Related papers: Exactly solvable Stuart-Landau models in arbitrary…
We consider co-rotational wave maps from the $(1+d)$-dimensional Minkowski space into the $d$-sphere for $d\geq 3$ odd. This is an energy-supercritical model which is known to exhibit finite-time blowup via self-similar solutions. Based on…
We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an…
Extending work of Texier and Zumbrun in the semilinear non-re ection symmetric case, we study O(2) transverse Hopf bifurcation, or \cellular instability," of viscous shock waves in a channel, for a class of quasilinear hyperbolic{parabolic…
The logistic two-gene negative-feedback oscillator is locally asymptotically stable for all biological parameter values, since the trace of the Jacobian is uniformly negative. Real biological oscillators (circadian rhythms, the segmentation…
In this paper, we analyse the dynamics of a pattern-forming system close to simultaneous Turing and Turing--Hopf instabilities, which have a 1:1 spatial resonance, that is, they have the same critical wave number. For this, we consider a…
Motivated by a certain type of unfolding of a Hopf-Hopf singularity, we consider a one-parameter family $(f_\gamma)_{\gamma\geq0}$ of $C^3$--vector fields in $\mathbb{R}^4$ whose flows exhibit a heteroclinic cycle associated to two periodic…
The real Ginzburg-Landau equation arises as a universal amplitude equation for the description of pattern-forming systems exhibiting a Turing bifurcation. It possesses spatially periodic roll solutions which are known to be stable against…
In this paper, we introduce a three-component Schnakenberg model. Its key feature is that it has a solution consisting of N spikes that undergoes a Hopf bifurcation with respect to N distinct modes nearly simultaneously. This results in…
We study a two-dimensional model of topological superconductor with equal spin pairing and repulsive Hubbard interaction. When the pairing gap equals to the hopping constant, half of the spectrum of this model are flat bands, which makes…
We consider co-rotational wave maps from (1+3)-dimensional Minkowski space into the three-sphere. This model exhibits an explicit blowup solution and we prove the asymptotic nonlinear stability of this solution in the whole space under…
We propose a new simple three-dimensional continuous autonomous model with two nonlinear terms and observe the dynamical behavior with respect to system parameters. This system changes the stability of fixed point via Hopf bifurcation and…
Oscillatory instabilities in Hamiltonian anharmonic lattices are known to appear through Hamiltonian Hopf bifurcations of certain time-periodic solutions of multibreather type. Here, we analyze the basic mechanisms for this scenario by…
We introduce an extended Kepler-Coulomb quantum model in spherical coordinates. The Schr\"{o}dinger equation of this Hamiltonian is solved in these coordinates and it is shown that the wave functions of the system can be expressed in terms…
In this paper we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate the periodic solutions of the complex cubic-quintic Ginzburg-Landau equation. The primary tools used here are Hopf…
We implement the geometric method proposed in ([9], [3], [16]) to analytically predict the sequence of bifurcations leading to a change of stability and/or the appearance of new periodic orbits in the secular 3D planetary three body…
For many physical systems the transition from a stationary solution to sustained small amplitude oscillations corresponds to a Hopf bifurcation. For systems involving impacts, thresholds, switches, or other abrupt events, however, this…
In this paper we use formal asymptotic arguments to understand the stability proper- ties of equivariant solutions to the Landau-Lifshitz-Gilbert model for ferromagnets. We also analyze both the harmonic map heatflow and Schrodinger map…
We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but…
We study actions of Lie supergroups, in particular, the hitherto elusive notion of orbits through odd (or more general) points. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general…
Coupled oscillator models where $N$ oscillators are identical and symmetrically coupled to all others with full permutation symmetry $S_N$ are found in a variety of applications. Much, but not all, work on phase descriptions of such systems…