Bogdanov Bifurcation Research Articles

Nonlinear rotating convection in a sparsely packed porous medium

We investigate linear and weakly nonlinear properties of rotating convection in a sparsely packed Porous medium. We obtain the values of Takens–Bogdanov bifurcation points and co-dimension two bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters relevant to rotating convection in a sparsely packed porous medium near a supercritical pitchfork bifurcation. We derive a nonlinear two-dimensional Landau–Ginzburg equation with real coefficients by using Newell–Whitehead method [16]. We investigate the effect of parameter values on the stability mode and show the occurrence of secondary instabilities viz., Eckhaus and Zigzag Instabilities. We study Nusselt number contribution at the onset of stationary convection. We derive two nonlinear one-dimensional coupled Landau–Ginzburg type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation and discuss the stability regions of standing and travelling waves.

Non-linear convection due to compositional and thermal buoyancy in Earth's outer core

The linear stability of convection due to compositional and thermal buoyancy in Earth's outer core has been investigated. We have obtained the values of Takens–Bogdanov bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters. We have derived a non-linear two-dimensional Landau–Ginzburg equation with real coefficients near the onset of stationary convection at a supercritical pitchfork bifurcation and two non-linear one-dimensional coupled Landau–Ginzburg type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation. We have studied Nusselt number contribution from a Landau–Ginzburg equation at the onset of stationary convection. We have discussed the stability regions of standing and travelling waves. We have also discussed the occurrence of secondary instabilities such as Eckhaus, zigzag and Benjamin–Feir instabilities. We have also derived the non-linear amplitude equation near the Takens–Bogdanov bifurcation point.

Takens–Bogdanov bifurcation of travelling-wave solutions in pipe flow

The appearance of travelling-wave-type solutions in pipe Poiseuille flow that are disconnected from the basic parabolic profile is numerically studied in detail. We focus on solutions in the twofold azimuthally-periodic subspace because of their special stability properties, but relate our findings to other solutions as well. Using time-stepping, an adapted Krylov–Newton method and Arnoldi iteration for the computation and stability analysis of relative equilibria, and a robust pseudo-arclength continuation scheme, we unfold a double-zero (Takens–Bogdanov) bifurcating scenario as a function of Reynolds number (Re) and wavenumber (κ). This scenario is extended, by the inclusion of higher-order terms in the normal form, to account for the appearance of supercritical modulated waves emanating from the upper branch of solutions at a degenerate Hopf bifurcation. We provide evidence that these modulated waves undergo a fold-of-cycles and compute some solutions on the unstable branch. These waves are shown to disappear in saddle-loop bifurcations upon collision with lower-branch solutions, in accordance with the bifurcation scenario proposed. The travelling-wave upper-branch solutions are stable within the subspace of twofold periodic flows, and their subsequent secondary bifurcations could contribute to the formation of the phase space structures that are required for turbulent dynamics at higher Re.

ANALYSIS OF THE TAKENS–BOGDANOV BIFURCATION ON m-PARAMETERIZED VECTOR FIELDS

Given an m-parameterized family of n-dimensional vector fields, such that: (i) for some value of the parameters, the family has an equilibrium point, (ii) its linearization has a double zero eigenvalue and no other eigenvalue on the imaginary axis, sufficient conditions on the vector field are given such that the dynamics on the two-dimensional center manifold is locally topologically equivalent to the versal deformation of the planar Takens–Bogdanov bifurcation.

HYPERCHAOTIC SYNCHRONIZATION UNDER SQUARE SYMMETRY

We study two identical hyperchaotic oscillators symmetrically coupled. Each oscillator represents a codimension-2 Takens–Bogdanov bifurcation under square symmetry, and was used to model a convection experiment in a time-dependent state. In the coupled system, the Lyapunov exponents behavior against the coupling parameter is used to detect changes in the dynamics, and the synchronization state is controlled by checking the phase planes. Complete synchronization is achieved without chaos suppression in a coupling parameter interval. Outside this window, complete synchronization cannot be generally achieved. As a consequence of a bubbling transition, synchronization is obtained only for some particular values of the initial conditions.

Stability diagram for the forced Kuramoto model

We analyze the periodically forced Kuramoto model. This system consists of an infinite population of phase oscillators with random intrinsic frequencies, global sinusoidal coupling, and external sinusoidal forcing. It represents an idealization of many phenomena in physics, chemistry, and biology in which mutual synchronization competes with forced synchronization. In other words, the oscillators in the population try to synchronize with one another while also trying to lock onto an external drive. Previous work on the forced Kuramoto model uncovered two main types of attractors, called forced entrainment and mutual entrainment, but the details of the bifurcations between them were unclear. Here we present a complete bifurcation analysis of the model for a special case in which the infinite-dimensional dynamics collapse to a two-dimensional system. Exact results are obtained for the locations of Hopf, saddle-node, and Takens–Bogdanov bifurcations. The resulting stability diagram bears a striking resemblance to that for the weakly nonlinear forced van der Pol oscillator.

Control of the Planar Takens–Bogdanov Bifurcation with Applications

It is well-known that on a versal deformation of the Takens–Bogdanov bifurcation is possible to find dynamical systems that undergo saddle-node, Hopf, and homoclinic bifurcations. In this document a nonlinear control system in the plane is considered, whose nominal vector field has a double-zero eigenvalue, and then the idea is to find under which conditions there exists a scalar control law such that be possible establish a priori, that the closed-loop system undergoes any of the three bifurcations: saddle-node, Hopf or homoclinic. We will say then that such system undergoes the controllable Takens–Bogdanov bifurcation. Applications of this result to the averaged forced van der Pol oscillator, a population dynamics, and adaptive control systems are discussed.

Homoclinic orbits and Hopf bifurcations in delay differential systems with T–B singularity

The paper carries the results on Takens–Bogdanov bifurcation obtained in [T. Faria, L.T. Magalhães, Normal forms for retarded functional differential equations and applications to Bogdanov–Takens singularity, J. Differential Equations 122 (1995) 201–224] for scalar delay differential equations over to the case of delay differential systems with parameters. Firstly, we give feasible algorithms for the determination of Takens–Bogdanov singularity and the generalized eigenspace associated with zero eigenvalue in R n . Next, through center manifold reduction and normal form calculation, a concrete reduced form for the parameterized delay differential systems is obtained. Finally, we describe the bifurcation behavior of the parameterized delay differential systems with T–B singularity in detail and present an example to illustrate the results.

Comparison of the bifurcation curves of a two-variable and a three-variable circadian rhythm model

Two dynamical systems describing the circadian fluctuation of two proteins (PER and TIM) in cells are compared. A simplified model with two variables has already been investigated. Detailed study of the possible bifurcation has been carried out. Periodic solutions of the differential equations with 24-h period have been obtained numerically. Here the general, more realistic model having three variables is investigated. The possible phase portraits and local bifurcations are studied in detail. The saddle-node and Hopf-bifurcation curves are determined in the plane of two parameters by using the parametric representation method. Using these curves the number and the type of the stationary points can be determined. The relation of the two bifurcation curves and the Takens–Bogdanov bifurcation points are also studied. The bifurcation curves are compared to those obtained for the simplified two-variable system.

RESONANCES OF PERIODIC ORBITS IN RÖSSLER SYSTEM IN PRESENCE OF A TRIPLE-ZERO BIFURCATION

This paper focuses on resonance phenomena that occur in a vicinity of a linear degeneracy corresponding to a triple-zero eigenvalue of an equilibrium point in an autonomous tridimensional system. Namely, by means of blow-up techniques that relate the triple-zero bifurcation to the Kuramoto–Sivashinsky system, we characterize the resonances that appear near the triple-zero bifurcation. Using numerical tools, the results are applied to the Rössler equation, showing a number of interesting bifurcation behaviors associated to these resonance phenomena. In particular, the merging of the periodic orbits appeared in resonances, the existence of two types of Takens–Bogdanov bifurcations of periodic orbits and the presence of Feigenbaum cascades of these bifurcations, joined by invariant tori curves, are pointed out.

Instabilities in spatially extended predator–prey systems: Spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations

We investigate the emergence of spatio-temporal patterns in ecological systems. In particular, we study a generalized predator–prey system on a spatial domain. On this domain diffusion is considered as the principal process of motion. We derive the conditions for Hopf and Turing instabilities without specifying the predator–prey functional responses and discuss their biological implications. Furthermore, we identify the codimension-2 Turing–Hopf bifurcation and the codimension-3 Turing–Takens–Bogdanov bifurcation. These bifurcations give rise to complex pattern formation processes in their neighborhood. Our theoretical findings are illustrated with a specific model. In simulations a large variety of different types of long-term behavior, including homogenous distributions, stationary spatial patterns and complex spatio-temporal patterns, are observed.

Symbolic computation for center manifolds and normal forms of Bogdanov bifurcation in retarded functional differential equations

In this paper, we present explicit formulas for computing the coefficients of a center manifold for the Bogdanov singularity in autonomous retarded functional differential equations with parameters. As a consequence, normal forms associated with the flow on a center manifold up to an arbitrary order are derived. The explicit formulas have been implemented using the computer algebra system Maple. The paper ends with some examples given in order to show the applicability of the methodology and the convenience of the computer software.

Spatial nonlinear dynamics near principal parametric resonance for a fluid-conveying cantilever pipe

Abstract Flow-induced vibrations of a fluid-conveying cantilever pipe are examined theoretically under the condition that the fluid velocity has a small harmonic pulsatile component. More specifically, the case of principal parametric resonance is considered for the pipe free to undergo three-dimensional motions. The mean flow is considered to be near the critical flow rate at which the tube undergoes a Hopf bifurcation into self-excited oscillations. When the governing equations of motion for the tube with steady flow are reduced to those on the center manifold in the neighborhood of Hopf bifurcation, the normal form equations are O(2)-equivariant. The weak harmonic fluctuations due to pulsatile flow result in symmetry-breaking terms in the normal form. The eigenvalues of an O(2)-equivariant system undergoing a symmetry-breaking Hopf bifurcation have multiplicity two. When an additive linear term, arising from time-periodic modulations of the original dynamic system, is introduced into the normal form, the symmetry-breaking bifurcation structure for the trivial solution splits into three categories: a steady-state bifurcation giving rise to standing wave fixed-point solutions, a Hopf bifurcation giving rise to two-frequency solutions, and an O(2)-Takens–Bogdanov bifurcation. The resulting dynamics in each case are studied along with secondary and tertiary bifurcations. The dynamics of the tube system are studied as a function of the mean flow rate and the frequency of flow fluctuations. Amplitude response diagrams constructed for a specific example tube system using the continuation and bifurcation analysis software package AUTO illustrate the variety of possible behaviors.

A bifurcation analysis of a simple electronic circuit

This paper presents a bifurcation analysis of a simple electronic device, with only one nonlinearity, exhibiting complex dynamics. We focus on the study of a Takens–Bogdanov bifurcation, which is degenerate for some parameter values. In this way, we detect several types of periodic and homoclinic dynamical behaviors.

SOME RESULTS ON CHUA'S EQUATION NEAR A TRIPLE-ZERO LINEAR DEGENERACY

In this work we study a wide class of symmetric control systems that has the Chua's circuit as a prototype. Namely, we compute normal forms for Takens–Bogdanov and triple-zero bifurcations in a class of symmetric control systems and determine the local bifurcations that emerge from such degeneracies. The analytical results are used as a first guide to detect numerically several codimension-three global bifurcations that act as organizing centres of the complex dynamics Chua's circuit exhibits in the parameter range considered. A detailed (although partial) bifurcation set in a three'parameter space is presented in this paper. We show relations between several high-codimension bifurcations of equilibria, periodic orbits and global connections. Some of the global bifurcations found have been neither analytically nor numerically treated in the literature.

Explicit centre manifold reduction and full bifurcation analysis for an astigmatic Maxwell–Bloch laser

Using an explicit centre manifold reduction, we set up a general framework to study the bifurcations of nearly degenerate modes of a rather general laser model. We then apply it specifically to the study of interactions of two modes of a Maxwell–Bloch laser with broken circular symmetry. Complex bifurcation sequences involving single-mode solutions, mode locking and mode beating regimes are predicted. These sequences are organised by a Z 2-symmetric Takens–Bogdanov bifurcation together with a zero-detuning Hopf bifurcation with 1:1 resonance that leads to an unexpected cyclic spatio-temporal symmetry of order 4. Numerical simulations of the original system show good agreement with theory.

Global bifurcations in the Takens–Bogdanov normal form with D4 symmetry near the O(2) limit

The dynamics of the normal form of the Takens–Bogdanov bifurcation with D 4 symmetry is governed by a one-dimensional map near the gluing bifurcation and near the O(2) integrable limit, rather than the three-dimensional map one would expect. This great simplification allows a quantitative description of the bifurcation sequence through which stability is transferred between invariant subspaces.

TAKENS–BOGDANOV BIFURCATIONS OF PERIODIC ORBITS AND ARNOLD'S TONGUES IN A THREE-DIMENSIONAL ELECTRONIC MODEL

In this paper we study Arnold's tongues in a ℤ2-symmetric electronic circuit. They appear in a rich bifurcation scenario organized by a degenerate codimension-three Hopf–pitchfork bifurcation. On the one hand, we describe the transition open-to-closed of the resonance zones, finding two different types of Takens–Bogdanov bifurcations (quadratic and cubic homoclinic-type) of periodic orbits. The existence of cascades of the cubic Takens–Bogdanov bifurcations is also pointed out. On the other hand, we study the dynamics inside the tongues showing different Poincaré sections. Several bifurcation diagrams show the presence of cusps of periodic orbits and homoclinic bifurcations. We show the relation that exists between two codimension-two bifurcations of equilibria, Takens–Bogdanov and Hopf–pitchfork, via homoclinic connections, period-doubling and quasiperiodic motions.

Relationships between the discriminant curve and other bifurcation diagrams

The parameter dependence of the number and type of the stationary points of an ODE is considered. The number of the stationary points is determined by the saddle-node (SN) bifurcation set and their type (e.g., stability) is given by another bifurcation diagram (e.g., Hopf bifurcation set). The relation between these bifurcation curves on the parmeter plane is investigated. It is shown that the ‘cross-shaped diagram’, when the Hopf bifurcation curve makes a loop around a cusp point of the SN curve, is typical in some sense. It is proved that the two bifurcation curves meet tangentially at their common points (Takens–Bogdanov point), and these common points persist as a third parameter is varied. An example is shown that exhibits two different types of 3-codimensional degenerate Takens–Bogdanov bifurcation.

ON THE HOPF–PITCHFORK BIFURCATION IN THE CHUA'S EQUATION

We study some periodic and quasiperiodic behaviors exhibited by the Chua's equation with a cubic nonlinearity, near a Hopf–pitchfork bifurcation. We classify the types of this bifurcation in the nondegenerate cases, and point out the presence of a degenerate Hopf–pitchfork bifurcation. In this degenerate situation, analytical and numerical study shows a diversity of bifurcations of periodic orbits. We find a secondary Hopf bifurcation of periodic orbits, where invariant torus appears. This secondary Hopf bifurcation is bounded by a Takens–Bogdanov bifurcation of periodic orbits. Here, a sequence of period-doubling bifurcations of invariant tori is detected. Resonance phenomena are also analyzed. In the case of strong resonance 1:4, we show a new sequence of period-doubling bifurcations of 4T invariant tori.