Enrique Tirapegui

A simple global characterization for normal forms of singular vector fields

We derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems. Using an appropriate scalar product in the space of homogeneous vector polynomials, we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator. This leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form. Explicit examples, and results obtained when there are additional symmetries, are also presented.

Faraday's instability in viscous fluid

We find a quantitative approximation which explains the appearance and amplification of surface waves in a highly viscous fluid when it is subjected to vertical accelerations (Faradays instability). Although stationary surface waves with frequency equal to half of the frequency of the excitation are observed in fluids of different kinematical viscosities we show here that the mechanism which produces the instability is very different for a highly viscous fluid as compared with a weakly viscous fluid. This is achieved by deriving an exact equation for the linear evolution of the surface which is non-local in time. We show that for a highly viscous fluid this equation becomes local and of second order and is then a Mathieu equation which is different from the one found for weak viscosity. Analysing the new equation we find an intimate relation with the Rayleigh–Taylor instability.

Normal form of a Hopf bifurcation with noise

Abstract It is proved that the stochastic normal form of a general system undergoing a Hopf bifurcation contains through stochastic resonance new types of terms which did not appear in the deterministic normal form.

Normal form reduction for time-periodically driven differential equations

Abstract We present a global characterisation for the normal form of a time-periodically driven differential equation describing the behaviour of a physical system in the neighborrhood of a multiple instability.

Additive Noise Induces Front Propagation

The propagation of a front connecting a stable homogeneous state with a stable periodic state in the presence of additive noise is studied. The mean velocity was computed both numerically and analitically. The numerics are in good agreement with the analitical prediction.

Analytical description of a state dominated by spiral defects in two-dimensional extended systems

Abstract We study the state of a diluted gas of spiral defects observed in a recent simulation of a two-dimensional extended system governed by the Ginzburg-Landau equation with complex coefficients. We show that this state, which is responsible for the disorganization of the system, can be described by an approximate solution in which the new variables are the positions of the spirals and a new global phase. The dynamics of these new variables and the conditions for the existence of the state are determined.

Finite coarse-graining and Chapman-Kolmogorov equation in conservative dynamical systems

Abstract The conditions for mapping the deterministic evolution of unstable conservative dynamical systems into a stochastic process through finite coarse-graining are derived. A Chapman-Kolmogorov equation giving rise to an H-theorem is deduced and its properties under time reversal and cell lumping of the coarse-graining partition are analyzed. The validity of the stochastic description is verified on the simple example of the automorphisms of the two-dimensional torus.

STATIONARY LOCALIZED SOLUTIONS IN THE SUBCRITICAL COMPLEX GINZBURG–LANDAU EQUATION

It is shown that pulses in the complete quintic one-dimensional Ginzburg–Landau equation with complex coefficients appear through a saddle-node bifurcation which is determined analytically through a suitable approximation of the explicit form of the pulses. The results are in excellent agreement with direct numerical simulations.

The Maxwell-Bloch description of 1/1 resonances

We discuss the generic 1/1 resonance of a reversible system weakly perturbed by dissipative terms. We show that the Maxwell-Bloch equations are the asymptotic normal form of the system when the energy is injected by coupling with a zero frequency mode.

THE STATIONARY INSTABILITY IN QUASI-REVERSIBLE SYSTEMS AND THE LORENZ PENDULUM

We study the resonance at zero frequency in presence of a neutral mode in quasi-reversible systems. The asymptotic normal form is derived and it is shown that in the presence of a reflection symmetry it is equivalent to the set of real Lorenz equations. Near the critical point an analytical condition for the persistence of an homoclinic curve is calculated and chaotic behavior is then predicted and its existence verified by direct numerical simulation. A simple mechanical pendulum is shown to be an example of the instability, and preliminary experimental results agree with the theoretical predictions.