then show that in the vicinity of this codimension-two bifurcation, the initial conditions decide whether a spatially uniform oscillatory or a spatially periodic steady pattern emerges in the long time limit. It is also possible to form a co-existing patterned and time-periodic state by fine-tuning the diffusivity ratio for moderate values, in qualitative agreement with recent experimental studies.
... A natural question is that whether nonlocal effect could induce codimension-two bifurcation for reaction-diffusion equations. For the reactiondiffusion equation, the interaction of a Turing instability (leading to spatially nonhomogeneous steady states) with a Hopf bifurcation (giving rise to temporal oscillations) has been observed and studied in chemical, biological and physical systems, see [42,43,49,15] and references therein. For example, Rovinsky and Menzinger [49] studied this Turing-Hopf interaction for three models of chemically active media by using Poincaré-Birkhoff method and shown the bistability of spatially nonhomogeneous steady states and homogeneous oscillations. ...
... For example, Rovinsky and Menzinger [49] studied this Turing-Hopf interaction for three models of chemically active media by using Poincaré-Birkhoff method and shown the bistability of spatially nonhomogeneous steady states and homogeneous oscillations. In the framework of amplitude equation formalism, De Wit et al. [15] investigated the bifurcation scenarios near the Turing-Hopf singularity. Recently, to show an accurate dynamic classification at this singularity, Song et al. [58] applied the normal form theory proposed by Faria [19] to a general reaction-diffusion equation, and obtained a series of explicit formulas for calculating the normal forms associated with the Turing-Hopf bifurcation. ...
... Recalling that F (0, 0, µ) = 0, D U F (0, 0, µ) = 0 and D U F (0, 0, µ) = 0, we have F 2 (U, U , α) = F 2 (U, U , 0). Plug (15) into (27) at w = 0, and then F 2 (U, U , α) becomes (15). By (19), we have ...
In this paper, we consider a general reaction-diffusion system with nonlocal effects and Neumann boundary conditions, where a spatial average kernel is chosen to be the nonlocal kernel. By virtue of the center manifold reduction technique and normal form theory, we present a new algorithm for computing normal forms associated with the codimension-two double Hopf bifurcation. The theoretical results are applied to a predator-prey model, and complex dynamic behaviors such as spatially nonhomogeneous periodic oscillations and spatially nonhomogeneous quasi-periodic oscillations could occur.
