Brusselator Equations Research Articles

GENERALIZED TEMPLATE FOR PERIOD DOUBLINGS

It is shown how one can derive the formula for the local crossing number with the help of the symbolic dynamics with two letters knowing that, on the extended (ξ, η)-template, any period-doubling sequences in continuous dynamical systems is characterized by the periodic block [Formula: see text] The formulas of the global crossing number and of the linking number are derived in terms of the local crossing number which can be extracted from the power spectrum of periodic orbits. The topological characterization of periodic orbits in continuous dynamical systems has come to be at hand for experimenters as well as for theorists. The formulas are applied to the situations of the numerical experiments for particular systems, and the results are investigated in connection with the concept of the irreducible and reducible templates. It is also shown how the formulas can be applicable to the forced Lorenz system, the Brusselator equation, the parametric pendulum, T(2, 5) resonant torus knot and P(7, 3, −2) pretzel knot.

A Numerical Liapunov-Schmidt Method with Applications to Hopf Bifurcation on a Square

We discuss an iterative method for calculating the reduced bifurcation equation of the Liapunov-Schmidt method and its numerical approximation. Using appropriate genericity assumptions (with symmetry), we derive a Taylor series for the reduced equation, where the bifurcation behavior is determined by its numerical approximation at a finite order of truncation. This method is used to calculate reduced equations at Hopf bifurcation of the two-dimensional Brusselator equations on a square with Neumann and Dirichlet boundary conditions. We examine several Hopf bifurcations within the three-parameter space. There are regions where we observe direct bifurcation to branches of periodic solutions with submaximal symmetry.

A numerical Liapunov-Schmidt method with applications to Hopf bifurcation on a square

We discuss an iterative method for calculating the reduced bifurcation equation of the Liapunov-Schmidt method and its numerical approximation. Using appropriate genericity assumptions (with symmetry), we derive a Taylor series for the reduced equation, where the bifurcation behavior is determined by its numerical approximation at a finite order of truncation. This method is used to calculate reduced equations at Hopf bifurcation of the two-dimensional Brusselator equations on a square with Neumann and Dirichlet boundary conditions. We examine several Hopf bifurcations within the three-parameter space. There are regions where we observe direct bifurcation to branches of periodic solutions with submaximal symmetry.

A Hopf bifurcation with Robin boundary conditions

We investigate Hopf bifurcation of an example reaction-diffusion system on a square domain with Robin boundary conditions; the Brusselator equations. By performing a smooth homotopy of boundary conditions from Neumann to Dirichlet type, we observe the creation of branches of periodic solutions with submaximal symmetry in codimension two bifurcations, although we do not fully calculate the branching behaviour. We also note that mode interactions behave generically on varying the boundary conditions. The investigation is performed using a numerical Liapunov-Schmidt reduction technique of Ashwin, Bohmer, and Mei (1994) and an analysis of Swift (1988).

Asymptotic and numerical study of Brusselator chaos

We investigate the Brusselator reaction–diffusion equations with periodic boundary conditions. We consider the range of values of the parameters used by Kuramoto in his study of chaotic concentration waves. We determine numerically the bifurcation diagram of the long-time travelling and standing wave solutions using a highly accurate Fourier pseudo-spectral method. For moderate values of the bifurcation parameter, we have found a sequence of instabilities leading either to periodic and quasiperiodic standing waves, or to chaotic regimes. However, for large values of the control parameter, we have found only uniform time-periodic solutions or time-periodic travelling wave solutions. Our numerical study motivates a new asymptotic analysis of the Brusselator equations for large values of the control parameter and small diffusion coefficients. This analysis explains the numerical predictions. The chaotic regime is limited to moderate values of the control parameter and periodic solutions are the only solutions for large values of the control parameter. We identify the stabilizing mechanism as the relaxation oscillations which appear when the control parameter is large. Our asymptotic result on the stability of periodic solutions is then generalized to a class of two-variable reaction-diffusion equations.

Global dynamics of the Brusselator equations

In this work the existence of a global attractor for the solution semio w of the Brusselator equations is proved. A new decomposition method is introduced to overcome the diculties in proving the asymptotical compact- ness of the coupled reaction-diusion equations whose nonlinearity does not possess dissipative property. It is proved that the Hausdor dimension and the fractal dimension of the global attractor are nite. The existence of a global attractor with nite dimensionality is also shown by the same approach for the Gray-Scott equations, the Glycolysis equations, and the Schnackenberg equations.