Bifurcation Equations Research Articles

On dynamics in a Keynesian model of monetary stabilization policy with debt effect

In this paper, a four-dimensional model of flexible prices with the central bank's stabilization policy, describing the development of the firms' private debt, the output, the expected rate of inflation and the rate of interest is analyzed. Questions concerning the existence of limit cycles around its normal equilibrium point are investigated. The bifurcation equation is found. The formulae for the calculation of its coefficients are gained. A numerical example is presented by means of numerical simulations.

Bifurcation characteristics of torsional-horizontal coupled vibration of rolling mill system

In this paper, the static bifurcation and dynamic bifurcation of rolling mill system with torsional-horizontal coupling vibration are studied. Firstly, the dynamic equation of torsional-horizontal coupled vibration of rolling mill main drive system is established. Rolling mill is driven by the gear pair, so the torsional vibration of the gear pair, the horizontal vibration and the friction factor are considered in the dynamic equation. Then the equivalent low-dimensional bifurcation equation, which can reveal the system nonlinear characteristics, is obtained using Lyapunov-Schmidt method, and the system static bifurcation characteristics are studied using singularity theory. Lastly, the bifurcation condition and stability of the system with dynamic Hopf bifurcation are studied using Hopf bifurcation theorem. Numerical simulation of the actual parameter values confirms the analytical results.

Double Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations

The double Hopf bifurcation of a composite laminated piezoelectric plate with combined external and internal excitations is studied. Using a multiple scale method, the average equations are obtained in two coordinates. The bifurcation response equations of the composite laminated piezoelectric plate with the primary parameter resonance, i.e., 1:3 internal resonance, are achieved. Then, the bifurcation feature of bifurcation equations is considered using the singularity theory. A bifurcation diagram is obtained on the parameter plane. Different steady state solutions of the average equations are analyzed. By numerical simulation, periodic vibration and quasi-periodic vibration responses of the composite laminated piezoelectric plate are obtained.

Generic unfolding of a degenerate heterodimensional cycle

This paper is devoted to the study of generic unfolding of a degenerate heterodimensional cycle under orbit flip condition. By setting up local moving frame systems in small tubular neighborhood of unperturbed heterodimensional cycle, we construct a Poincare return map and further obtain the bifurcation equations. Based on the bifurcation analysis, different bifurcation phenomena are discussed under small perturbations. An analytical example is also given to demonstrate our main results.

Equivariant bifurcation in a coupled complex-valued neural network rings

Network with interacting loops and time delays are common in physiological systems. In the past few years, the dynamic behaviors of coupled interacting loops neural networks have been widely studied due to their extensive applications in classification of pattern recognition, signal processing, image processing, engineering optimization and animal locomotion, and other areas, see the references therein. In a large amount of applications, complex signals often occur and the complex-valued recurrent neural networks are preferable. In this paper, we study a complex value Hopfield-type network that consists of a pair of one-way rings each with four neurons and two-way coupling between each ring. We discuss the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. The existence of multiple branches of bifurcating periodic solution is obtained. We also found that the spatio-temporal patterns of bifurcating periodic oscillations alternate according to the change of the propagation time delay in the coupling, i.e., different ranges of delays correspond to different patterns of neural network oscillators. The oscillations of corresponding neurons in the two loops can be in phase or anti-phase depending on the parameters and delay. Some numerical simulations support our analysis results.

Equivariant bifurcations in a non-local model of ferromagnetic materials

In this work we study the bifurcations from the trivial equilibrium of the equation ∂u∂t(x,t)=−u(x,t)+tanh(β(J∗u)(x,t)), in the space of 2τ periodic functions. This is accomplished with the help of the equivariant branching lemma, which allows us to take into account the symmetries present in the model. We show that the phenomenon of ‘spontaneous symmetry-breaking’ occurs here, that is, the bifurcating solutions are less symmetric than the trivial one. We also prove that, under certain conditions, these equilibria can be globally continued.

Nonlinear Dynamics Analysis for the Taut Inclined Cable Excited by Deck Vibration

AbstractA simple mechanical model is proposed to describe the dynamics system of the taut inclined cable excited by the deck vibration. Using Galerkin method, the dynamics system is simplified into a one-degree of freedom nonlinear system. Average method is used to obtain the average equations and bifurcation equation. Bifurcation maps are obtained and used to reveal the evolution mechanism of period-1 motion to period-2 motion. Several motion forms are discussed and evolution progress between different motions is studied. Several kinds of parametric resonance are revealed in various parameter regions. The conclusions indicate that the parametric resonance excited by the frequency ratio 3:2 is significant should be paid more attention.

Saddle-node bifurcations of multiple homoclinic solutions in ODES

We study codimension 3 degenerate homoclinic bifurcations under periodic perturbations. Assume that among the 3 bifurcation equations, one is due to the homoclinic tangecy along the orbital direction. To the lowest order, the bifurcation equations become 3 quadratic equations. Under generic conditions on perturbations of the normal and tangential directions of the homoclinic orbit, up to 8 homoclinic orbits can be created through saddle-node bifurcations. Our results generate the homoclinic tangency bifurcation in Guckenheimer and Holmes [ 8 ].

CODIAGONALIZATION OF MATRICES AND EXISTENCE OF MULTIPLE HOMOCLINIC SOLUTIONS

The purpose of this paper is twofold. First, we use Lagrange's method and the generalized eigenvalue problem to study systems of two quadratic equations. We find exact conditions so the system can be codiagonalized and can have up to 4 solutions. Second, we use this result to study homoclinic bifurcations for a periodically perturbed system. The homoclinic bifurcation is determined by 3 bifurcation equations. To the lowest order, they are 3 quadratic equations, which can be simplified by the codiagonalization of quadratic forms. We find that up to 4 transverse homoclinic orbits can be created near the degenerate homoclinic orbit.

Bifurcations in Delay Differential Equations: An Algorithmic Approach in Frequency Domain

In this work we study local oscillations in delay differential equations with a frequency domain methodology. The main result is a bifurcation equation from which the existence and expressions of local periodic solutions can be determined. We present an iterative method to obtain the bifurcation equation up to a fixed arbitrary order. It is shown how this method can be implemented in symbolic math programs.

Bifurcation analysis of reduced rotor model based on nonlinear transient POD method

The singularity theory is applied to study the bifurcation behaviors of a reduced rotor model obtained by nonlinear transient POD method in this paper. A six degrees of freedom (DOFs) rotor model with cubically nonlinear stiffness supporting at both ends is established by the Newton's second law. The nonlinear transient POD method is used to reduce a six-DOFs model to a one-DOF one. The reduced model reserves the dynamical characteristics and occupies most POM energy of the original one. The singularity of the reduced system is analyzed, which replaces the original system. The bifurcation equation of the reduced model indicates that it is a high co-dimension bifurcation problem with co-dimension 6, and the universal unfolding (UN) is provided. The transient sets of six unfolding parameters, the bifurcation diagrams between the bifurcation parameter and the state variable are plotted. The results obtained in this paper present a new kind of method to study the UN theory of multi-DOFs rotor system.

Equivariant bifurcations in four-dimensional fixed point spaces

ABSTRACTIn this paper we continue the study of group representations which are counterexamples to the Ize conjecture. As in previous papers we find new infinite series of finite groups leading to such counterexamples. These new series are quite different from the previous ones, for example the group orders do not form an arithmetic progression. However, as before we find Lie groups which contain all these groups. This additional structure was observed, but not used in the previous studies of this problem. Here we also investigate the related bifurcations. To a large extent, these are closely related to the presence of mentioned compact Lie group containing the finite groups. This might give a tool to study the bifurcations related to all low dimensional counterexamples of the Ize conjecture. It also gives an indication of where we can expect to find examples where the bifurcation behaviour is different from what we have seen in the known examples.

Bifurcation and stability analysis for 3SPS+1PS parallel hip joint manipulator based on unified theory

For a parallel hip joint manipulator, the unified kinematics and stiffness model are established based on a novel unified theory, and then the bifurcation and stability are analyzed under the same unified theory framework. In bifurcation analysis, a chaos method is first applied to solve the non-linear bifurcation equations in order to get the full configuration of the parallel hip joint manipulator, which improves the convergence rate and accuracy. Based on the full-configuration solution, the single-parameter and double-parameter simulation for the bifurcation and stability of the parallel hip joint manipulator is performed. The bifurcation simulation results show that the configuration only changes along the corresponding path but cannot change to other paths when the configuration of the parallel hip joint manipulator is at a certain path. The stability simulation results show that when the parallel hip joint manipulator enters into an uncontrolled domain of a bifurcation posture along different paths, the posture component which changes dramatically will lose control first, and the other posture components will move along the changed configuration. In this paper, the kinematics, stiffness, bifurcation and stability of the parallel hip joint manipulator are solved under the same theory framework, which improves the solving efficiency and enriches the mechanical theory for the parallel manipulators.

Equivariant Hopf-Pitchfork Bifurcation of Symmetric Coupled Neural Network with Delay

This paper is concerned with how the symmetry and singularity of a system of differential equations affect generic dynamics and bifurcations. By computation of Hopf-pitchfork point in a two-parameter nonlinear problem satisfying a [Formula: see text]-symmetry condition, the mode interactions in two-parameter bifurcations with a single zero and two pairs of imaginary roots are considered. The codimension two normal form with equivariant Hopf-pitchfork bifurcations are given. Through analyzing the unfolding structure, local classification in the neighborhood of equivariant Hopf-pitchfork bifurcations point for the [Formula: see text]-symmetric is undertaken. A rich variety of dynamical and bifurcation behaviors is pointed out. Beyond a stable fixed point or a pair of stable fixed points, some interesting phenomena are also found, such as the coexistence of two periodic solutions which are verified both theoretically and numerically.

Standing Waves in Near-Parallel Vortex Filaments

A model derived in (Klein et al., J Fluid Mech 288:201–248, 1995) for n near-parallel vortex filaments in a three dimensional fluid region takes into consideration the pairwise interaction between the filaments along with an approximation for motion by self-induction. The same system of equations appears in descriptions of the fine structure of vortex filaments in the Gross–Pitaevski model of Bose–Einstein condensates. In this paper we construct families of standing waves for this model, in the form of n co-rotating near-parallel vortex filaments that are situated in a central configuration. This result applies to any pair of vortex filaments with the same circulation, corresponding to the case n = 2. The model equations can be formulated as a system of Hamiltonian PDEs, and the construction of standing waves is a small divisor problem. The methods are a combination of the analysis of infinite dimensional Hamiltonian dynamical systems and linear theory related to Anderson localization. The main technique of the construction is the Nash–Moser method applied to a Lyapunov–Schmidt reduction, giving rise to a bifurcation equation over a Cantor set of parameters.

Postcritical behavior of a discrete Nicolai column

The Nicolai paradox, concerning the loss of stability of a column subjected to an evanescent follower torque, so far analyzed in the literature in the linear field, is here studied in the nonlinear regime. A paradigmatic, discrete, minimal mechanical system is introduced, roughly describing the column. The bifurcation equations ruling the dynamics around a semi-simple Hopf bifurcation are derived via the Multiple Scale Method, and the postcritical behavior of the system is investigated. A slight asymmetry of the model, which splits the critical eigenvalues, is also taken into account. It is shown that, while the trivial equilibrium configuration of the system is rendered unstable by vanishingly small circulatory forces, nonlinearities can, under certain circumstances, limit the amplitude of the oscillations. The existence of unsafe initial conditions is discussed, and the dangerous effects of the follower torque, also in nonlinear regime, are highlighted.

Stability analysis of a nonlinear electromechanical coupling transmission system with time delay feedback

The static bifurcation characteristic and the stability of a nonlinear electromechanical coupling system with time delay are studied in this paper. Considering the energy in air-gap field of AC motor, the dynamic equation of electromechanical coupling transmission system is deduced by using the dissipation Lagrange equation. The equivalent low-dimensional bifurcation equation is obtained by using the method of Lyapunov–Schmidt reduction. And the bifurcation characteristic is analyzed by the method of the singularity theory. In order to control the dynamic behavior of the system, a time delay feedback is introduced and the distribution of the roots is studied. The analytical conditions that determine the feedback gain and time delay are given based on the delay differential equation stability theory. Under certain conditions, it is shown that the zero solution is locally asymptotically stable when the time delay is suitably small, while the change of stability of zero solution will cause a bifurcating periodic solution as the time delay passes through a certain critical value. Numerical simulations are also performed, which confirm the analytical results.

Degenerate bifurcation of the rotating patches

In this paper we study the existence of doubly-connected rotating patches for Euler equations when the classical non-degeneracy conditions are not satisfied. We prove the bifurcation of the V-states with two-fold symmetry, however for higher m-fold symmetry with m≥3 the bifurcation does not occur. This answers to a problem left open in [10]. Note that, contrary to the known results for simply-connected and doubly-connected cases where the bifurcation is pitchfork, we show that the degenerate bifurcation is actually transcritical. These results are in agreement with the numerical observations recently discussed in [10]. The proofs stem from the local structure of the quadratic form associated to the reduced bifurcation equation.

Synchronization of traveling waves in a dispersive system of weakly coupled equations

The system of weakly coupled differential equations describing traveling waves in dispersive media is considered. The Lyapunov — Schmidt construction is used to study the branching of cnoidal-type periodic solutions. The analysis of bifurcation equations uses the group symmetry and cosymmetry of original equations. Sufficient condition for existence of the phase-shifted modes of cnoidal waves is formulated in terms of the Pontryagin's function determined by the nonlinear perturbation terms

Bifurcation and chaos for electromechanical coupling torsional vibration in rolling mill system driven by DC motor

Bifurcation and chaos for electromechanical coupling torsional vibration in rolling mill system driven by DC motor is studied in this paper. Considering the electromechanical devices under unsaturated magnetic circuit, the dynamical equation of electromechanical coupling vibration in rolling mill drive system is deduced by using the dissipation Lagrange equation. The equivalent low-dimensional bifurcation equation, which can reveal the influence of system parameters on the nonlinear dynamic characteristics, is obtained by reducing the dimensionality system using the method of Lyapunov-Schmidt reduction, and the static bifurcation characteristic is analyzed by using the singularity theory. The torsional rigidity of rotating shaft and harmonic excitation are chosen as the bifurcation parameters. Finally, the simulation is carried out with actual parameters, the bifurcation diagram, Poincare map and the maximum Lyapunov exponent are also given, which show the influence of bifurcation parameters on the chaotic motions of the system.