We extend the invariant manifold method for analyzing the asymptotics of dissipative partial differential equations on unbounded spatial domains to treat equations in which the linear part has order greater than two. One important example of this type of equation which we analyze in some detail is the Cahn-Hilliard equation. We analyze the marginally stable solutions of this equation in some de...

Many interesting models in science and engineering involve forced or coupled oscillators. The most striking feature of such systems is the transition between phase locking and quasi-periodicity. Phase locking produces a periodic solution that generically persists under variation of parameters. In contrast, quasi-periodicity is a codimension-one phenomenon, which is thus generically destroyed by...

We study the dynamics of the Forced Logistic Map in the cylinder. We compute a bifurcation diagram in terms of the dynamics of the attracting set. Different properties of the attracting set are considered, as the Lyapunov exponent and, in the case of having a periodic invariant curve, its period and its reducibility. This reveals that the parameter values for which the invariant curve doubles i...

Recent experiments [1] on two–frequency parametrically excited surface waves exhibit an intriguing “superlattice” wave pattern near a codimension–two bifurcation point where both subharmonic and harmonic waves onset simultaneously, but with different spatial wavenumbers. The superlattice pattern is synchronous with the forcing, spatially periodic on a large hexagonal lattice, and exhibits small...

We study two identical hyperchaotic oscillators symmetrically coupled. Each oscillator represents a codimension-2 Takens–Bogdanov bifurcation under square symmetry, and was used to model a convection experiment in a time-dependent state. In the coupled system, the Lyapunov exponents behavior against the coupling parameter is used to detect changes in the dynamics, and the synchronization state ...

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 obtain the amplitude equations for steady-state and Hopf bifurcation from the equilibrium state with a uniform phase distribution. When the population is describ...

When a vector field in R is discontinuous on a smooth codimension one surface, it may be simultaneously tangent to both sides of the surface at generic isolated points (singularities). For a piecewise-smooth dynamical system governed by the vector field, we show that the local dynamics depends on a single quantity—the jump in direction of the vector field through the singularity. This quantity ...

In this paper, a bifurcation analysis is undertaken for a discrete-time Hopfield neural network of two neurons with two different delays and self-connections. Conditions ensuring the asymptotic stability of the null solution are found, with respect to two characteristic parameters of the system. It is shown that for certain values of these parameters, fold or Neimark-Sacker bifurcations occur, ...

We consider the codimension-three phenomenon of homoclinic bifurcations of flows containing a pair of orbits homoclinic to a saddle point whose principal eigenvalues are in resonance. We concentrate upon the simplest possible configuration, the so-called “figure-of-eight,” and reduce the dynamics near the homoclinic connections to those on a two-dimensional locally invariant centre manifold. Th...

We observe traveling waves emitted from Turing spots in the chlorine dioxide-iodine-malonic acid reaction. The newborn waves are continuous, but they break into segments as they propagate, and the propagation of these segments ultimately gives rise to spatiotemporal chaos. We model the wave-breaking process and the motion of the chaotic segments. We find stable segmented spirals as well. We att...