Stochastic P-bifurcation in a tri-stable Van der Pol system with fractional derivative under Gaussian white noise

In this paper, we study the tri-stable stochastic P-bifurcation problem of a generalized Van der Pol system with fractional derivative under Gaussian white noise excitation. Firstly, using the principle for minimal mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be transformed into an equivalent integer order system. Secondly, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by the stochastic averaging, and using the singularity theory, we find the critical parametric conditions for stochastic P-bifurcation of amplitude of the system, which can make the system switch among the three steady states. Finally, we analyze different types of the stationary PDF curves of the system amplitude qualitatively by choosing parameters corresponding to each region divided by the transition set curves, and the system response can be maintained at the small amplitude near the equilibrium by selecting the appropriate unfolding parameters. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order controller to adjust the response of the system

The dynamics of the pendulum suspended on the forced Duffing oscillator

We investigate the dynamics of the pendulum suspended on the forced Duffing oscillator. The detailed bifurcation analysis in two parameter space (amplitude and frequency of excitation) which presents both oscillating and rotating periodic solutions of the pendulum has been performed. We identify the areas with low number of coexisting attractors in the parameter space as the coexistence of different attractors has a significant impact on the practical usage of the proposed system as a tuned mass absorber.Comment: Accepte

Evolutionary-based sparse regression for the experimental identification of duffing oscillator

In this paper, an evolutionary-based sparse regression algorithm is proposed and applied onto experimental data collected from a Duffing oscillator setup and numerical simulation data. Our purpose is to identify the Coulomb friction terms as part of the ordinary differential equation of the system. Correct identification of this nonlinear system using sparse identification is hugely dependent on selecting the correct form of nonlinearity included in the function library. Consequently, in this work, the evolutionary-based sparse identification is replacing the need for user knowledge when constructing the library in sparse identification. Constructing the library based on the data-driven evolutionary approach is an effective way to extend the space of nonlinear functions, allowing for the sparse regression to be applied on an extensive space of functions. The results show that the method provides an effective algorithm for the purpose of unveiling the physical nature of the Duffing oscillator. In addition, the robustness of the identification algorithm is investigated for various levels of noise in simulation. The proposed method has possible applications to other nonlinear dynamic systems in mechatronics, robotics, and electronics

FORCED NONLINEAR OSCILLATOR IN A FRACTAL SPACE

A critical hurdle of a nonlinear vibration system in a fractal space is the inefficiency in modelling the system. Specifically, the differential equation models cannot elucidate the effect of porosity size and distribution of the periodic property. This paper establishes a fractal-differential model for this purpose, and a fractal Duffing-Van der Pol oscillator (DVdP) with two-scale fractal derivatives and a forced term is considered as an example to reveal the basic properties of the fractal oscillator. Utilizing the two-scale transforms and He-Laplace method, an analytic approximate solution may be attained. Unfortunately, this solution is not physically preferred. It has to be modified along with the nonlinear frequency analysis, and the stability criterion for the equation under consideration is obtained. On the other hand, the linearized stability theory is employed in the autonomous arrangement. Consequently, the phase portraits around the equilibrium points are sketched. For the non-autonomous organization, the stability criteria are analyzed via the multiple time scales technique. Numerical estimations are designed to confirm graphically the analytical approximate solutions as well as the stability configuration. It is revealed that the exciting external force parameter plays a destabilizing role. Furthermore, both of the frequency of the excited force and the stiffness parameter, execute a dual role in the stability picture