Investigation On The Bifurcation Phenomena Of Several Types Of Fractional Multistable Systems

The random and nonlinear factors exist widely in nature and society,it is because of these factors that increase the complexity of the systems,so it is necessary to study the random dynamic behavior of the nonlinear system.The definition of fractional derivatives contain convolution,which can express a memory effect well and show a cumulative effect over time.Comparing with the integer-order calculus,the system with fractional calculus has more advantages and it is a powerful mathematical tool for describing the memory characteristics,thus,it is necessary and significant to study the mechanical characteristics and the fractional parametric influences on systems.For the parametric design of fractional nonlinear dynamic systems,based on the equivalent expression of fractional derivative and taking the singularity theory as the mainly analytical tool,in terms of the calculation method,deterministic and stochastic bifurcation of fractional dynamic systems,the following five aspects of work are carried out:(1)A matrix method for solving fractional differential equations with different orders is proposed,several kinds of equations with analytic solutions are taken as examples to verify the correctness of this method.Taking fractional Duffing equation as an example,the possible problems of replacing Caputo algorithm with GL algorithm in some literatures are illustrated.Using GL algorithm to calculate the differential equation with Caputo derivative,except that there is no problem under zero initial condition,due to the different initial conditions,the calculation results may have the following problems:(1)only the transition process is different,the steady-state result is correct;(2)both the transition process and the steady-state result are different.The reason for this phenomenon is whether the initial condition is close to the boundary of the basin of attraction.(2)The 3:1 superharmonic resonance and stochastic P-bifurcation of Duffing oscillator with fractional derivative are studied.Firstly,regarding the system's amplitude-frequency response equation as the bifurcation equation,the critical parameters for bifurcation of the system are obtained by singularity theory.The components of the 1X and 3X frequencies of the system's time history are extracted by the spectrum analysis,by comparing them with the approximate solutions of the system,the correctness of the analytical results obtained is verified.Secondly,according to the principle of minimal mean square error,the Duffing system under multiplicative noise excitation is transformed into an integer-order system with linear stiffness.The critical parametric conditions for the stochastic P-bifurcation of the system are obtained by the stochastic averaging method and singularity theory,so that the motion state of the system can be controlled by choosing the appropriate unfolding parameters.(3)The tri-stable stochastic P-bifurcation of a generalized Van der Pol system with fractional derivative under Gaussian white noise excitation is studied.The original system is transformed into an equivalent integer-order system,the stationary Probability Density Function(PDF)of the system's amplitude is obtained by stochastic averaging method.Using singularity theory,the critical parametric conditions for stochastic P-bifurcation of the system are obtained,the stochastic P-bifurcation behaviors of the system under additive noise and the combined excitation of additive and multiplicative Gaussian white noises together are discussed respectively.The system's response can be maintained at the small amplitude near the equilibrium or monostability by selecting the corresponding unfolding parameters,which can provide the theoretical guidance for system design and avoid the damage and instability caused by the system's nonlinear jump phenomenon or large amplitude vibration.The numerical results by Monte Carlo simulation of the original system also verify the theoretical results obtained in this paper.(4)The bistable stochastic P-bifurcation of a generalized Van der Pol system with the fractional time-delay feedback under Gaussian white noise excitation is studied.According to the principle of minimal mean square error,the original system is transformed into an equivalent integer-order system,and using the stochastic averaging method and singularity theory,the critical parametric conditions for the system's stochastic P-bifurcation are obtained.The stochastic P-bifurcation behaviors of the system under additive noise,multiplicative noise and the combined excitation of additive and multiplicative Gaussian white noises together are discussed respectively.The system's response can be maintained at the small amplitude near the equilibrium by selecting the appropriate unfolding parameters,which can provide the theoretical guidance for the design of fractional controller.(5)The bistable stochastic P-bifurcation behaviors for axially moving of a viscoelastic beam model with fractional derivatives of high order nonlinear terms under Gaussian white noise and colored noise are studied respectively.The ordinary differential equation for simply supported viscoelastic beam of transverse vibration is deduced by d'Alambert principle and the Galerkin discrete method.By using the stochastic averaging method and singularity theory to the equivalent integer-order system,the critical parametric conditions for stochastic P-bifurcation of the system are obtained,the system's response can be maintained at the small amplitude near the equilibrium by selecting the appropriate unfolding parameters,providing theoretical guidance for system design in practical engineering.