CODIMENSION TWO BIFURCATIONS AND HOPF BIFURCATIONS OF AN IMPACTING VIBRATING

ＸｉｅＪｉａｎｈｕａ（谢建华）（ＲｅｃｅｉｖｅｄＯｃｔ．５，１９９４；ＣｏｍｍｕｎｉｃａｔｅｄｂｙＬｉＬｉ）ＣＯＤＩＭＥＮＳＩＯＮＴＷＯＢＩＦＵＲＣＡＴＩＯＮＳＡＮＤＨＯＰＦＢＩＦＵＲＣＡＴＩＯＮＳＯＦＡＮＩＭＰＡＣＴＩＮＧＶＩＢＲＡＴＩＮＧＳＹＳＴ...     

One-to-three resonant Hopf bifurcations of a maglev system

This paper studies the dynamics of a maglev system around 1:3 resonant Hopf–Hopf bifurcations. When two pairs of purely imaginary roots exist for the corresponding characteristic equation, the maglev system has an interaction of Hopf–Hopf bifurcations at the intersection of two bifurcation curves in the feedback control parameter and time delay space. The method of multiple time scales is employed to drive the bifurcation equations for the maglev system by expressing complex amplitudes in a combined polar-Cartesian representation. The dynamics behavior in the vicinity of 1:3 resonant Hopf–Hopf bifurcations is studied in terms of the controller’s parameters (time delay and two feedback control gains). Finally, numerical simulations are presented to support the analytical results and demonstrate some interesting phenomena for the maglev system.     

The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations

This paper considers the computation of the simplest parameterized normal forms (SPNF) of Hopf and generalized Hopf bifurcations. Although the notion of the simplest normal form has been studied for more than two decades, most of the efforts have been spent on the systems that do not involve perturbation parameters due to the restriction of the computational complexity. Very recently, two singularities – single zero and Hopf bifurcation – have been investigated, and the SPNFs for these two cases have been obtained. This paper extends a recently developed method for Hopf bifurcation to compute the SPNF of generalized Hopf bifurcations. The attention is focused on a codimension-2 generalized Hopf bifurcation. It is shown that the SPNF cannot be obtained by using only a near-identity transformation. Additional transformations such as time and parameter rescaling are further introduced. Moreover, an efficient recursive formula is presented for computing the SPNF. Examples are given to demonstrate the applicability of the new method.     

Local Integrability and Linearizability of a $$$$ Resonant Quadratic System

In this paper we study the local integrability and linearizability of quadratic three dimensional systems of the form

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First, we obtain necessary and sufficient conditions for the complete integrability and linearizability of this system. Then, we discuss the problem of existence of one first integral of the form \(\psi ^{(1)}(x,y,z)=xy+O(|x,y,z|^3)\). Computation of resonant focus quantities and the decomposition of the variety of the ideal that they generate in the ring of polynomials of parameters \(a_{ij},b_{ij},c_{ij}\) of the system were used to obtain necessary conditions of integrability and linearizability. The theory of Darboux integrability and some other methods are used to show the sufficiency. In the investigation of the conditions for the existence of one first integral the decomposition of the variety mentioned above was performed using modular computations, its consequences are discussed.

     

Prediction of bifurcations by varying critical parameters of COVID-19

Nonlinear Dynamics - Coronavirus disease 2019 is a recent strong challenge for the world. In this paper, an epidemiology model is investigated as a model for the development of COVID-19. The...     

Prediction of period doubling bifurcations in harmonically forced memristor circuits

Nonlinear Dynamics - The paper studies bifurcations and complex dynamics in a class of nonautonomous oscillatory circuits with a flux-controlled memristor and harmonic forcing term. It is first...     

On the two bifurcations of a white-noise excited Hopf bifurcation system

I.IntroductionNonlinearanalysiseffortsmainlyincluderesearchesonthestablemotionofasystem.investigationsonitsstabilityfeaturesandtheinstantaneousmotionofadynamicalsystemwhenchangesoccurtoitsgoverningparameters.Theso-calledstochasticbifurcationimpliesthetran…     

Global bifurcations and multi-pulse orbits of a parametric excited system with autoparametric resonance

We consider an autoparametric system which consists of an oscillator coupled with a parametrically excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in principal parametric resonance. The system contains the most general type of quadratic and cubic non-linearities. The method of second-order averaging is used to yield a set of autonomous equations of the second-order approximations to the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Shilnikov-type multi-pulse homoclinic orbits in the averaged equations. The results obtained above mean the existence of amplitude-modulated chaos in the Smale horseshoe sense in the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse chaotic motions of the parametric excited system with autoparametric resonance are also found by using numerical simulation.     

Control of catastrophic bifurcations of SSR in a hybrid series compensated system

Nonlinear Dynamics - Series compensation of the transmission line increases the power flow capability of the system. Hybrid series compensation is a combination of active and passive series...     

Mechanisms of strange nonchaotic attractors in a nonsmooth system with border-collision bifurcations

Nonlinear Dynamics - It is not very clear to understand genesis and mechanisms for the creation of strange nonchaotic attractors (SNAs) due to the nonsmooth bifurcations in the nonsmooth systems. A...     

Local stability and Hopf bifurcations analysis of the Muthuswamy-Chua-Ginoux system

The three-dimensional Muthuswamy–Chua–Ginoux (MCG, for short) circuit system based on a thermistor is a generalization of the classical Muthuswamy–Chua circuit differential system. At present, there are only partial numerical simulations for the qualitative analysis of the MCG circuit system. In this work, we study local stability and Hopf bifurcations of the MCG circuit system depending on 8 parameters. The emerging of limit cycles under zero-Hopf bifurcation and Hopf bifurcation is investigated in detail by using the averaging method and the center manifolds theory, respectively. We provide sufficient conditions for a class of the circuit systems to have a prescribed number of limit cycles bifurcating from the zero-Hopf equilibria by making use of the third-order averaging method, as well as the methods of Gröbner basis and real solution classification from symbolic computation. Such algebraic analysis allows one to study the zero-Hopf bifurcation for any other differential system in dimension 3 or higher. After, the classical Hopf bifurcation of the circuit system is analyzed by computing the first three focus quantities near the Hopf equilibria. Some examples and numerical simulations are presented to verify the established theoretical results.

     

Stability and Hopf bifurcations of an optoelectronic time-delay feedback system

The local dynamics around the trivial solution of an optoelectronic time-delay feedback system is investigated in the paper, and the effect of the feedback strength on the stability is addressed. The linear stability analysis shows that as the feedback strength varies, the system undergoes exactly two times of stability switch from a stable status to an unstable status or vice versa, and at each of the two end points of the stable interval, a Hopf bifurcation occurs. To gain insight of the bifurcated periodic solution, the Lindstedt–Poincaré method that involves easy computation, rather than the center manifold reduction that involves a great deal of tedious computation as done in the literature, is used to calculate the bifurcated periodic solution, and to determine the direction of the bifurcation. Two case studies are made to demonstrate the efficiency of the method.     

Hopf bifurcations in a predator-prey system of population allelopathy with a discrete delay and a distributed delay

A delayed Lotka?CVolterra predator-prey system of population allelopathy with discrete delay and distributed maturation delay for the predator population described by an integral with a strong delay kernel is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.     

Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system

In this paper, we address the problem of the bifurcation control of a delayed fractional-order dual model of congestion control algorithms. A fractional-order proportional–derivative (PD) feedback controller is designed to control the bifurcation generated by the delayed fractional-order congestion control model. By choosing the communication delay as the bifurcation parameter, the issues of the stability and bifurcations for the controlled fractional-order model are studied. Applying the stability theorem of fractional-order systems, we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation. Additionally, the critical value of time delay is figured out, where a Hopf bifurcation occurs and a family of oscillations bifurcate from the equilibrium. It is also shown that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD controller. Finally, numerical simulations are given to validate the main results and the effectiveness of the control strategy.

     

Global bifurcations and chaos in a Van der Pol-Duffing-Mathieu system with three-well potential oscillator

Semi-analytical and semi-numerical method is used to investigate the global bifurcations and chaos in the nonlinear system of a Van der Pol-Duffing-Mathieu oscillator. Semi-analytical and semi-numerical method means that the autonomous system, called Van der Pol-Duffing system, is analytically studied to draw all global bifurcations diagrams in parameter space. These diagrams are called basic bifurcation diagrams. Then fixing parameter in every space and taking parametrically excited amplitude as a bifurcation parameter, we can observe the evolution from a basic bifurcation diagram to chaotic pattern by numerical methods. The project supported by the National Natural Science Foundation of China     

Modified post-bifurcation dynamics and routes to chaos from double-Hopf bifurcations in a hyperchaotic system

In order to understand the onset of hyperchaotic behavior recently observed in many systems, we study bifurcations in the modified Chen system leading from simple dynamics into chaotic regimes. In particular, we demonstrate that the existence of only one fixed point of the system in all regions of parameter space implies that this simple point attractor may only be destabilized via a Hopf or double Hopf bifurcation as system parameters are varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The normal form immediately following double Hopf bifurcations is constructed analytically by the method of multiple scales. Analysis of this generalized double Hopf normal form along standard lines reveals possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. However, considering these more carefully, we find that only certain combinations or sequences of these dynamical regimes are possible, while others derived and considered in earlier work are in fact mathematically impossible. We also discuss the post-bifurcation dynamics in the context of two intermittent routes to chaos (routes following either (i) subcritical or (ii) supercritical Hopf or double Hopf bifurcations). In particular, the route following supercritical bifurcations is somewhat subtle. Such behavior following repeated Hopf bifurcations is well-known and widely observed, including in the classical Ruelle?CTakens and quasiperiodic routes to chaos. However, to the best of our knowledge, it has not been considered in the context of the double-Hopf normal form, although it has been numerically observed and tracked in the post-double Hopf regime. Numerical simulations are employed to corroborate these various predictions from the normal form. They reveal the existence of stable periodic and toroidal attractors in the post-supercritical-Hopf cases, and either attractors at infinity or bounded chaotic dynamics following subcritical Hopf bifurcations. Future work will map out the remainder of the routes into the chaotic regimes, including further bifurcations of the post-supercritical-Hopf two- and three-tori via either torus doubling or breakdown.     

Basins and bifurcations of a delayed feedback control system and its experimental verification for a DC bus circuit

Nonlinear Dynamics - Stochastic synchrony, also known as noise-induced synchronization that leads to phase coherence, arises when a set of uncoupled neurons synchronizes to a common white noise...