Abstract
This paper is concerned with the computation of the normal form and its application to a viscoelastic moving belt. First, a new computation method is proposed for significantly refining the normal forms for high-dimensional nonlinear systems. The improved method is described in detail by analyzing the four-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain three different cases, that are, (i) two pairs of pure imaginary eigenvalues, (ii) one nonsemisimple double zero and a pair of pure imaginary eigenvalues, and (iii) two nonsemisimple double zero eigenvalues. Then, three explicit formulae are derived, herein, which can be used to compute the coefficients of the normal form and the associated nonlinear transformation. Finally, employing the present method, we study the nonlinear oscillation of the viscoelastic moving belt under parametric excitations. The stability and bifurcation of the nonlinear vibration system are studied. Through the illustrative example, the feasibility and merit of this novel method are also demonstrated and discussed.
Highlights
Bifurcation and stability analysis of nonlinear differential equations is one of the challenging problems of mathematicians and engineers
An efficient method for computing the normal form of highdimensional nonlinear systems is presented in this paper
Based on the current studies, it is found that the newly developed computation method improves the classical normal form. It is a further reduction of the classical normal form
Summary
Bifurcation and stability analysis of nonlinear differential equations is one of the challenging problems of mathematicians and engineers. Zhang and Leung [19] considered a general four-dimensional normal form of a double Hopf bifurcation Yu and his associates [20,21,22] developed efficient computing methods for parametric normal forms. They applied the new method to consider controlling bifurcations of the nonlinear dynamical systems. Buono and Belair [29] studied the normal form of a vector field, which is generated by a scalar delay-differential equations at nonresonant double Hopf bifurcation points. We will develop an efficient method for computing the normal forms directly for general fourdimension systems and apply the method to consider controlling bifurcations.
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