Related papers: A normal form for excitable media
Standing slow-mode waves have been recently observed in flaring loops by the Atmospheric Imaging Assembly (AIA) of the Solar Dynamics Observatory (SDO). By means of the coronal seismology technique transport coefficients in hot ($\sim$10…
In this paper, we report the bifurcations of mode-locked periodic orbits occurring in maps of three or higher dimensions. The `torus' is represented by a closed loop in discrete time, which contains stable and unstable cycles of the same…
We obtain a parametric normal form for any non-degenerate perturbation of the generalized saddle-node case of Bogdanov--Takens singularity. Explicit formulas are derived and greatly simplified for an efficient implementation in any computer…
We investigate the origin of various convective patterns using bifurcation diagrams that are constructed using direct numerical simulations. We perform two-dimensional pseudospectral simulations for a Prandtl number 6.8 fluid that is…
It is shown, that at weakly nonlinear interaction of waves are possible as modes with chaotic dynamics, and with increasing degree of coherence. Conditions are found at which they arise. One of the types of such interaction is decays. The…
We present a non-perturbative technique to study pulse dynamics in excitable media. The method is used to study propagation failure in one-dimensional and two-dimensional excitable media. In one-dimensional media we describe the behaviour…
We consider a delay differential equation that occurs in the study of chronic myelogenous leukemia. After shortly reminding some previous results concerning the stability of equilibrium solutions, we concentrate on the study of stability of…
The phase-space structure of two families of galactic potentials is approximated with a resonant detuned normal form. The normal form series is obtained by a Lie transform of the series expansion around the minimum of the original…
It has been reported that traveling waves propagate periodically and stably in sub-excitable systems driven by noise [Phys. Rev. Lett. \textbf{88}, 138301 (2002)]. As a further investigation, here we observe different types of traveling…
A recent work by the authors on the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in delay differential equations motivates the derivation of periodic normal forms. In this paper, we prove the…
We study networks of theta neurons arranged on a ring with delayed interactions. In the continuum limit the systems are described by next generation neural field models with delays. We consider distributed delays with both finite and…
We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type $$ \partial^2_t u(t,x)- a(x,\lambda)^2\partial_x^2u(t,x)= b(x,\lambda,u(t,x),u(t-\tau,x),\partial_tu(t,x),\partial_xu(t,x)), \; x…
We investigate the formation of steady states in one-dimensional Bose-Einstein condensates of repulsively interacting ultracold atoms loaded into a quasiperiodic potential created by two incommensurate periodic lattices. We study the…
In this paper, we study the realizability problem for retarded functional differential equations near an equilibrium point undergoing a nonlinear mode interaction between a saddle-node bifurcation and a non-resonant multiple Hopf…
Nonlinear normal modes are periodic orbits that survive in nonlinear many-body Hamiltonian systems, and their instability is crucial for relaxation dynamics. Here, we study the instability process of the $\pi/3$-mode in the…
We consider a one-dimensional Swift-Hohenberg equation coupled to a conservation law, where both equations contain additional dispersive terms breaking the reflection symmetry $x \mapsto -x$. This system exhibits a Turing instability and we…
We analyze the nonlinear dynamics near the incoherent state in a mean-field model of coupled oscillators. The population is described by a Fokker-Planck equation for the distribution of phases, and we apply center-manifold reduction to…
Travelling-wave solutions are shown to bifurcate from relative periodic orbits in plane Poiseuille flow at Re = 2000 in a saddle-node infinite period bifurcation. These solutions consist in self-sustaining sinuous quasi-streamwise streaks…
We present experimental results on hydrothermal traveling-waves dynamics in long and narrow 1D channels. The onset of primary traveling-wave patterns is briefly presented for different fluid heights and for annular or bounded channels,…
Chaotic attractors commonly contain periodic solutions with unstable manifolds of different dimensions. This allows for a zoo of dynamical phenomena not possible for hyperbolic attractors. The purpose of this Letter is to demonstrate these…