The First-Integral Method for Duffing-Van Der Pol-Type Oscillator System

In this thesis, we restrict our attention to nonlinear Duffing–van der Pol–type oscillator system by means of the First-integral method. This system has physical relevance as a model in certain flow-induced structural vibration problems, which includes the van der Pol oscillator and the damped Duffing oscillator etc as particular cases. Firstly we apply the Division Theorem for two variables in the complex domain, which is based on the ring theory of commutative algebra, to explore a quasi-polynomial first integral to an equivalent autonomous system. Then through a certain parametric condition, we derive a more general first integral of the Duffing–van der Pol–type oscillator system

Study of the Van der Pol Oscillator with Fractional Derivatives

In this paper we propose a modified version of the classical Van der Pol oscillator by introducing fractional-order time derivatives into the state-space model. The resulting fractional-order Van der Pol oscillator is analyzed in the time and frequency domains, by using phase portraits, spectral analysis and bifurcation diagrams. The fractional-order dynamics is illustrated through numerical simulations of the proposed schemes by using approximations to fractional-order operators. Finally, the analysis is extended to the forced Van der Pol oscillator.N/

The Forced van der Pol Equation II: Canards in the reduced system

This is the second in a series of papers about the dynamics of the forced van der Pol oscillator [J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 1–35]. The first paper described the reduced system, a two dimensional flow with jumps that reflect fast trajectory segments in this vector field with two time scales. This paper extends the reduced system to account for canards, trajectory segments that follow the unstable portion of the slow manifold in the forced van der Pol oscillator. This extension of the reduced system serves as a template for approximating the full nonwandering set of the forced van der Pol oscillator for large sets of parameter values, including parameters for which the system is chaotic. We analyze some bifurcations in the extension of the reduced system, building upon our previous work in [J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 1–35]. We conclude with computations of return maps and periodic orbits in the full three dimensional flow that are compared with the computations and analysis of the reduced system. These comparisons demonstrate numerically the validity of results we derive from the study of canards in the reduced system