تجزیه و تحلیل حساسیت از بروسلیتر تحمیل شده بصورت تصادفی و دوره ای

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.

Many phenomena of statistical physics (lasers, radio-frequency generators), biological systems and chemical reactions are due to the interaction of couplet oscillations. This is the first mechanical example of “mode locking” that has been observed and studied experimentally at least since the time of Huygens, who discovered “synchronization” of two pendulum clocks mounted on a common wall. An investigation the periodically forced nonlinear oscillators was started by van der Pol. Subsequent experiments with forced van der Pol and other nonlinear oscillators have discovered the existence of a variety of a periodic, quasiperiodic and aperiodic (chaotic) behaviors (see [1] and [2]). The various transitions (“bifurcations”) from periodic to more complicated regimes are a central problem in modern nonlinear dynamic theory. There are many papers devoted to a qualitative analysis of the periodically forced oscillations (see [3] and [4]). Fundamental results based on mappings analysis of the circumference on to itself were imported by Poincare [5], Denjoy [6] and Arnold [7]. A standard quantification of qualitative understanding of phenomena mentioned above is the bifurcation diagram. Most of the results on bifurcation analysis have been produced for a space of frequency–amplitude parameters of an external periodic force with a fixed nonlinear oscillator. Many researchers have demonstrated a very complicated overlapping structure of entrainment zones (Arnold tongues) of the forced oscillators. Nevertheless, there is obvious interest in the forced result dependence on the parameters of initial system itself. In this paper we are interested in the limit cycle zone of the nonlinear oscillator parameters. It is clear that the response of nonlinear oscillator for the same external periodic force may have an essential difference for different regions of this zone. For some regions the fixed periodic stimulus slightly deforms the initial unforced limit cycle. For other regions it leads to a qualitative change of the oscillation behavior. In these circumstances it would be desirable to have more detailed description of the limit cycle zone of parameters. It needs a quantitative description reflecting the difference in the response of different regions of a cycle sensitivity level to external disturbances. The analytical basis of such description should be some quantitative measure of sensitivity of cycle. Historically, the Lyapunov exponent was the first value describing the degree of cycle stability. For many systems with auto-oscillations, the mathematical model is the nonlinear deterministic system equation(1.1) with T-periodic solution x=ξ(t) (phase trajectory Γ is a limit cycle). The classical analysis of stability of periodic solutions is based on a linear system of the first approximation and search of its Lyapunov exponents. As one of the exponents is equal to unity, the given approach in the case of system on a plane tends to a unique quantitative parameter of stability cycle equation(1.2) An inequality ρ<1 is the necessary and sufficient condition of an exponential orbital stability. The value ρ specifies (asymptotically) a stability degree of cycles for an initial moment single disturbance. In most cases, the real systems are forced by various disturbances during the motion time. Under these circumstances, ρ is too rough an integrated characteristic to allow to distinguish and compare the stability degrees of various orbit pieces with nonvanishing disturbances. A system of stochastic differential equations equation(1.3) is a traditional mathematical model allowing to study quantitative description of results of nonvanishing disturbances. Here w(t) is Wiener process, σ(x) is disturbances matrix function, and ε is a parameter of its intensity. The analysis of the auto-oscillations of nonlinear systems with respect to stochastic disturbances was initiated by Andronov et al. [8] and continued by many researchers, see e.g. [9], [10], [11], [12] and [13]. Kolmogorov–Fokker–Planck equation gives the most detailed description of the probabilistic behavior of system (1.3). However, using this equation directly is very difficult even for the simplest situations. It is important to note that when stochastic disturbances are small, they lead to famous problems of analysis of equations with small coefficients near higher derivatives. A new approach connected with the using of the quasipotential function in stochastic analysis is being actively developed (see e.g. [14], [15], [16], [17] and [18]). For the first time the quasipotential function has appeared in papers by Wentzell and Freidlin in connection with the decision of a known Kolmogorov ‘exit problem’ of the random trajectory from the vicinity of a steady point of rest. Note that quasipotential function is used widely by Graham and Tel [19] and [20] for analysis of statistical nonequilibrium thermodynamics problem. Quasipotential v(x) gives the following asymptotic of stationary density distributions ρ(x,ε): For a small ε it gives the following approximation: equation(1.4) The quasipotential allows (see [16] and [21]) to find some important characteristics: average time of exit of a random trajectory from domain and the most probable point of an exit. In Section 2, we give some definitions and brief main results by Ludwig [22], Naeh et al. [14], Mil'shtein and Ryashko [23], Ryashko [24] concerning a local approximation v(x) in the small vicinity of the cycle Γ. Due to some parametrization the construction of the first approximation of quasipotential v(x) is reduced to the search for T-periodic solution of the linear matrix differential equation. For the plane orbit case using singularity of this matrix equation the approximation problem is reduced to the solution of some scalar linear equation. The solution of this equation – a T-periodic function – is found in explicit form. This scalar function plays the role of sensitivity function allowing to compare the stability levels of the different pieces of orbit. The value of sensitivity function maximum is introduced as a new sensitivity parameter of a limit cycle. A parameter of cycle stiffness is discussed. In Section 3, we demonstrate analytical possibilities of sensitivity function to predict some peculiarities of dynamic for stochastically forced Brusselator. An example of sensitivity analysis based on sensitivity function is shown. From this analysis the critical value of Brusselator parameter is found. For this critical value the peculiarities of the random trajectories arrangement are demonstrated. The sensitivity function is a useful analytical tool for the prediction of singular responses both to stochastic and to periodic disturbances. In Section 4, we demonstrate dynamics of periodic forced Brusselator for values of parameter close to critical one. An essential difference in Brusselator response for the same periodic disturbances is shown. For critical value of Brusselator parameter the period doubling regime of the transition to chaos scenario is demonstrated.