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This paper deals with the classification of transition phenomena in the most basic dissipative system possible, namely, the 1D reaction-diffusion equation. The emphasis is on the relation between the linear and nonlinear terms and the effect of the boundaries which influence the first transitions. We consider the cases where the linear part is self-adjoint with second-order and fourth-order derivatives which is the case which most often arises in applications. We assume that the nonlinear term depends on the unknown function and its first derivative which is basically the semilinear case for the second-order reaction-diffusion system. As for the boundary conditions, we consider the typical Dirichlet, Neumann, and periodic boundary settings. In all the cases, the equations admit a trivial steady state which loses stability at a critical parameter. We aim to classify all possible transitions and bifurcations that take place. Our analysis shows that these systems display all three types of transitions: continuous, jump and mixed. Moreover they exhibit transcritical, supercritical bifurcations with bifurcated states such as finitely many equilibria, circle of equilibria, and slowly rotating limit cycle. Many applications found in the literature are basically corollaries of our main results. We apply our results to classify the first transitions of the Chaffee-Infante equation, the Fisher-KPP equation, the Kuramoto-Sivashinsky equation, and the Swift-Hohenberg equation.

The structural bifurcation of a 2D divergence free vector field \begin{document}$ \mathbf{u}(\cdot, t) $\end{document} when \begin{document}$ \mathbf{u}(\cdot, t_0) $\end{document} has an interior isolated singular point \begin{document}$ \mathbf{x}_0 $\end{document} of zero index has been studied by Ma and Wang [ 23 ]. Although in the class of divergence free fields which undergo a local bifurcation around a singular point, the ones with index zero singular points are generic, this class excludes some important families of symmetric flows. In particular, when \begin{document}$ \mathbf{u}(\cdot, t_0) $\end{document} is anti-symmetric with respect to \begin{document}$ \mathbf{x}_0 $\end{document} , or symmetric with respect to the axis located on \begin{document}$ \mathbf{x}_0 $\end{document} and normal to the unique eigendirection of the Jacobian \begin{document}$ D\mathbf{u}(\cdot, t_0) $\end{document} , the vector field must have index 1 or -1 at the singular point. Thus we study the structural bifurcation when \begin{document}$ \mathbf{u}(\cdot, t_0) $\end{document} has an interior isolated singular point \begin{document}$ \mathbf{x}_0 $\end{document} with index -1, 1. In particular, we show that if such a vector field with its acceleration at \begin{document}$ t_0 $\end{document} both satisfy the aforementioned symmetries then generically the flow will undergo a local bifurcation. Under these generic conditions, we rigorously prove the existence of flow patterns such as pairs of co-rotating vortices and double saddle connections. We also present numerical evidence of the Stokes flow in a rectangular cavity showing that the bifurcation scenarios we present are indeed realizable.

In this article, we study the dynamics of large-scale motion in atmosphere and ocean governed by the 3D quasi-geostrophic potential vorticity (QGPV) equation with a constant stratification. It is shown that for a Kolmogorov forcing on the first energy shell, there exist a family of exact solutions that are dissipative Rossby waves. The nonlinear stability of these exact solutions are analyzed based on the assumptions on the growth rate of the forcing. In the absence of forcing, we show the existence of selective decay states for the 3D QGPV equation. The selective decay states are the 3D Rossby waves traveling horizontally at a constant speed. All these results can be regarded as the expansion of that of the 2D QGPV system and in the case of 3D QGPV system with isotropic viscosity. Finally, we present a geometric foundation for the model as a general equation for nonequilibrium reversible-irreversible coupling.

This study examines the instability and dynamical transitions of the two-layer western boundary currents represented by the Munk profile in the upper layer and a motionless bottom layer in a closed rectangular domain. First, a bound on the intensity of the Munk profile below which the western boundary currents are locally nonlinearly stable is provided. Second, by reducing the infinite dimensional system to a finite dimensional one via the center manifold reduction, non-dimensional transition numbers are derived, which determine the types of dynamical transitions both from a pair of simple complex eigenvalues as well as from a double pair of complex conjugate eigenvalues as the Reynolds number crosses a critical threshold. We show by careful numerical estimations of the transition numbers that the transitions in both cases are continuous at the critical Reynolds number. After the transition from a pair of simple complex eigenvalue, the western boundary layer currents turn into a periodic circulation, whereas a quasi-periodic or possibly a chaotic circulation emerges after the transition from a pair of double complex eigenvalues. Finally, a comparison between the transitions exhibited in one-layer and two-layer models is provided, which demonstrates the fundamental differences between the two models.

The main objective of this article is to study the nonlinear stability and dynamic transitions of the basic (zonal) shear flows for the three-dimensional continuously stratified rotating Boussinesq model. The model equations are fundamental equations in geophysical fluid dynamics, and dynamics associated with their basic zonal shear flows play a crucial role in understanding many important geophysical fluid dynamical processes, such as the meridional overturning oceanic circulation and the geophysical baroclinic instability. In this paper, first we derive a threshold for the energy stability of the basic shear flow, and obtain a criteria for nonlinear stability in terms of the critical horizontal wavenumbers and the system parameters such as the Froude number, the Rossby number, the Prandtl number and the strength of the shear flow. Next we demonstrate that the system always undergoes a dynamic transition from the basic shear flow to either a spatiotemporal oscillatory pattern or circle of steady states, as the shear strength $\Lambda$ of the basic flow crosses a critical threshold $\Lambda_c$. Also we show that the dynamic transition can be either continuous or catastrophic, and is dictated by the sign of a transition parameter $A$, fully characterizing the nonlinear interactions of different modes. A systematic numerical method is carried out to explore transition in different flow parameter regimes. We find that the system admits only critical eigenmodes with horizontal wave indices $(0,m_y)$. Such modes, horizontally have the pattern consisting of $m_y$-rolls aligned with the x-axis. Furthermore, numerically we encountered continuous transitions to multiple steady states, continuous and catastrophic transitions to spatiotemporal oscillations., Comment: 20 pages, 7 figures

In this paper, the effect of e-cigarettes on smoking cessation is studied using the tools of dynamical systems theory. The purpose here is to examine this efficacy by representing and analyzing a non-linear ODE system modeling potential smokers, tobacco smokers, e-cigarette smokers and quitters. The transition from smoking class to e-cigarette smoking class is represented by the “peer pressure”. The model exhibits three possible equilibrium solutions which are the smoking-free equilibrium, e-cigarette smoking-free equilibrium and endemic equilibrium. It is shown that the smoking free equilibrium always exists. Moreover, this equilibrium is stable if the basic reproduction number R 0 is less than unity. When R 0 > 1 , it is shown that the smoking free equilibrium is unstable, and of the other two equilibria is stable. The global stability analysis of the smoking-free equilibrium is also given. Some numerical simulations are plotted using the data obtained from the literature. The theoretical results are also confirmed by numerical results.

In this article, the spectral instability and the associated bifurcations of the shear flows of the 2D quasi-geostrophic equation with a generalized Kolmogorov forcing are investigated. To determine the linear instability of the basic shear flow, we write the corresponding eigenvalue problem as a system of finite difference equations whose nontrivial solutions are expressed in the form of continued fractions. By a rigorous analysis of these continued fractions, we prove the existence of a number R c such that if the control parameter R , which is proportional to the Reynolds number and the intensity of the curl of forcing, is over R c , then the basic shear flow loses its stability. To shed light on the bifurcation involved in the loss of stability of the basic shear flow, a natural method is used to reduce the quasi-geostrophic equation to a system of ordinary differential equations. Based on numerical experiments on the coefficients of this reduced system, we show that both supercritical and subcritical Hopf bifurcations occur depending on the frequency of the generalized Kolmogorov forcing. Moreover, we investigate the double Hopf bifurcations which occur at critical aspect ratios. Our results show that in the double Hopf bifurcation case, two periodic solutions, one stable and the other unstable, bifurcate on R > R c .

In this article, the viscously-damped instability arising in the shear jet of west boundary layer governed by the two-layer quasi-geostrophic equation with a layered topography is analyzed. First, the nonlinear stability and the exponential stability of the shear jet is studied. More precisely, we derive an upper bound on the Reynolds number Re below which the shear jet is not only locally nonlinearly stable but also globally exponentially stable. Second, it is shown that there exists a critical value of the Reynolds number Re above which the shear jet will become linearly unstable and there exists a dynamic transition in the west boundary layer. To shed light on the type of the dynamic transition, we reduce the two-layer quasi-geostrophic equation to a system of ODEs by making use of the technique of center manifold reduction. Then, we infer from this system of ODEs that the dynamic transition is of continuous type, leading to a stable periodic oscillation of west boundary layer currents. Finally, we investigate the effect of the slope of the bottom topography on the stability and transition of the shear jet. We find that although a large slope stabilizes the shear jet, it has no impact on the transition type.

The main aim of the paper is to investigate the transitions of the thermohaline circulation in a spherical shell in a parameter regime which only allows transitions to multiple equilibria. We find that the first transition is either continuous (Type-I) or drastic (Type-II) depending on the sign of the transition number. The transition number depends on the system parameters and $$l_c$$ , which is the common degree of spherical harmonics of the first critical eigenmodes, and it can be written as a sum of terms describing the nonlinear interactions of various modes with the critical modes. We obtain the exact formulas of this transition number for $$l_c=1$$ and $$l_c=2$$ cases. Numerically, we find that the main contribution to the transition number is due to nonlinear interactions with modes having zero wave number and the contribution from the nonlinear interactions with higher frequency modes is negligible. In our numerical experiments we encountered both types of transition for $$\text {Le}1$$ . In the continuous transition scenario, we rigorously prove that an attractor in the phase space bifurcates which is homeomorphic to the 2 $$l_c$$ dimensional sphere and consists entirely of degenerate steady state solutions.

The main aim of this paper is to study the dynamic transitions in flows described by the two-dimensional, barotropic vorticity equation in a periodic zonal channel. In [Z.-M. Chen et al., SIAM J. Appl. Math., 64 (2003), pp. 343--368], the existence of a Hopf bifurcation in this model as the Reynolds number crosses a critical value was proven. In this paper, we extend these results by addressing the stability problem of the bifurcated periodic solutions. Our main result is the explicit expression of a nondimensional parameter $\gamma$ which controls the transition behavior. We prove that depending on $\gamma$, the modeled flow exhibits either a continuous (Type I) or catastrophic (Type II) transition. Numerical evaluation of $\gamma$ for a physically realistic region of parameter space suggests that a catastrophic transition is preferred in this flow, which may lead to chaotic flow regimes.

The main objective of this article is to study the three-dimensional Rayleigh-Benard convection in a rectangular domain from a pattern formation perspective. It is well known that as the Rayleigh number crosses a critical threshold, the system undergoes a Type-I transition, characterized by an attractor bifurcation. The bifurcated attractor is an (m-1)-dimensional homological sphere where m is the multiplicity of the first critical eigenvalue. When m=1, the structure of this attractor is trivial. When m=2, it is known that the bifurcated attractor consists of steady states and their connecting heteroclinic orbits. The main focus of this article is then on the pattern selection mechanism and stability of rolls, rectangles and mixed modes (including hexagons) for the case where m=2. We derive in particular a complete classification of all transition scenarios, determining the patterns of the bifurcated steady states, their stabilities and the basin of attraction of the stable ones. The theoretical results lead to interesting physical conclusions, which are in agreement with known experimental results. For example, it is shown in this article that only the pure modes are stable whereas the mixed modes are unstable.

The main objective of this article is to study the order-disorder phase transition and pattern formation for systems with long-range repulsive interactions. The main focus is on the Cahn-Hilliard model with a nonlocal term in the corresponding energy functional, representing the long-range repulsive interaction. First, we show that as soon as the linear problem loses stability, the system always undergoes a dynamic transition to one of the three types, forming different patterns/structures. The types of transition are then dictated by a nondimensional parameter, measuring the interactions between the long-range repulsive term and the quadratic and cubic nonlinearities in the model. The derived explicit form of this parameter offers precise information for the phase diagrams. Second, we obtain a novel and explicit pattern selection mechanism associated with the competition between the long-range repulsive interaction and the short-range attractive interactions. In particular, the hexagonal pattern is unique to the long-range interaction, and is associated with a novel two-dimensional reduced transition equations on the center manifold generated by the unstable modes, consisting of (degenerate) quadratic terms and non-degenerate cubic terms. Finally, explicit information on the metastability and basin of attraction of different disordered/ordered states and patterns are derived as well.

In this study we consider the stability and transitions for the Poiseuille flow of a second grade fluid which is a model for non-Newtonian fluids. We restrict our attention to flows in an infinite pipe with circular cross section that are independent of the axial coordinate. We show that unlike the Newtonian ($\epsilon=0$) case, in the second grade model ($\epsilon \neq 0$ case), the time independent base flow exhibits transitions as the Reynolds number $R$ exceeds the critical threshold $R_c \approx 4.124 \epsilon^{-1/4}$ where $\epsilon$ is a material constant measuring the relative strength of second order viscous effects compared to inertial effects. At $R=R_c$, we find that generically the transition is either continuous or catastrophic and a small amplitude, time periodic flow with 3-fold azimuthal symmetry bifurcates. The time period of the bifurcated solution tends to infinity as $R$ tends to $R_c$. Our numerical calculations suggest that for low $\epsilon$ values, the system prefers a catastrophic transition where the bifurcation is subcritical. We also find that there is a Reynolds number $R_E$ with $R_E < R_c$ such that for $R, Comment: 19 pages, 5 figures

We study the Rayleigh–Benard convection in a 2D rectangular domain with no-slip boundary conditions for the velocity. The main mathematical challenge is due to the no-slip boundary conditions, because the separation of variables for the linear eigenvalue problem, which works in the free-slip case, is no longer possible. It is well known that as the Rayleigh number crosses a critical threshold Rc, the system bifurcates to an attractor, which is an (m − 1)-dimensional sphere, where m is the number of eigenvalues, which cross zero as R crosses Rc. The main objective of this article is to derive a full classification of the structure of this bifurcated attractor when m = 2. More precisely, we rigorously prove that when m = 2, the bifurcated attractor is homeomorphic to a one-dimensional circle consisting of exactly four or eight steady states and their connecting heteroclinic orbits. In addition, we show that the mixed modes can be stable steady states for small Prandtl numbers. Copyright © 2014 John Wiley & Sons, Ltd.

The main objective of this paper is to describe the dynamic transition of the incompressible MHD equations in a three dimensional (3D) rectangular domain from a perspective of pattern formation. We aim to classify the formations of roll, rectangle and hexagonal patterns at the first critical Rayleigh number. When the first eigenvalue of the linearized operator is real and simple, the critical eigenvector has either a roll structure or a rectangle structure. In both cases we find that the transition is continuous or jump depending on a non-dimensional number computed explicitly in terms of system parameters. When the critical eigenspace has dimension two corresponding to two real eigenvalues, we study the transitions of hexagonal pattern. In this case, we show that all three types of transitions--continuous, jump and mixed--can occur in eight different transition scenarios. Finally, we study the case where the first eigenvalue is complex, simple and corresponding eigenvector has a roll structure. In this case, we find that both continuous and jump transitions are possible. We give several bounds on the parameters which separate the parameter space into regions of different transition scenarios.

The main objectives of this article are two-fold. First, we study the effect of the nonlinear Onsager mobility on the phase transition and on the well-posedness of the Cahn-Hilliard equation modeling a binary system. It is shown in particular that the dynamic transition is essentially independent of the nonlinearity of the Onsager mobility. However, the nonlinearity of the mobility does cause substantial technical difficulty for the well-posedness and for carrying out the dynamic transition analysis. For this reason, as a second objective, we introduce a systematic approach to deal with phase transition problems modeled by quasilinear partial differential equation, following the ideas of the dynamic transition theory developed recently by Ma and Wang.