Splay states and two-cluster states in ensembles of excitable units

Focusing on systems of sinusoidally coupled active rotators, we study the emergence and stability of periodic collective oscillations for systems of identical excitable units with repulsive all-to-all interaction. Special attention is put on splay states and two-cluster states. Recently, it has been shown that one-parameter families of such systems, containing the parameter values at which the Watanabe–Strogatz integrability takes place, feature an instantaneous non-local exchange of stability between splay and two-cluster states. Here, we illustrate how in the extended families that circumvent the Watanabe–Strogatz dynamics, this abrupt transition is replaced by the “gradual transfer” of stability between the 2-cluster and the splay states, mediated by mixed-type solutions. We conclude our work by recovering the same kind of dynamics and transfer of stability in an ensemble of voltage-coupled Morris–Lecar neurons.

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>

Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems

The problem of linear instability of a nonlinear travelling wave in a canonical Hamiltonian system with translational symmetry subject to superharmonic perturbations is discussed. It is shown that exchange of stability occurs when energy is stationary as a function of wave speed. This generalizes a result proved by Saffman (J. Fluid Mech., vol. 159, 1985, pp. 169–174) for travelling wave solutions exhibiting a wave profile with reflectional symmetry. The present argument remains true for any non-canonical Hamiltonian system that can be cast in Darboux form, i.e. a canonical Hamiltonian form on a submanifold defined by constraints, such as a two-dimensional surface wave on a constant shearing flow, revealing a general feature of Hamiltonian dynamics.

Linearized stability implies dynamic stability for equilibria of 1-dimensional, p-Laplacian boundary value problems

We consider the parabolic, initial-boundary value problem 1$$\matrix{ {\displaystyle{{\partial v} \over {\partial t}} = \Delta _p(v) + f(x,v),} & {{\rm in}({\rm - 1},{\rm 1}) \times ({\rm 0},\infty ),} \cr {v( \pm 1,t) = 0,} \hfill \hfill \hfill & {{\rm t}\in [{\rm 0},\infty ),} \hfill \hfill \cr {v = v_0\in C_0^0 ([-1,1]),} & {{\rm in}[{\rm - 1},{\rm 1}] \times \{ {\rm 0}\} ,} \cr } $$ where Δp denotes the p-Laplacian on ( − 1, 1), with p > 1, and the function f:[ − 1, 1] × ℝ → ℝ is continuous, and the partial derivative fv exists and is continuous and bounded on [ − 1, 1] × ℝ. It will be shown that (under certain additional hypotheses) the ‘principle of linearized stability’ holds for equilibrium solutions u0 of (1). That is, the asymptotic stability, or instability, of u0 is determined by the sign of the principal eigenvalue of a suitable linearization of the problem (1) at u0. It is well-known that this principle holds for the semilinear case p = 2 (Δ2 is the linear Laplacian), but has not been shown to hold when p ≠ 2.We also consider a bifurcation type problem similar to (1), having a line of trivial solutions. We characterize the stability or instability of the trivial solutions, and the bifurcating, non-trivial solutions, and show that there is an ‘exchange of stability’ at the bifurcation point, analogous to the well-known result when p = 2.

Stability and selection of lamellar eutectic growth with anisotropic surface tension

The morphological stability of lamellar eutectic growth with the anisotropic effect of surface tension is studied by means of the interfacial wave (IFW) theory developed by Xu in the 1990s. We solve the related linear eigenvalue problem for the case that the Peclet number is small and the segregation coefficient parameter is close to the unit. The stability criterion of lamellar eutectic growth with the anisotropic surface tension is obtained. The linear stability analysis reveals that the stability of lamellar eutectic growth depends on a stability critical number [Formula: see text]. Similar to the case of isotropic surface tension, the system involves two types of global instability mechanisms: the “exchange of stability” invoked by the non-oscillatory, unstable modes and the “global wave instabilities” invoked by four types of oscillatory unstable modes, namely antisymmetric–antisymmetric (AA-), symmetric–symmetric (SS-), antisymmetric–symmetric (AS-) and symmetric–antisymmetric (SA-) modes. The anisotropic surface tension, by decreasing the corresponding stability critical number [Formula: see text], stabilizes the “exchange of stability” mechanism and “global wave instability” mechanism invoked by AA-, SA- and SS-modes. However, by increasing the corresponding stability critical number [Formula: see text], the anisotropic surface tension destabilizes the “global wave instability” mechanism invoked by AS-mode.

Compressible Analysis of Bénard Convection of Magneto Rotatory Couple-Stress Fluid

Thermal Instability (Benard’s Convection) in the presence of uniform rotation and uniform magnetic field (separately) is studied. Using the linearized stability theory and normal mode analyses the dispersion relation is obtained in each case. In the case of rotatory Benard’s stationary convection compressibility and rotation postpone the onset of convection whereas the couple-stress have duel character onset of convection depending on rotation parameter. While in the absence of rotation couple-stress always postpones the onset of convection. On the other hand, magnetic field on thermal instability problem on couple-stress fluid for stationary convection couple-stress parameter and magnetic field postpones the onset of convection. The effect of compressibility also postpones the onset of convection in both cases as rotation and magnetic field. Graphs have been plotted by giving numerical values to the parameters to depict the stationary characteristics. Further, the magnetic field and rotation are found to introduce oscillatory modes which were non-existent in their absence and then the principle of exchange of stability is valid. The sufficient conditions for non-existence of overstability are also obtained.

Thermomagnetic Convection in Porous Media: Effect of Anisotropy and Local Thermal Nonequilibrium

The onset of thermomagnetic convection in an anisotropic layer of Darcy porous medium in the presence of a uniform vertical magnetic field is investigated using a local thermal nonequilibrium (LTNE) model for energy equation representing the solid and fluid phases separately. Anisotropies in permeability as well as in fluid and solid thermal conductivities are considered. The principle of exchange of stability is shown to be valid. Asymptotic solutions for the Rayleigh number for both small and large values of scaled interphase heat transfer coefficient are presented and the comparison of results with those computed numerically shows good agreement. The mechanical and thermal anisotropy parameters have opposing influence on the stability characteristics of the system. Besides, the influence of magnetic parameters on the instability of the system is also reported.

Effect of Rotation and Magnetic Field on Thermal Stability of Ferromagnetic Fluid

In this paper, the effect of rotation and magnetic field on thermal stability of a layer of ferromagnetic fluid heated from below has been investigated. For a fluid layer between two free boundaries, an exact solution is obtained using a linearized stability theory and normal mode analysis. For the case of stationary convection, it is found that magnetic field and rotation have stabilizing effect on the thermal stability of the system. The principle of exchange of stability is not valid for the problem under consideration, whereas in the absence of rotation and magnetic field, it is valid.