Nonvariational Systems Research Articles

A new dynamical approach of Emden-Fowler equations and systems

We give a new approach to general Emden-Fowler equations and systems of the form \begin{equation*} (E_{\varepsilon })-\Delta _{p}u=-{\rm div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u)=\varepsilon \left\vert x\right\vert ^{a}u^{Q}, \end{equation*} \begin{equation*} (G)\left\{ \begin{array}{c} -\Delta _{p}u=-{\rm div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u)=\varepsilon _{1}\left\vert x\right\vert ^{a}u^{s}v^{\delta }, \\ -\Delta _{q}v=-{\rm div}(\left\vert \nabla v\right\vert ^{q-2}\nabla u)=\varepsilon _{2}\left\vert x\right\vert ^{b}u^{\mu }v^{m},% \end{array}% \right. \end{equation*}% where $p,q,Q,\delta, \mu, s,m,$ $a,b$ are real parameters, $p,q\neq 1,$ and $% \varepsilon, \varepsilon _{1},\varepsilon _{2}=\pm 1.$ In the radial case we reduce the problem $(G)$ to a quadratic system of four coupled first-order autonomous equations of Kolmogorov type. In the scalar case the two equations $(E_{\varepsilon })$ with source ($\varepsilon =1)$ or absorption (% $\varepsilon =-1)$ are reduced to a unique system of order 2. The reduction of system $(G)$ allows us to obtain new local and global existence or nonexistence results. We consider in particular the case $\varepsilon _{1}=\varepsilon _{2}=1.$ We describe the behaviour of the ground states when the system is variational. We give a result of existence of ground states for a nonvariational system with $p=q=2$ and $s=m>0,$ that improves the former ones. It is obtained by introducing a new type of energy function. In the nonradial case we solve a conjecture of nonexistence of ground states for the system with $p=q=2$, $\delta =m+1$ and $\mu =s+1.$

Synchronization in Coupled Phase Oscillators

We make a short review about the synchronization in coupled phase oscillator models. Next, we study the common-noise-induced synchronization among active rotators. At an intermediate noise strength, the noise-induced synchronization takes place most effectively, which is analogous to the stochastic resonance. Finally, we study the synchronization of coupled phase oscillators with nonvariational interaction on scale-free networks. We find a sharp transition and a weak hysteresis in the nonvariational systems. The sharp transition is found also in the mean-field approximation.

Existence of positive solutions for nonlocal and nonvariational elliptic systems

In the paper we prove a result on the existence of positive solutions for a class of nonvariational elliptic system with nonlocal source by Galerkin methods and a fixed point theorem in finite dimensions. We establish another existence result by the super and subsolution method and a monotone iteration.

The Slow Dynamics of Two-Spike Solutions for the Gray--Scott and Gierer--Meinhardt Systems: Competition and Oscillatory Instabilities

The dynamics and instability mechanisms of both one- and two-spike solutions to the Gierer--Meinhardt (GM) and Gray--Scott (GS) models are analyzed on a bounded one-dimensional spatial domain. For each of these nonvariational two-component systems, the semistrong spike-interaction limit where the ratio $O(\varepsilon^{-2})$ of the two diffusion coefficients is asymptotically large is studied. In this limit, differential equations for the spike locations, with speed $O(\varepsilon^2) \ll 1$, are derived. To determine the stability of the spike patterns, nonlocal eigenvalue problems, which depend on the instantaneous spike locations, are derived and analyzed. For these nonlocal eigenvalue problems, it is shown that eigenvalues can enter into the unstable right half-plane either along the real axis or through a Hopf bifurcation, leading to either a competition instability or an oscillatory instability of the spike pattern, respectively. Competition instabilities occur only for two-spike patterns and numerically are shown to lead to the annihilation of a spike. Oscillatory instabilities occur for both one- and two-spike solutions. For two-spike solutions this instability typically synchronizes the oscillations of the spike amplitudes. Since the nonlocal eigenvalue problems depend on the instantaneous spike locations, the key feature of these instabilities is that they can be dynamic in nature and can be triggered at some point during the slow evolution of a spike pattern that is initially stable at t=0. The asymptotic theory is compared with full numerical simulations and previous theoretical results.

Thermodynamic potentials for non-equilibrium systems

The notion of non-equilibrium potential for systems far from equilibrium is reviewed and the relation to the reversed process is examined. The potential is constructed in the neighborhood of the homogeneous attractors for a non-variational extended system, namely the subcritical complex Ginzburg–Landau equation. This construction is the second known example of a Lyapunov functional for a non-variational system.

Decay of Metastable Rest State in Excitable Reaction-Diffusion System

The time-evolution of an excitable reaction-diffusion system of the Bonhoffer-van der Pol type equation is investigated by adding an external noise term. In some parameter regime where the system is monostable. the excited domains can spontaneously nucleate and grow at the expense of the uniform metastable rest state. We formulate a statistical theory of nucleation for this non-variational system where no Lyapunov functional exists. The nucleation rate is calculated approximately as a function of critical nucleation radius.