Dynamic Transition Theory Research Articles

First transition dynamics of reaction–diffusion equations with higher order nonlinearity

AbstractIn this study, the dynamics of a reaction–diffusion equation on a bounded interval at the first dynamic transition point is investigated. The main difference of this study from the existing ones in the dynamic transition literature is that in this work, no specific form of the nonlinear operator is assumed except that it is an analytic function of and . The linear operator is also assumed to be a general operator with sinusoidal eigenvectors. Moreover, we assume that there is a first transition as a real simple eigenvalue changes sign. With this general framework, we make a rigorous analysis of the dependence of the first transition dynamics on the coefficients of the Taylor expansion of the nonlinear operator. The main tool we use in this study is the center manifold reduction combined with the classification of the dynamic transition theory. This study is a generalization of a recent work where the nonlinear operator contains only low‐order (quadratic and cubic) nonlinearities. The current generalization requires certain technical difficulties such as the validity of the genericity conditions on the Taylor coefficients of the nonlinear operator and a bootstrapping argument using these genericity conditions. The results of this work can be generalized in various directions, which are discussed in the conclusions section.

Dynamical transition and bifurcation of a diffusive predator–prey model with an Allee effect on prey

We study the stability and dynamic transitions of a diffusive predator–prey model with Allee effect on prey in a two-dimensional domain. Our tool includes the recently developed dynamic transition theory by Ma and Wang (2019). We first verify the principle of exchange of stability(PES), then we obtain rigorously the nonlinear transition behavior of the reaction–diffusion system. For single eigenvalue crossing, global transitions where solutions jump away from origin are also discovered along with local pitch-fork and Hopf bifurcations. We also investigates the rarely studied double eigenvalue crossing case which shows more complex bifurcations. All these dynamic behaviors are classified according to a nondimensional transition number and detailed orbital changes are illustrated. Further numerical studies are included to reflect the theoretical results and display intricate relationships between solution patterns and parameters, among which the distinct role of the Allee effect is analyzed in detail.

Dynamic Transition Analysis for Activator-Substrate System

The main objective of this article is to study the dynamic transition associated with the activator-substrate system. Two criteria are derived to describe the transition from real eigenvalues or complex eigenvalues and the types of transition. Notably, we get two parameters b_{1} and b_{2}, which can determine the the types of transitions for the two criteria respectively. The analysis is carried out using dynamic transition theory developed recently by Ma and Wang (Phase transition dynamics, Springer, New York, 2013, Bifurcation Theory and Applications, World Scientific, Singapore, 2005, Stability and Bifurcation of Nonlinear Evolutions Equations, Science Press, Beijing, China, 2007).

Theory of Dynamical Phase Transitions in Quantum Systems with Symmetry-Breaking Eigenstates.

We present a theory for the two kinds of dynamical quantum phase transitions, termed DPT-I and DPT-II, based on a minimal set of symmetry assumptions. In the special case of collective systems with infinite-range interactions, both are triggered by excited-state quantum phase transitions. For quenches below the critical energy, the existence of an additional conserved charge, identifying the corresponding phase, allows for a nonzero value of the dynamical order parameter characterizing DPTs-I, and precludes the main mechanism giving rise to nonanalyticities in the return probability, trademark of DPTs-II. We propose a statistical ensemble describing the long-time averages of order parameters in DPTs-I, and provide a theoretical proof for the incompatibility of the main mechanism for DPTs-II with the presence of this additional conserved charge. Our results are numerically illustrated in the fully connected transverse-field Ising model, which exhibits both kinds of dynamical phase transitions. Finally, we discuss the applicability of our theory to systems with finite-range interactions, where the phenomenology of excited-state quantum phase transitions is absent. We illustrate our findings by means of numerical calculations with experimentally relevant initial states.

Transition and bifurcation analysis for chemotactic systems with double eigenvalue crossings

<abstract><p>Our main objective of this research is to study the dynamic transition for diffusive chemotactic systems modeled by Keller-Segel equations in a rectangular domain. The main tool used is the recently developed dynamic transition theory. Through a reduction analysis and focusing on systems with certain symmetry where double eigenvalue crossing occurs during the instability process, it is shown that the chemotactic system can undergo both continuous and jump type transitions from the steady states, depending on non-dimensional parameters $ \alpha $, $ \mu $ and the side length $ L_1 $ and $ L_2 $ of the container. Detailed dynamic structures during transition, including metastable and stable states and orbital connections between them, are rigorously obtained. This result extends the previous work with only one eigenvalue crossing at critical parameters and offers more complex insights given the symmetry of our settings.</p></abstract>

Dynamic phase transition theory

Thermodynamic conventions suffer from describing dynamical distinctions, especially when the structural and energetic changes induced by rare events are insignificant. By using the ensemble theory in the trajectory space, we present a statistical approach to address this problem. Rather than spatial particle-particle interaction which dominates thermodynamics, the temporal correlation of events dominates the dynamics. The zeros of dynamic partition function mark phase transitions in the space-time, i.e., dynamic phase transition (DPT), as Yang and Lee formulate traditional phase transitions, and hence determine dynamic phases on both sides of the zeros. Analogous to the role of temperature (pressure) as thermal (mechanical) potential, we interpret the controlling variable of DPT, i.e., dynamic field, as the dynamical potential. These findings offer possibility towards a unified picture of phase and phase transition.

Transitions of zonal flows in a two-layer quasi-geostrophic ocean model

We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that a conjugate pair of modes loses stability and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing related midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case, we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to \(S^3\). The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains (stable or unstable) time-periodic solutions and a quasi-periodic solution.

Stability and dynamic transition of vegetation model for flat arid terrains

<p style='text-indent:20px;'>In this paper, we aim to investigate the dynamic transition of the Klausmeier-Gray-Scott (KGS) model in a rectangular domain or a square domain. Our research tool is the dynamic transition theory for the dissipative system. Firstly, we verify the principle of exchange of stability (PES) by analyzing the spectrum of the linear part of the model. Secondly, by utilizing the method of center manifold reduction, we show that the model undergoes a continuous transition or a jump transition. For the model in a rectangular domain, we discuss the transitions of the model from a real simple eigenvalue and a pair of simple complex eigenvalues. our results imply that the model bifurcates to exactly two new steady state solutions or a periodic solution, whose stability is determined by a non-dimensional coefficient. For the model in a square domain, we only focus on the transition from a real eigenvalue with algebraic multiplicity 2. The result shows that the model may bifurcate to an <inline-formula><tex-math id="M1">\begin{document}$ S^{1} $\end{document}</tex-math></inline-formula> attractor with 8 non-degenerate singular points. In addition, a saddle-node bifurcation is also possible. At the end of the article, some numerical results are performed to illustrate our conclusions.</p>

Dynamical transition and chaos for a five‐dimensional Lorenz model

In this paper, we study the dynamical transition and chaos for a five‐dimensional Lorenz system. Based on the eigenvalue analysis, the principle of exchange of stabilities conditions is obtained. By using the dynamical transition theory, three different types of dynamical transition for the five‐dimensional Lorenz system are derived. More precisely, when the control parameter , the system has a continuous transition and bifurcates to two stable steady states. As r further increases, the system undergoes two successive transitions. That is, under some condition, the transition is continuous and a stable limit cycle is bifurcated; if not, the system undergoes a jump transition and an unstable periodic orbit occurs. Especially, the chaotic orbits occur when . Finally, numerical results are given to illustrate our theoretical analysis.

Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion

<p style='text-indent:20px;'>The Swift-Hohenberg equation is ubiquitous in the study of bistable dynamics. In this paper, we study the dynamic transitions of the Swift-Hohenberg equation with a third-order dispersion term in one spacial dimension with a periodic boundary condition. As a control parameter crosses a critical value, the trivial stable equilibrium solution will lose its stability, and undergoes a dynamic transition to a new physical state, described by a local attractor. The main result of this paper is to fully characterize the type and detailed structure of the transition using dynamic transition theory [<xref ref-type="bibr" rid="b7">7</xref>]. In particular, employing techniques from center manifold theory, we reduce this infinite dimensional problem to a finite one since the space on which the exchange of stability occurs is finite dimensional. The problem then reduces to analysis of single or double Hopf bifurcations, and we completely classify the possible phase changes depending on the dispersion for every spacial period.</p>

Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food

<p style='text-indent:20px;'>The article aims to investigate the dynamic transitions of a toxin-producing phytoplankton zooplankton model with additional food in a 2D-rectangular domain. The investigation is based on the dynamic transition theory for dissipative dynamical systems. Firstly, we verify the principle of exchange of stability by analysing the corresponding linear eigenvalue problem. Secondly, by using the technique of center manifold reduction, we determine the types of transitions. Our results imply that the model may bifurcate two new steady state solutions, which are either attractors or saddle points. In addition, the model may also bifurcate a new periodic solution as the control parameter passes critical value. Finally, some numerical results are given to illustrate our conclusions.

Unbalanced Langmuir kinetics affects TASEP dynamical transitions: mean-field theory

In a previous study we developed a mean-field theory of dynamical transitions for the totally-asymmetric simple-exclusion process with open boundaries and Langmuir kinetics, in the so-called balanced regime, characterized by equal binding and unbinding rates. Here we show that simply including the possibility of unbalanced rates gives rise to an unexpectedly richer dynamical phase diagram. In particular, the current work predicts an unusual type of dynamical transition, which exhibits certain similarities with first-order phase transitions of equilibrium systems. We also point out that different types of dynamical transition are accompanied by different structural changes in the (mean-field) relaxation spectrum.

Dynamical transition theory of hexagonal pattern formations

The main goal of this paper is to understand the formation of hexagonal patterns from the dynamical transition theory point of view. We consider the transitions from a steady state of an abstract nonlinear dissipative system. To shed light on the formation of mixed mode patterns such as the hexagonal pattern, we consider the case where the linearized operator of the system has two critical real eigenvalues, at a critical value λc of a control parameter λ with associated eigenmodes having a roll and rectangular pattern. By using center manifold reduction, we obtain the reduced equations of the system near the critical transition value λc. By a through analysis of these equations, we fully characterize all possible transition scenarios when the coefficients of the quadratic part of the reduced equations do not vanish. We consider three problems, two variants of the 2D Swift-Hohenberg equation and the 3D surface tension driven convection, to demonstrate that all the main theoretical results we obtain here are indeed realizable.

On the stability and transition of the Cahn-Hilliard/Allen-Cahn system

In this paper, the main objective is to study the stability and transition of the Cahn-Hilliard/Allen-Cahn system. By using the dynamic transition theory, combining with the spectral theorem for general linear completely continuous fields, we prove that the system undergoes a continuous transition and bifurcates from a trivial solution to an attractor as the control parameter crosses a certain critical value. In addition, for some special cases, i.e., the domain is \begin{document}$ n $\end{document} -dimensional box \begin{document}$ (n = 1,2,3) $\end{document} , we not only obtain the stability of the singular points of the attractors, the topological structure of the attractors is also illustrated.

On the stability and transition for the Navier‐Stokes‐alpha model

The main aim of this paper is to investigate the stability and transition of the Navier‐Stokes‐alpha model. By using the continued‐fraction method, combining with the dynamic transition theory, we show the existence of a Hopf bifurcation in this model as Reynolds number crosses a critical value. Upon deriving the explicit expression of a non‐dimensional number P called transition number, which is a function of the critical Reynolds number and the aspect ratio, we further analyze the transition associated with the Hopf bifurcation. More precisely, it is shown that the modeled flow exhibits either a continuous or catastrophic transition at the critical Reynolds number, whose specific type of the transition is determined by the sign of the real part of P at the critical Reynolds number, and the spatio‐temporal structure of the limit cycle bifurcated that corresponds to a wave that propagates slowly westward and is symmetric about the mid‐axis of the channel.

Dynamical transitions of a low-dimensional model for Rayleigh–Bénard convection under a vertical magnetic field

In this article, we study the dynamic transitions of a low-dimensional dynamical system for the Rayleigh–Bénard convection subject to a vertically applied magnetic field. Our analysis follows the dynamical phase transition theory for dissipative dynamical systems based on the principle of exchange of stability and the center manifold reduction. We find that, as the Rayleigh number increases, the system undergoes two successive transitions: the first one is a well-known pitchfork bifurcation, whereas the second one is structurally more complex and can be of different type depending on the system parameters. More precisely, for large magnetic field, the second transition is of continuous type and gives to a stable limit cycle; on the other hand, for low magnetic field or small height-to-width aspect ratio, a jump transition occurs where an unstable periodic orbit eventually collides with the stable steady state, leading to the loss of stability at the critical Rayleigh number. Finally, numerical results are presented to corroborate the analytic predictions.

Constructing effective free energies for dynamical quantum phase transitions in the transverse-field Ising chain

The theory of dynamical quantum phase transitions represents an attempt to extend the concept of phase transitions to the far from equilibrium regime. While there are many formal analogies to conventional transitions, it is a major question to which extent it is possible to formulate a nonequilibrium counterpart to a Landau-Ginzburg theory. In this work we take a first step in this direction by constructing an effective free energy for continuous dynamical quantum phase transitions appearing after quantum quenches in the transverse-field Ising chain. Due to unitarity of quantum time evolution this effective free energy becomes a complex quantity transforming the conventional minimization principle of the free energy into a saddle-point equation in the complex plane of the order parameter, which as in equilibrium is the magnetization. We study this effective free energy in the vicinity of the dynamical quantum phase transition by performing an expansion in terms of the complex magnetization and discuss the connections to the equilibrium case. Furthermore, we study the influence of perturbations and signatures of these dynamical quantum phase transitions in spin correlation functions.

Dynamical transition for S‐K‐T biological competing model with cross‐diffusion

The main objective of this paper is to study the dynamical transition for the S‐K‐T biological competition system with cross‐diffusion. Based on the spectral analysis, the principle of exchange of stabilities conditions for eigenvalues are obtained. By using the dynamical transition theory, 2 different types of dynamical transition for the S‐K‐T model are also derived. In addition, an example is given to illustrate our main results.

Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain

The main objective of this article is to study the dynamic transitions of the FitzHugh-Nagumo equations on a finite domain with the Neumann boundary conditions and with uniformly injected current. We show that when certain parameter conditions are satisfied, the system undergoes a continuous dynamic transition to a limit cycle. A mixed type transition is also found when other conditions are imposed on the parameters. The main method used here is Ma & Wang's dynamic transition theory, which can be used generally on different set-ups for the FitzHugh-Nagumo equations.

Thermodynamical potentials of classical and quantum systems

The aim of the paper is to systematically introduce thermodynamic potentials for thermodynamic systems and Hamiltonian energies for quantum systems of condensates. The study is based on the rich previous work done by pioneers in the related fields. The main ingredients of the study consist of 1) SO(3) symmetry of thermodynamical potentials, 2) theory of fundamental interaction of particles, 3) the statistical theory of heat developed recently [ 23 ], 4) quantum rules for condensates that we postulate in Quantum Rule 4.1, and 5) the dynamical transition theory developed by Ma and Wang [ 20 ]. The statistical and quantum systems we study in this paper include conventional thermodynamic systems, thermodynamic systems of condensates, as well as quantum condensate systems. The potentials and Hamiltonian energies that we derive are based on first principles, and no mean-field theoretic expansions are used.