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.

     

Bifurcations of backbone curves for systems of coupled nonlinear two mass oscillator

This paper considers the dynamic response of coupled, forced and lightly damped nonlinear oscillators with two degree-of-freedom. For these systems, backbone curves define the resonant peaks in the frequency–displacement plane and give valuable information on the prediction of the frequency response of the system. Previously, it has been shown that bifurcations can occur in the backbone curves. In this paper, we present an analytical method enabling the identification of the conditions under which such bifurcations occur. The method, based on second-order nonlinear normal forms, is also able to provide information on the nature of the bifurcations and how they affect the characteristics of the response. This approach is applied to a two-degree-of-freedom mass, spring, damper system with cubic hardening springs. We use the second-order normal form method to transform the system coordinates and identify which parameter values will lead to resonant interactions and bifurcations of the backbone curves. Furthermore, the relationship between the backbone curves and the complex dynamics of the forced system is shown.     

Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model

In this paper, we analyze the codimension-2 bifurcations of equilibria of a two-dimensional Hindmarsh–Rose model. By using the bifurcation methods and techniques, we give a rigorous mathematical analysis of Bautin bifurcation. The main result is that no more than two limit cycles can be bifurcated from the equilibrium via Hopf bifurcation; sufficient conditions for the existence of one or two limit cycles are obtained. This paper also shows that the model undergoes a Bogdanov–Takens bifurcation which includes a saddle-node bifurcation, an Andronov–Hopf bifurcation, and a homoclinic bifurcation. In some case, the globally asymptotical stability is discussed.     

Asymptotic Resonance, Interaction of Modes and Subharmonic Bifurcation

We study the existence of small amplitude oscillations near elliptic equilibria of autonomous systems, which mix different normal modes. The reference problem is the Fermi-Pasta-Ulam β-model: a chain of nonlinear oscillators with nearest-neighborhood interaction. We develop a new bifurcation approach that locates secondary bifurcations from the unimodal primary branches. Two sufficient conditions for bifurcation are given: one involves only the arithmetic properties of the eigenvalues of the linearized system (asymptotic resonance), while the other takes into account the nonlinear character of the interaction between normal modes (nonlinear coupling). Both conditions are checked for the Fermi-Pasta-Ulam problem.     

Stability analysis and rich oscillation patterns in discrete-time FitzHugh–Nagumo excitable systems with delayed coupling

In this paper, we give a detailed study of the stable region in discrete-time FitzHugh–Nagumo delayed excitable Systems, which can be divided into two parts: one is independent of delay and the other is dependent on delay. Two different new states are to be observed, which are new steady states (equilibria-the excitable FitzHugh–Nagumo) or limit cycles/higher periodic orbits (the FitzHugh–Nagumo oscillators) as the origin loses its stability, and usually, one is synchronized and the other asynchronized. We also find out that there exist critical curves through which there occur fold bifurcations, flip bifurcations, Neimark–Sacker bifurcations and even higher-codimensional bifurcations etc. It is also shown that delay can play an important role in rich dynamics, such as the occurrence of chaos or not, by means of Lyapunov exponents, Lyapunov dimensions, and the sensitivity to the initial conditions. Multistability phenomena are also discussed including the coexistence of synchronized and asynchronized oscillators, or synchronized/asynchronized oscillators and multiple stable nontrivial equilibria etc.     

转子系统振动变参控制中的瞬态响应

本文以可变参数的挤压油膜阻尼器作为控制元件，研究了转子在稳态转速及加速运动过程中进行变参控制时的瞬态响应问题。结果说明了对转子系统的振动进行分段变参控制，无论是在稳态还是在加速运动过程中，一般都可以取得满意的控制效果，不仅可以减小转子系统的振动，而且还可以使转子系统平稳地通过具有较大振动的共振区，但变参位置不应在多值转速区内。     

Dynamical analysis of the generalized Sprott C system with only two stable equilibria

A generalized Sprott C system with only two stable equilibria is investigated by detailed theoretical analysis as well as dynamic simulation, including some basic dynamical properties, Lyapunov exponent spectra, fractal dimension, bifurcations, and routes to chaos. In the parameter space where the equilibria of the system are both asymptotically stable, chaotic attractors coexist with period attractors and stable equilibria. Moreover, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated. Periodic solutions and chaotic attractors can be found when these cycles disappear.     

Some new insights into the Liu system

The present work is devoted to giving new insights into the Liu chaotic system. The local dynamical entities, such as the number of equilibria, the stability of hyperbolic equilibria, and the stability of the nonhyperbolic equilibrium obtained by using the center manifold theorem, the pitchfork bifurcation, the degenerate pitchfork bifurcation, and Hopf bifurcations, are all analyzed when the parameters are varied in the space of parameters. All the closed orbits of the system are also proven rigorously to be nonplanar but only to be curves in space. Moreover, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated.     

Analysis of Hopf and Takens–Bogdanov Bifurcations in a Modified van der Pol–Duffing Oscillator

We analyze a modified van der Pol–Duffing electronic circuit, modeled by a tridimensional autonomous system of differential equations with Z₂-symmetry. Linear codimension-one and two bifurcations of equilibria give rise to several dynamical behaviours, including periodic, homoclinic and heteroclinic orbits. The local analysis provides, in first approximation, the different bifurcation sets. These local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Hopf bifurcation shows the presence of cusps of saddle-node bifurcations of periodic orbits. The existence of a codimension-four Hopf bifurcation is also pointed out. In the case of the Takens–Bogdanov bifurcation, several degenerate situations of codimension-three are analyzed in both homoclinic and heteroclinic cases. The existence of a Hopf–Shil'nikov singularity is also shown.     

Effects of viscoelasticity on the stability and bifurcations of nonlinear energy sinks

Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally, but the lack of studies of real environmental conditions on these absorbers is felt. The present work investigates the effect of viscoelasticity on the stability and bifurcations of a system attached to a nonlinear energy sink(NES). In this paper, the Burgers model is assumed for the viscoelasticity in an NES, and a linear oscillator system is considered for inve...     

Bifurcations of a Single-Support Elastic Thin-Walled Rotor

The bifurcations of a thin-walled shell rotor during simple and complex rotation are analyzed. The similarity and difference of the problem formulations and solution techniques are pointed out. In both cases, the buckling mode is described by the first circumferential harmonic. The dependence of rotor bifurcations on natural frequencies is studied__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 127–134, March 2005.     

Dynamics analysis of a non-smooth Filippov pest-natural enemy system with time delay

In this paper, we propose a non-smooth Filippov system that describes the interaction of the pest and natural enemy with considering time delay, which represents the change in the growth rate of natural enemies before it is released to prey on pests. When the number of the pest is below the threshold, no control is applied; otherwise, control measures will be adopted. We discuss the stability of the equilibria and the existence of Hopf bifurcation. The results show that the Hopf bifurcation occurs when the time delay passes through some critical values. By applying the Filippov convex method, we obtain the dynamics of the sliding mode. The solutions of the system eventually tend toward the regular equilibrium, the pseudo-equilibrium or a standard periodic solution. Numerical simulations show that time delay plays an important role in local and global sliding bifurcations. We can obtain boundary focus bifurcations from boundary node bifurcations by varying time delay. Furthermore, touching, buckling and crossing bifurcations can be obtained frequently by increasing time delay. The results can provide some insights in pest control.

     

Bifurcations in the attitude dynamics of a spacecraft in a gravity field

A normalized averaged (integrable) Hamiltonian approximation is used to study the attitude dynamics of a rigid body satellite in a gravity field, with an emphasis on bifurcations. The phase portrait is represented in a Mercator map and on a 3D sphere. Pitchfork bifurcations and degeneracies (a dense set of equilibria) are found.     

Sharp-Interface Nematic-Isotropic Phase Transformations With Flow

We develop a sharp-interface theory for phase transformations between the isotropic and uniaxial nematic phases of a flowing liquid crystal. Aside from conventional evolution equations for the bulk phases and corresponding interface conditions, the theory includes a supplemental interface condition expressing the balance of configurational momentum. As an idealized illustrative application of the theory, we consider the problem of an evolving spherical droplet of the isotropic phase surrounded by the nematic phase in a radially-oriented state. For this problem, the bulk and interfacial equations collapse to a single nonlinear second-order ordinary differential equation for the radius of the droplet—an equation which, in essence, expresses the balance of configurational momentum on the interface. This droplet evolution equation, which closely resembles a previously derived and extensively studied equation for the expansion of contraction of a spherical gas bubble in an incompressible viscous liquid, includes terms accounting for the curvature elasticity and viscosity of the nematic phase, interfacial energy, interfacial viscosity, and the ordering kinetics of the phase transformation. We determine the equilibria of this equation and study their stability. Additionally, we find that motion of the interface generates a backflow, without director reorientation, in the nematic phase. Our analysis indicates that a backflow measurement has the potential to provide an independent means to determine the density difference between the isotropic and uniaxial nematic phases.     

Bifurcation and path-following continuation analysis of periodic orbits of an extended Fermi oscillator model

In this research, we offer eigenvalue analysis and path following continuation to describe the impact, stick, and non-stick between the particle and boundaries to understand the nonlinear dynamics of an extended Fermi oscillator. The principles of discontinuous dynamical systems will be utilized to explain the moving process in such an extended Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. The major parameters are the frequency, amplitude in periodic excitation force, and the gap between the top and bottom boundary. We employ path-following analysis to illustrate the bifurcations that lead to solution destabilization. We present the evolution of the period solutions of the extended Fermi oscillator as the parameter varies. From the viewpoint of eigenvalue analysis, the essence of period-doubling, saddle-node, and Torus bifurcation is revealed. Numerical continuation methods are used to do a complete one- and two-parameter bifurcation analysis of the extended Fermi oscillator. The presence of codimension-one bifurcations of limit cycles, such as saddle-node, period-doubling, and Torus bifurcations, is shown in this work. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the extended Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters.

     

Lagrangian block diagonalization

Relative equilibria, i.e., steady motions associated to specified group motions, are an important class of steady motions of Hamiltonian and Lagrangian systems with symmetry. Relative equilibria can be identified by means of a variational principle on the tangent space of the configuration manifold. We show that relative equilibria can also be found by means of a variational principle on the configuration manifold itself. Formal stability of a relative equilibrium corresponds to definiteness of the second variation of the energymomentum functional, which is a specified combination of the total energy and the group momentum, on an appropriate subspace. We decompose this subspace into three subspaces by means of the Legendre transformation and the group action and show that the second variation block diagonalizes with respect to these subspaces. The techniques employed here are a generalization of the reduced energy-momentum method of Simoet al. (1991), which applies only to simple mechanical systems, to a more general class of conservative systems, including systems on which the symmetry group does not act freely. We briefly discuss a generalization of a result due to Patrick (1990) that provides conditions under which formal stability implies nonlinear orbital stability. Several simple examples, including natural mechanical systems, are used to illustrate the block diagonalization procedure.     

Nonstationary response near generic bifurcations

We consider in this paper nonstationary response near generic bifurcations of equilibria under one control parameter. The bifurcations treated are the transcritical, super- and subcritical Hopf, and the fold all in their simplest, generic normal forms. The nonstationary is generated by varying the control parameter, either linearly or sinusoidally. Exact analytical solutions are obtained, and local and global stability is discussed     

Steady states and nonlinear buckling of cable-suspended beam systems

This paper deals with the equilibria of an elastically-coupled cable-suspended beam system, where the beam is assumed to be extensible and subject to a compressive axial load. When no vertical load is applied, necessary and sufficient conditions in order to have nontrivial solutions are established, and their explicit closed-form expressions are found. In particular, the stationary solutions are shown to exhibit at most two non-vanishing Fourier modes and the critical values of the axial-load parameter which produce their pitchfork bifurcation (buckling) are established. Depending on two dimensionless parameters, the complete set of resonant modes is devised. As expected, breakdown of the pitchfork bifurcations under perturbation is observed when a distributed transversal load is applied to the beam. In this case, both unimodal and bimodal stationary solutions are studied in detail. Finally, the more complex behavior occurring when trimodal solutions are involved is briefly sketched.     

Sliding modes of two interacting frictional interfaces

In the context of rate-and-state friction, we report an extensive analysis of stability of the quasi-static frictional sliding of two parallel interfaces dividing a linear elastic solid sheared at a constant rate. One possibility for the frictional sliding is that the interfaces slip at equal rates, a steady state described as symmetric. However a steady-state friction law that is non-monotonic allows the competing possibility of an asymmetric steady state in which the interfaces slide at different rates. A rate-and-state law that delivers such behaviour and agrees with the experimental results of Heslot et al. [1994. Creep, stick-slip, and dry-friction dynamics: experiments and a heuristic model. Phys. Rev. E 49, 4973-4988] is proposed. Analytical results combined with numerical investigations performed with the continuation package Auto and direct time integration are used to compile the complete picture of the many bifurcations that exist between the diverse steady and oscillatory sliding modes. In addition to the control parameters corresponding to the driving velocity and the stiffness of the medium, we find that the geometrical details of the steady-state friction law determine the occurrence and nature of bifurcations. Pitchfork bifurcations from the symmetric to asymmetric steady states coincide with the extrema of the friction law; Hopf bifurcations occur in the velocity weakening regime of the friction law. Torus and period-doubling bifurcations of periodic orbits also occur, and lead to complicated dynamics. We also present results of numerical computations that illustrate the complex and versatile dynamics of the two-interface system. We anticipate that the dynamics found in our model should be verifiable by experiments.     

Bifurcations and equilibria in the extended N-body ring problem

We consider the motion of an infinitesimal particle under the gravitational field of (n+1) bodies in ring configuration, that consist of n primaries of equal mass m placed at the vertices of a regular polygon, plus another primary of mass m₀=βm located at the geometric center of the polygon. We analyze the phase flow, determine the equilibria of the system, their linear stability and the bifurcations depending on the mass of the central primary (parameter β).This study is extended to the case when the central body is an ellipsoid or a radiation source. In this case, the topology of the problem is modified.