Sensitivity analysis of the stochastically and periodically forced Brusselator

The problem of sensitivity of nonlinear system limit cycle with respect to small stochastic and periodic disturbances is considered. Sensitivity analysis on the basis of quasipotential function is performed. The quasipotential is used widely in statistical physics (for instance by Graham for analysis of nonequilibrium thermodynamics problem). We consider an application of quasipotential technique to sensitivity problem. For the plane orbit case an approximation of quasipotential is expressed by some scalar function. This function (sensitivity function) is introduced as a base tool of a quantitative description for a system response on the external disturbances. New cycle characteristics (sensitivity factor, parameter of stiffness) are considered. The analysis of the forced Brusselator based on sensitivity function is shown. From this analysis the critical value of Brusselator parameter is found. The dynamics of forced Brusselator for this critical value is investigated. For small stochastic disturbances the burst of response amplitude is shown. For small periodic disturbances the period doubling regime of the transition to chaos scenario is demonstrated.

[1]  Regularity of boundary quasi-potentials for planar systems , 1994 .

[2]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[3]  H. Poincaré,et al.  Sur les courbes définies par les équations différentielles(III) , 1885 .

[4]  Bernard J. Matkowsky,et al.  A direct approach to the exit problem , 1990 .

[5]  Lev Ryashko,et al.  THE STABILITY OF STOCHASTICALLY PERTURBED ORBITAL MOTIONS , 1996 .

[6]  G. Mil’shtein,et al.  A first approximation of the quasipotential in problems of the stability of systems with random non-degenerate perturbations , 1995 .

[7]  D. Ludwig.  Persistence of Dynamical Systems under Random Perturbations , 1975 .

[8]  Vadim N. Smelyanskiy,et al.  Topological features of large fluctuations to the interior of a limit cycle , 1997 .

[9]  Graham,et al.  Steady-state ensemble for the complex Ginzburg-Landau equation with weak noise. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[10]  Grégoire Nicolis,et al.  Self-Organization in nonequilibrium systems , 1977 .

[11]  A. Denjoy,et al.  Sur les courbes définies par les équations différentielles à la surface du tore , 1932 .

[12]  Earl H. Dowell,et al.  Parametric Random Vibration , 1985 .

[13]  I. Bendixson.  Sur les courbes définies par des équations différentielles , 1901 .

[14]  M. Levi.  Qualitative Analysis of the Periodically Forced Relaxation Oscillations , 1981 .

[15]  H. Saunders,et al.  Random Vibration of Elastic Systems , 1987 .

[16]  J. Flaherty,et al.  Frequency Entrainment of a Forced van der pol Oscillator. , 1977 .

[17]  Graham,et al.  Nonequilibrium potential for coexisting attractors. , 1986, Physical review. A, General physics.

[18]  R. L. Stratonovich,et al.  Topics in the theory of random noise , 1967 .

[19]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.