Double-zero Bifurcation Research Articles

A double-zero bifurcation in a Lorenz-like system

The Lorenz system presents a double-zero bifurcation (a double-zero eigenvalue with geometric multiplicity two). However, its study by means of standard techniques is not possible because it occurs for a non-isolated equilibrium. To circumvent this difficulty, we add in the third equation a new term, Dz2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Dz^2$$\\end{document}. In this Lorenz-like system, the analysis of the double-zero bifurcation of the equilibrium at the origin guarantees, for certain values of the parameters, the existence of a heteroclinic cycle between the two equilibria located on the z-axis. The numerical continuation in parameter space of the locus of heteroclinic connections allows to detect various degeneracies of codimension two and three, some of which have not been previously studied in the literature. These bifurcations are organizing centers of the complicated dynamics exhibited by this system. Furthermore, studying how the bifurcation sets evolve when D tends to zero, we are able to explain, in the Lorenz system, the origin of several global connections which are related to T-point heteroclinic loops.

On the Double-Zero Bifurcation of Jerk Systems

In this paper, we construct approximate normal forms of the double-zero bifurcation for a two-parameter jerk system exhibiting a non-degenerate fold bifurcation. More precisely, using smooth invertible variable transformations and smooth invertible parameter changes, we obtain normal forms that are also jerk systems. In addition, we discuss some of their parametric portraits.

Double-zero degeneracy and heteroclinic cycles in a perturbation of the Lorenz system

In this paper we consider a 3D three-parameter unfolding close to the normal form of the triple-zero bifurcation exhibited by the Lorenz system. First we study analytically the double-zero degeneracy (a double-zero eigenvalue with geometric multiplicity two) and two Hopf bifurcations. We focus on the more complex case in which the double-zero degeneracy organizes several codimension-one singularities, namely transcritical, pitchfork, Hopf and heteroclinic bifurcations. The analysis of the normal form of a Hopf-transcritical bifurcation allows to obtain the expressions for the corresponding bifurcation curves. A degenerate double-zero bifurcation is also considered. The theoretical information obtained is very helpful to start a numerical study of the 3D system. Thus, the presence of degenerate heteroclinic and homoclinic orbits, T-point heteroclinic loops and chaotic attractors is detected. We find numerical evidence that, at least, four curves of codimension-two global bifurcations are related to the triple-zero degeneracy in the system analyzed.

Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition

A diffusive predator-prey model with nonlocal prey competition and the homogeneous Neumann boundary conditions is considered, to explore the effects of nonlocal reaction term. Firstly, conditions of the occurrence of Hopf, Turing, Turing-Turing and double zero bifurcations, are established. Then, several concise formulas of computing normal form at a double zero singularity for partial functional differential equations, are provided. Next, via analyzing normal form derived by utilizing these formulas, we find that diffusive predator-prey system admits interesting spatiotemporal dynamics near the double zero singularity, like tristable phenomenon that a stable spatially inhomogeneous periodic solution with the shape of \begin{document}$ \cos\omega_0 t\cos\frac{x}{l}- $\end{document} like which is unstable in model without nonlocal competition and also greatly different from these with the shape of \begin{document}$ \cos\omega_0 t+\cos\frac{x}{l}- $\end{document} like resulting from Turing-Hopf bifurcation, coexists with a pair of spatially inhomogeneous steady states with the shape of \begin{document}$ \cos\frac{x}{l}- $\end{document} like. At last, numerical simulations are shown to support theory analysis. These investigations indicate that nonlocal reaction term could stabilize spatially inhomogeneous periodic solutions with the shape of \begin{document}$ \cos\omega_0 t\cos\frac{kx}{l}- $\end{document} like for reaction-diffusion systems subject to the homogeneous Neumann boundary conditions.

Approximations of the Heteroclinic Orbits Near a Double-Zero Bifurcation with Symmetry of Order Two. Application to a Liénard Equation

A dynamical system possessing an equilibrium point with two zero eigenvalues is considered. We assume that a degenerate Bogdanov–Takens bifurcation with symmetry of order two is present and, in the parameter space, a curve of heteroclinic bifurcation values emerges from the codimension two bifurcation point. Using a blow-up transformation and a perturbation method, we obtain second order approximations both for the heteroclinic orbits and for the curve of heteroclinic bifurcation values. Applications of our results for the truncated normal form and for a Liénard equation are presented. Some numerical simulations illustrating the accuracy of our results are performed.

Homoclinic and Heteroclinic Orbits for a Liénard System

A 2D dynamical system exhibiting a double-zero bifurcation with symmetry of order two is considered. This bifurcation involves the presence in the parameter space of a curve corresponding either to double homoclinic or to heteroclinic bifurcations. In this paper we derive second order approximations for the homoclinic orbits and for the curve of homoclinic bifurcation values considering the system truncated up to five order terms and parameter-dependent coefficients. These approximations were obtained using the regular perturbation method. These formulae are applied to a Lienard system, which develops a double-zero bifurcation with symmetry of order two for some parameters values. Second order approximations for the heteroclinic orbits of this system are also given. The analytical results are very accurate and they are in good accordance with the numerical ones.

Approximations of the Homoclinic Orbits Near a Double-Zero Bifurcation with Symmetry of Order Two

The two-dimensional system of differential equations corresponding to the normal form of the double-zero bifurcation with symmetry of order two is considered. This is a codimension two bifurcation. The associated dynamical system exhibits, among others, a homoclinic bifurcation. In this paper, we obtain second order approximations both for the curve of parametric values of homoclinic bifurcation and for the homoclinic orbits. To perform this task, we reduce first the normal form to a perturbed Hamiltonian system, using a blow-up technique. Then, by means of a perturbation method, we determine explicit first and second order approximations of the homoclinic orbits. The solutions obtained theoretically are compared with those obtained numerically for several cases. Finally, an application of the obtained results is presented.

Computation of universal unfolding of the double-zero bifurcation in -symmetric systems by a homological method

We present a homological method and explicit computational formulae for normal form and universal unfolding of the double-zero bifurcation in any autonomous ordinary differential equation system with -symmetry. We take the Chua's circuit as an example to show advantages of this method.

Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads

The combined effects of conservative and non-conservative loads on the mechanical behavior of an unshearable and inextensional visco-elastic beam, close to bifurcation, are investigated. The equations of motion and boundary conditions are derived via a constrained variational principle, and the Lagrange multiplier successively condensed, to get integro-differential equations. These latter, with the mechanical boundary conditions appended, are put in an operator-form, amenable to perturbation analysis. A linear stability analysis is carried out in the space of the two loading parameter, displaying the existence of codimension-1 and codimension-2 bifurcations. The influence of both internal and external damping on this scenario is thoroughly investigated. A post-critical analysis is carried out around a double-zero bifurcation, by using an adapted version of the multiple scale method, based on fractional series expansions in the perturbation parameter. The integro-differential problem is directly attacked, so that any a priori discretization is avoided. Emphasis is given to the interaction between the two damping coefficients. This reveals the existence, also in the non-linear range, of a phenomenon of destabilization, so far known only in the linear range.

Computation of universal unfolding of the double zero bifurcation in the Z2-symmetric system

We present explicit formulae of universal unfolding of the double zero bifurcation for any autonomous ordinary differential equation systems (ODEs) with Z2-symmetry. These formulae only need to use coefficients of the Taylor expansion of the vector field at its equilibria and are equally suitable for both numerical and symbolic computations, and they construct the relationship between original parameters and topological structures for the double zero bifurcation analysis in any autonomous ODEs. They can be directly used to check whether there exist the original parameters such that a system can exhibit the versal double zero bifurcation. Moreover, we can use them to compute the corresponding bifurcation curves with high precision. We take Chua's circuit as an example to demonstrate the advantages of these formulae.

Bifurcation, stability, and cluster formation of multi-strain infection models

Clustering behaviours have been found in numerous multi-strain transmission models. Numerical solutions of these models have shown that steady-states, periodic, or even chaotic motions can be self-organized into clusters. Such clustering behaviours are not a priori expected. It has been proposed that the cross-protection from multiple strains of pathogens is responsible for the clustering phenomenon. In this paper, we show that the steady-state clusterings in existing models can be analytically predicted. The clusterings occur via semi-simple double zero bifurcation from the quotient networks of the models and the patterns which follow can be predicted through the stability analysis of the bifurcation. We calculate the stability criteria for the clustering patterns and show that some patterns are inherently unstable. Finally, the biological implications of these results are discussed.

BIFURCATION ANALYSIS OF DAMPED VISCO-ELASTIC PLANAR BEAMS UNDER SIMULTANEOUS GRAVITATIONAL AND FOLLOWER FORCES

The mechanical behavior of a non-conservative non-linear beam, internally and externally damped, undergoing codimension-1 (static or dynamic) and codimension-2 (double-zero) bifurcations, is analyzed. The system consists of a purely flexible, planar, visco-elastic beam, fixed at one end, loaded at the tip by a follower force and a dead load, acting simultaneously. An integro-differential equation of motion in the transversal displacement, with relevant boundary conditions, is derived. Then, the linear stability diagram of the trivial rectilinear configuration is built-up in the space of the two loading parameters. Attention is then focused on the double-zero bifurcation, for which a post-critical analysis is carried out without any a-priori discretization. An adapted version of the Multiple Scale Method, based on a fractional series expansion in the perturbation parameter, is employed to derive the bifurcation equations. Finally, bifurcation charts are evaluated, able to illustrate the system behavior around the codimension-2 bifurcation point.

Divergence, Hopf and double-zero bifurcations of a nonlinear planar beam

The multiple scale method is directly applied to a one-dimensional continuous model to derive the equations governing the system asymptotic dynamic around a bifurcation point. The theory is illustrated with reference to a specific example, namely an internally constrained planar beam, equipped with a lumped visco-elastic device and loaded by a follower force. Nonlinear, integro-differential equations of motion are derived and expanded upto cubic terms in the transversal displacements and velocities of the beam. The linear stability of the trivial equilibrium is first studied. It reveals the existence of divergence and Hopf and double-zero bifurcations. The spectral properties of the linear operator are studied at the bifurcation points by obtaining closed-form expressions. Notably, the system posses an incomplete system of eigenvectors at the double-zero point (i.e. it is defective or nilpotent), thus entailing the need to find a generalized eigenvector. A multiple-scale analysis is then performed for the three bifurcations and the relevant bifurcation equations are derived directly in their normal forms. Finally, they are numerically studied to furnish a comprehensive scenario of the postcritical behaviour around the bifurcations.

Flexural–torsional bifurcations of a cantilever beam under potential and circulatory forces II. Post-critical analysis

The post-critical behavior of a cantilever beam with rectangular cross-section, under simultaneous action of conservative and non-conservative loads, is analyzed. An internally constrained Cosserat rod model is adopted to describe the dynamics of the beam in finite displacement regime. The bifurcation equations for simple buckling (divergence), simple flutter (Hopf) and double-zero (Takens–Bogdanova–Arnold) bifurcations are derived by means of the multiple time scales method. Due to the nilpotent eigenvalue at the double-zero critical point, the evaluation of the generalized Keldysh's eigenfunctions is required. Finally, some numerical results are shown and the bifurcation scenario of the beam is discussed.

Control of oscillations from the k-zero bifurcation

In this paper, an analytical method for the analysis and control of oscillations in non-linear control systems, whose linearization around the origin has k eigenvalues zero, is presented. The main idea consists in exploit, for the particular case of the double-zero (or Takens–Bognadov) bifurcation, the existence of a curve of Hopf bifurcation points on its versal deformation, to control oscillations. Then the general case is reduced to the double-zero case through a change of coordinates and a change in the input control. The method is illustrated with the pendubot, an underactuated robot manipulator of two degrees of freedom.

Bifurcation Equations Through Multiple-Scales Analysis for a Continuous Model of a Planar Beam

The Multiple-Scale Method is applied directly to a one-dimensional continuous model to derive the equations governing the asymptotic dynamic of the system around a bifurcation point. The theory is illustrated with reference to a specific example, namely an internally constrained planar beam, equipped with a lumped viscoelastic device and loaded by a follower force. Nonlinear, integro-differential equations of motion are derived and expanded up to cubic terms in the transversal displacements and velocities of the beam. They are put in an operator form incorporating the mechanical boundary conditions, which account for the lumped viscoelastic device; the problem is thus governed by mixed algebraic-integro-differential operators. The linear stability of the trivial equilibrium is first studied. It reveals the existence of divergence, Hopf and double-zero bifurcations. The spectral properties of the linear operator and its adjoint are studied at the bifurcation points by obtaining closed-form expressions. Notably, the system is defective at the double-zero point, thus entailing the need to find a generalized eigenvector. A multiple-scale analysis is then performed for the three bifurcations and the relevant bifurcation equations are derived directly in their normal forms. Preliminary numerical results are illustrated for the double-zero bifurcation.

ON THE BEHAVIOR OF A SELF-EXCITING FARADAY DISK HOMOPOLAR DYNAMO WITH A VARIABLE NONLINEAR SERIES MOTOR

In this paper we seek to bridge the gap between the study of a self-exciting Faraday disk homopolar dynamo with a linear series motor [Hide et al., 1996] and the case when the torque acting on the armature of the motor is proportional to the square of the current flowing through the dynamo [Hide, 1998]. We also focus on the issue of when the nonlinear quenching of oscillatory solutions can occur. The present study is a special case of the more general problem when azimuthal eddy currents are permitted to flow [Moroz & Hide, 2000] and shares with that problem the existence of multiple steady states and Hopf bifurcations. This results in distinct double-zero bifurcations for the trivial and the nontrivial equilibrium states as well as other codimension-two bifurcations, which leads to the suppression of oscillatory solutions.

Quadratic Stabilization of Nonlinear Control Systems with a Double-Zero Bifurcation

In this paper, we determine the normal forms for control systems with a double-zero bifurcation. Based on the normal forms, we find invariants, stabilizability conditions and synthesize a quadratic stabilizing controller.

EFFECTS DUE TO INDUCED AZIMUTHAL EDDY CURRENTS IN A SELF-EXCITING FARADAY DISK HOMOPOLAR DYNAMO WITH A NONLINEAR SERIES MOTOR II: THE GENERAL CASE

This paper forms the second part of a two-part study into the effects of azimuthal eddy currents in the Faraday disk self-exciting homopolar dynamo, connected in series with the coil when the applied couple driving the disk is steady. The Lorentz couple driving the armature of the motor is a general quadratic function I(1 - ε + εSI) of the current I(t), where 0 ≤ ε ≤ 1. Here we investigate how cases with 0 < ε < 1 relate to the two special cases of ε = 0 and ε = 1, considered in Part I of our study [Hide & Moroz, 1999]. One key difference is that the lack of reflectional symmetry in the general ε problem means that the linear stability curves for the onset of both steady and oscillatory behavior for both the trivial and the nontrivial equilibrium solutions no longer coincide. This results in distinct Takens–Bogdanov double-zero bifurcations for these states, as well as multiple branches to the Hopf bifurcation curves, associated with bifurcations from the nontrivial equilibrium states. Multiple bifurcations involving simultaneous steady and nondegenerate oscillatory solutions are also possible.

Sensitivities and Linear Stability Analysis Around a Double-Zero Eigenvalue

A general, multiparameter system admitting a double-zero eigenvalue at a critical equilibrium point is considered. A sensitivity analysis of the critical eigenvalues is performed to explore the neighborhood of the critical point in the parameter space. Because the coalescence of the eigenvalues implies that the Jacobian matrix is defective (or nilpotent ), well-suited techniques of perturbation analysis must be employed to evaluate the eigenvalues and the eigenvector sensitivities. Different asymptotic methods are used, based on perturbations both of the eigenvalue problem and the characteristic equation. The analysis reveals the existence of a generic (nonsingular ) case and of a nongeneric (singular) case. However, even in the generic case, a codimension-1 subspace exists in the parameter space on which a singularity occurs. By the use of the relevant asymptotic expansions, linear stability diagrams arebuiltup, and different bifurcation mechanisms (divergence‐Hopf, doubledivergence, doubledivergence ‐Hopf, degenerateHopf )arehighlighted. The problem of e nding a uniqueexpression uniformly valid in the wholespaceis then addressed. It isfound that a second-degreealgebraic equation governsthebehaviorofthecriticaleigenvalues. It also permits clarie cation of the geometrical meaning of the unfolding parameters, which has been discussed in literature for the Takens ‐Bogdanova bifurcation. Finally, a mechanical system loaded by nonconservative forces and exhibiting a double-zero bifurcation is studied as an example.