Singularity Analysis of Composite Laminated Piezoelectric Rectangular Plate Structure with 1 : 2 Internal Resonance

This study investigates the dynamical behavior of the composite laminated piezoelectric rectangular plate with 1 : 2 internal resonance near the singularity using the extended singularity theory method. Based on the previous four-dimensional averaged equations of polar coordinates where the partial derivative terms are not equal to zero, the universal unfolding with codimension 3 of the proposed system is given. The main material parameters that affect the dynamic behavior of the laminated piezoelectric rectangular composite plate near the singularity under transverse excitation are revealed by the transition set of universal unfolding with codimension 3. In addition, the plots of the transition set in three bifurcation parameters space are discussed. These numerical results can show that the stability near the singularity of the proposed system is better when period ratio is less than zero.

Bursting oscillations in a slow-varying periodically excited vector field with Bogdanov–Takens bifurcation

The main purpose of the article is to classify all the possible bursting oscillations in a vector field with Bogdanov–Takens bifurcation at the origin. Based on the universal unfolding of the normal form of the vector field, the topological structure in the neighborhood of the bifurcation point on the unfolding parameters is presented. Replacing one of the unfolding parameters by a slow-varying periodic exciting term, the coupling of two scales in frequency domain involves the vector field, which may lead to the bursting oscillations. According to the bifurcation analysis, we focus on three typical cases to investigate the dynamical evolution with the increase of the exciting amplitude. By introducing the transformed phase portrait, the mechanism of bursting oscillations can be presented. Three types of bifurcations, that is, fold, Hopf, and saddle on the limit cycle bifurcations may cause the alternations of the trajectory between the quiescent states and the spiking states, different combinations of which may result in different bursting attractors. Furthermore, the inertia of the movement may result in the delay effect of the bifurcation, which may lead to the disappearance of the bifurcation influence and the corresponding spiking oscillations.

On the Universal Unfolding of Vector Fields in One Variable: A Proof of Kostov’s Theorem

In this note we present variants of Kostov’s theorem on a versal deformation of a parabolic point of a complex analytic 1-dimensional vector field. First we provide a self-contained proof of Kostov’s theorem, together with a proof that this versal deformation is indeed universal. We then generalize to the real analytic and formal cases, where we show universality, and to the $${\mathcal {C}}^\infty $$ C ∞ case, where we show that only versality is possible.

Open Saito Theory for A and D Singularities

A well-known construction of B. Dubrovin and K. Saito endows the parameter space of a universal unfolding of a simple singularity with a Frobenius manifold structure. In our paper, we present a generalization of this construction for the singularities of types $A$ and $D$ that gives a solution of the open WDVV equations. For the $A$-singularity, the resulting solution describes the intersection numbers on the moduli space of $r$-spin disks, introduced recently in a work of the 2nd author, E. Clader and R. Tessler. In the 2nd part of the paper, we describe the space of homogeneous polynomial solutions of the open WDVV equations associated to the Frobenius manifolds of finite irreducible Coxeter groups.

Bifurcation Analysis of Composite Laminated Piezoelectric Rectangular Plate Structure in the Case of 1:2 Internal Resonance

In this paper, the authors study the bifurcation problems of the composite laminated piezoelectric rectangular plate structure with three bifurcation parameters by singularity theory in the case of 1:2 internal resonance. The sign function is employed to the universal unfolding of bifurcation equations in this system. The proposed approach can ensure the nondegenerate conditions of the universal unfolding of bifurcation equations in this system to be satisfied. The study presents that the proposed system with three bifurcation parameters is a high codimensional bifurcation problem with codimension 4, and 6 forms of universal unfolding are given. Numerical results show that the whole parametric plane can be divided into several persistent regions by the transition set, and the bifurcation diagrams in different persistent regions are obtained.

On the Singularity Theory Applied in Rail Vehicle Bifurcation Problem

The application of Hopf bifurcation is essential to rail vehicle dynamics because it corresponds to the linear critical speed. In engineering, researchers always wonder which vehicle parameters are sensitive to it. With the nonlinear singularity theory's development, it has been widely applied in many other engineering areas. This paper mainly studies the singularity theory applied in nonlinear rail vehicle dynamics. First, the bifurcation norm forms of wheelset and bogie system are, respectively, deduced. Then the universal unfolding is obtained and the influences of perturbation on bifurcation are investigated. By the analysis of a simple bar-spring system, the relationship between the unfolding and original perturbation parameters can be found. But this may be difficult to calculate for the case in vehicle system because of higher degrees-of-freedom (DOFs) and indicate that can explain the influence of all possible parameters perturbations on vehicle bifurcation.

Multiple Buckling and Codimension-Three Bifurcation Phenomena of a Nonlinear Oscillator

In this paper, we investigate the global bifurcations and multiple bucklings of a nonlinear oscillator with a pair of strong irrational nonlinear restoring forces, proposed recently by Han et al. [2012]. The equilibrium stabilities of multiple snap-through buckling system under static loading are analyzed. It is found that complex bifurcations are exhibited of codimension-three with two parameters at the catastrophe point. The universal unfolding for the codimension-three bifurcation is also found to be equivalent to a nonlinear viscous damped system. The bifurcation diagrams and the corresponding codimension-three behaviors are obtained by employing subharmonic Melnikov functions for the existing singular closed orbits of homoclinic, tangent homoclinic, homo-heteroclinic and cuspidal heteroclinic, respectively.