Codimension-three Bifurcation Research Articles

Bifurcation scenario in the two-dimensional laminar flow past a rotating cylinder

Abstract

Elliptic Bubbles in Moser's 4D Quadratic Map: The Quadfurcation

Moser derived a normal form for the family of four-dimensional, quadratic, symplectic maps in 1994. This six-parameter family generalizes H\'enon's ubiquitous 2D map and provides a local approximation for the dynamics of more general 4D maps. We show that the bounded dynamics of Moser's family is organized by a codimension-three bifurcation that creates four fixed points---a bifurcation analogous to a doubled, saddle-center---which we call a quadfurcation. In some sectors of parameter space a quadfurcation creates four fixed points from none, and in others it is the collision of a pair of fixed points that re-emerge as two or possibly four. In the simplest case the dynamics is similar to the cross product of a pair of H\'enon maps, but more typically the stability of the created fixed points does not have this simple form. Up to two of the fixed points can be doubly-elliptic and be surrounded by bubbles of invariant two-tori; these dominate the set of bounded orbits. The quadfurcation can also create one or two complex-unstable (Krein) fixed points. Special cases of the quadfurcation correspond to a pair of weakly coupled H\'enon maps near their saddle-center bifurcations. The quadfurcation also occurs in the creation of accelerator modes in a 4D standard map.

Moser’s Quadratic, Symplectic Map

In 1994, J\"urgen Moser generalized H\'enon's area-preserving quadratic map to obtain a normal form for the family of four-dimensional, quadratic, symplectic maps. This map has at most four isolated fixed points. We show that the bounded dynamics of Moser's six parameter family is organized by a codimension-three bifurcation, which we call a quadfurcation, that can create all four fixed points from none. The bounded dynamics is typically associated with Cantor families of invariant tori around fixed points that are doubly elliptic. For Moser's map there can be two such fixed points: this structure is not what one would expect from dynamics near the cross product of a pair of uncoupled H\'enon maps, where there is at most one doubly elliptic point. We visualize the dynamics by escape time plots on 2D planes through the phase space and by 3D slices through the tori.

Study of the Hopf bifurcation in the Lorenz, Chen and Lü systems

In this paper, we perform a complete study of the Hopf bifurcations in the three-parameter Lorenz system, $$\dot{x} = \sigma (y-x),\,\dot{y} = \rho x - y - xz,\,\dot{z} = -bz + xy$$ , with $$\sigma , \rho , b \in \mathbb {R}$$ . On the one hand, we reobtain the results found in the literature for the Lorenz model when the three parameters are positive. On the other hand, we completely determine the loci of all the degeneracies exhibited by the Hopf bifurcation of the origin and of the nontrivial equilibria. In this last case, we demonstrate, among other things, that the first two Lyapunov coefficients simultaneously vanish in two codimension-three bifurcation points, giving rise in both cases to the coexistence of three periodic orbits involved in a cusp bifurcation. The analytical study that we carry out, where several complicated expressions have to be handled, successfully closes the problem of the Hopf bifurcations in the Lorenz system. Moreover, from our results, it is easy to obtain all the information on the Hopf bifurcations in the Chen and Lu systems, taking into account that they are, generically, particular cases of the Lorenz system, as can be seen by means of a linear scaling in time and state variables.

Multiple Buckling and Codimension-Three Bifurcation Phenomena of a Nonlinear Oscillator

In this paper, we investigate the global bifurcations and multiple bucklings of a nonlinear oscillator with a pair of strong irrational nonlinear restoring forces, proposed recently by Han et al. [2012]. The equilibrium stabilities of multiple snap-through buckling system under static loading are analyzed. It is found that complex bifurcations are exhibited of codimension-three with two parameters at the catastrophe point. The universal unfolding for the codimension-three bifurcation is also found to be equivalent to a nonlinear viscous damped system. The bifurcation diagrams and the corresponding codimension-three behaviors are obtained by employing subharmonic Melnikov functions for the existing singular closed orbits of homoclinic, tangent homoclinic, homo-heteroclinic and cuspidal heteroclinic, respectively.

Codimension-three bifurcations in a Bénard-Marangoni problem

This Brief Report studies the linear stability of a thermoconvective problem in an annular domain for relatively low (~1) Prandtl (viscosity effects) and Biot (heat transfer) numbers. The four possible patterns for the instabilities, namely, hydrothermal waves of first and second class, longitudinal rolls, and corotating rolls, are present in a small region of the Biot-Prandtl plane. This region can be split in four zones, depending on the sort of instability found. The boundary of these four zones is composed of codimension-two points. Authors have also found two codimension-three points, where some of the former curves intersect. Results shown in this Brief Report clarify some reported experiments, predict new instabilities, and, by giving a deeper insight into how physical parameters affect bifurcations, open a gateway to control those instabilities.

Numerical analysis of codimension-one, -two and -three bifurcations in a periodically-forced impact oscillator with two discontinuity surfaces

We analyse a model of a periodically-forced impact oscillator with two discontinuity surfaces. This model describes a pair of meshing gears, where the discontinuities arise from impacts between the gear teeth. A classical approach of basin-of-attraction computations and bifurcation diagrams is used in conjunction with the recently developed discontinuity-geometry methodology to provide new insights into the extremely rich dynamical behaviour observed. In particular, we show that all periodic solutions with impacts emanate from a codimension-three bifurcation.

Modal decomposition and normal form for hydrodynamic flows: Examples from cellular flame patterns

Spatio-temporal complexity of hydrodynamic flows may be reduced through modal decomposition, especially in systems with symmetries. The symmetries of the most significant modes can then be used to deduce normal form equations associated with the observed state. In turn, the normal form equations can be used to deduce bifurcations to and from the given state. We illustrate this process using two spatio-temporal cellular states on a circular flame front. The first example contains a pair of uniformly rotating cells. Principle component analysis shows that two coherent structures capture most of the dynamics and suggests that the state is a broken-parity traveling mode. Other experimentally observed states, such as modulated rotating states and a heteroclinic cycle between two spatially orthonormal states result from secondary bifurcations from the rotating state. The second example, referred to as the hopping mode, visually appears to have significantly more complicated dynamics. However, modal decomposition shows that it consists of two parity broken states moving at different angular velocities. The corresponding normal form contains a codimension-three steady-state bifurcation leading to a homoclinic cycle whose spatio-temporal characteristics are similar to those of hopping states. We use these examples to propose a methodology to combine coherent structures that form a single, possibly time-dependent entity which we refer to as a generalized coherent structure. The process can reduce the number of entities needed to expand complex spatio-temporal states. The paper is dedicated to the memory of Michael Gorman, whose experiments on cellular flame fronts and relentless demands for better theoretical understanding of the patterns motivated the study.

ANALYSIS OF THE T-POINT–HOPF BIFURCATION WITH ℤ2-SYMMETRY: APPLICATION TO CHUA'S EQUATION

The aim of this work is twofold — on the one hand, to perform a theoretical analysis of the global behavior organized by a T-point–Hopf in ℤ2-symmetric systems; on the other hand, to apply the obtained results for a numerical study of Chua's equation, where for the first time this bifurcation is considered.In a parameterized three-dimensional system of autonomous differential equations, a T-point is a point of the parameter space where a special kind of codimension-two heteroclinic cycle occurs. A more degenerate scenario appears when one of the equilibria involved in such a cycle undergoes a Hopf bifurcation. This degeneration, which corresponds to a codimension-three bifurcation, is called T-point–Hopf and has been recently studied for a generic system. However, the presence of ℤ2-symmetry may lead to the existence of a double T-point–Hopf heteroclinic cycle, which is responsible for the appearance of interesting global behavior that we will study in this paper.The theoretical models proposed for two different situations are based on the construction of a Poincaré map. The existence of certain kinds of homoclinic and heteroclinic connections between equilibria and/or periodic orbits is proved and their organization close to the T-point–Hopf bifurcation is described. The numerical phenomena found in Chua's equation strongly agree with the results deduced from the models.

Double Hopf Bifurcation with Strong Resonances in Delayed Systems with Nonlinerities

An efficient method is proposed to study delay-induced strong resonant double Hopf bifurcation for nonlinear systems with time delay. As an illustration, the proposed method is employed to investigate the 1 : 2 double Hopf bifurcation in the van der Pol system with time delay. Dynamics arising from the bifurcation are classified qualitatively and expressed approximately in a closed form for either square or cubic nonlinearity. The results show that 1 : 2 resonance can lead to codimension-three and codimension-two bifurcations. The validity of analytical predictions is shown by their consistency with numerical simulations.

THE CUSP–HOPF BIFURCATION

The coalescence of a Hopf bifurcation with a codimension-two cusp bifurcation of equilibrium points yields a codimension-three bifurcation with rich dynamic behavior. This paper presents a comprehensive study of this cusp-Hopf bifurcation on the three-dimensional center manifold. It is based on truncated normal form equations, which have a phase-shift symmetry yielding a further reduction to a planar system. Bifurcation varieties and phase portraits are presented. The phenomena include all four cases that occur in the codimension-two fold–Hopf bifurcation, in addition to bistability involving equilibria, limit cycles or invariant tori, and a fold–heteroclinic bifurcation that leads to bursting oscillations. Uniqueness of the torus family is established locally. Numerical simulations confirm the prediction from the bifurcation analysis of bursting oscillations that are similar in appearance to those that occur in the electrical behavior of neurons and other physical systems.

Codimension-three bifurcations: Explanation of the complex one-, two-, and three-dimensional bifurcation structures in nonsmooth maps

Many physical and engineering systems exhibit cascades of periodic attractors arranged in period increment and period adding sequences as a parameter is varied. Such systems have been found to yield piecewise smooth maps, and in some cases the obtained map is discontinuous. By investigating the normal form of such maps, we have detected a type of codimension-three bifurcation which serves as the organizing center of periodic and aperiodic dynamics in the parameter space. The results will help in understanding the occurrence and structure of such cascades observed in many nonsmooth systems in science and engineering.

Bifurcation of Nonclassical Viscous Shock Profiles from the Constant State

We determine the bifurcation from the constant solution of nonclassical transitional and overcompressive viscous shock profiles, in regions of strict hyperbolicity. Whereas classical shock waves in systems of conservation laws involve a single characteristic field, nonclassical waves involve two fields in an essential way. This feature is reflected in the viscous profile differential equation, which undergoes codimension-three bifurcation of the kind studied by Dumortier et al., as opposed to the codimension-one bifurcation occurring in the classical case. We carry out a complete bifurcation analysis for systems of two quadratic conservation laws with constant, strictly parabolic viscosity matrices by reducing to a canonical form introduced by Fiddelaers. We show that all such systems, except possibly those on a codimension-one variety in parameter space, give rise to nonclassical shock waves, and we classify the number and types of their bifurcation points. One consequence of our analysis is that weak transitional waves arise in pairs, with profiles forming a 2-cycle configuration previously shown to lead to nonuniqueness of Riemann solutions and to nontrivial asymptotic dynamics of the conservation laws. Another consequence is that appearance of weak nonclassical waves is necessarily associated with change of stability in constant solutions of the parabolic system of conservation laws, rather than with change of type in the associated hyperbolic system.

The non-transverse Shil’nikov–Hopf bifurcation: uncoupling of homoclinic orbits and homoclinic tangencies

It is known that in a neighbourhood of a codimension-two Shil’nikov–Hopf bifurcation, primary periodic orbits lie on a single wiggly curve in period–parameter space, with accumulation points at parameter values of a pair of homoclinic tangencies to a periodic orbit. In contrast, it has recently been shown by Hirschberg and Laing that primary periodic orbits lie on an infinity of isolas in a neighbourhood of certain degenerate homoclinic tangency to a periodic orbit. This paper analyses the codimension-three bifurcation caused by a non-transverse (i.e., degenerately parametrically unfolded) Shil’nikov–Hopf bifurcation, which contains nearby dynamics akin to both degeneracies. Two cases are classified as being downward pointing or upward pointing depending on whether the variation of a third parameter causes either the annihilation of a locus of saddle-focus homoclinic orbits to equilibria, or the uncoupling of this locus from the locus of Hopf bifurcations. We undertake a heuristic analysis of the unfolding, showing that in both cases it contains codimension-two non-transversal homoclinic orbits to equilibria and non-transversal homoclinic tangencies to periodic orbits. Unfolding the former non-transverse orbit, is shown to cause two wiggly curves to coalesce and leave finitely many isolas of periodic orbits. Unfolding the latter causes two wiggly curves to coalesce into first infinitely many and then finitely many isolas. Asymptotic expressions are given for the accumulation of two types of isola-forming bifurcations. The implications of Z 2 -equivariance on the unfolding is discussed. Finally, numerical evidence is presented for both upward and downward pointing non-transverse Shil’nikov–Hopf bifurcations occuring in a model of an autonomous non-linear electronic circuit, both with respect to a Z 2 -equivariant and a Z 2 -non-equivariant equilibrium. Numerical computation of curves of periodic orbits, homoclinic orbits and homoclinic tangencies to periodic orbits are shown to agree broadly with the theory but to uncover extra complications.

On a codimension-three bifurcation arising in a simple dynamo model

In this paper we investigate the dynamics associated with a degenerate codimension-two Takens-Bogdanov bifurcation which arises in a recently derived model for self-exciting dynamo action introduced by Hide et al. [R. Hide, A.C. Skeldon, D.J. Acheson, A study of two novel self-exciting single-disk homopolar dynamos: theory, Proc. R. Soc. Lond. A 452 (1996) 1369–1395]. The general unfolding of such a codimension-three bifurcation has already been discussed in an abstract setting by Li and Rousseau [Codimension-2 symmetric homoclinic bifurcations and application to 1 : 2 resonance, Can J. Math. 42 (1990) 191–212]. Here we describe the unfolding scenario in the context of the dynamo problem. In particular we compare the behaviour predicted by the normal form analysis with a bifurcation study of the full dynamo equations in the neighbourhood of the codimension-three point.

Symmetry breaking and overturning oscillations in thermohaline-driven flows

The bifurcation structure of thermohaline-driven flows is studied within one of the simplest zonally averaged models which captures thermohaline transport: a Boussinesq model of surface-forced thermohaline flow in a two-dimensional rectangular basin. Under mixed boundary conditions, i.e. prescribed surface temperature and fresh-water flux, it is shown that symmetry breaking originates from a codimension-two singularity which arises through the intersection of the paths of two symmetry-breaking pitchfork bifurcations. The physical mechanism of symmetry breaking of both the thermally and salinity dominated symmetric solution is described in detail from the perturbation structures near bifurcation. Limit cycles with an oscillation period in the order of the overturning time scale arise through Hopf bifurcations on the branches of asymmetric steady solutions. The physical mechanism of oscillation is described in terms of the most unstable mode just at the Hopf bifurcation. The occurrence of these oscillations is quite sensitive to the shape of the prescribed fresh-water flux. Symmetry breaking still occurs when, instead of a fixed temperature, a Newtonian cooling condition is prescribed at the surface. There is only quantitative sensitivity, i.e. the positions of the bifurcation points shift with the surface heat transfer coefficient. There are no qualitative changes in the bifurcation diagram except in the limit where both the surface heat flux and fresh-water flux are prescribed. The bifurcation structure at large aspect ratio is shown to converge to that obtained by asymptotic theory. The complete structure of symmetric and asymmetric multiple equilibria is shown to originate from a codimension-three bifurcation, which arises through the intersection of a cusp and the codimension-two singularity responsible for symmetry breaking.

Chaos in magnetoconvection

The partial differential equations (PDEs) for two-dimensional incompressible convection in a strong vertical magnetic field have a codimension-three bifurcation when the parameters are chosen so that the bifurcations to steady and oscillatory convection coincide and the limit of narrow rolls is taken. The third-order set of ordinary differential equations (ODEs) that govern the behaviour of the PDEs near this bifurcation are derived using perturbation theory. These ODEs are the normal form of the codimension-three bifurcation; as such, they prove to be an excellent predictor of the behaviour of the PDEs. This is the first time that a detailed comparison has been made between the chaotic behaviour of a set of PDEs and that of the corresponding set of model ODEs, in a parameter regime where the ODEs are expected to provide accurate approximations to solutions of the PDEs. Most significantly, the transition from periodic orbits to a chaotic Lorenz attractor predicted by the ODEs is recovered in the PDEs, making this one of the few situations in which the nature of chaotic oscillations observed numerically in PDEs can be established firmly. Including correction terms obtained from the perturbation calculation enables the ODEs to track accurately the bifurcations in the PDEs over an appreciable range of parameter values. Numerical calculations suggest that the T-point (where there are heteroclinic connections between a saddle point and a pair of saddle-foci), which is associated with the transition from a Lorenz attractor to a quasi-attractor in the normal form, is also found in the PDEs. Further numerical simulations of the PDEs with square rolls confirm the existence of chaotic oscillations associated with a heteroclinic connection between a pair of saddle-foci.

ON THE "CROSSROAD AREA–SADDLE AREA" AND "CROSSROAD AREA–SPRING AREA" TRANSITIONS

Let T be a one-dimensional or two-dimensional map. The three considered areas are related to three different configurations of fold and flip bifurcation curves, centred at a cusp point of a fold curve in the T parameter plane (b, c). The two transitions studied here occur via a codimension-three bifurcation defined in each case, when varying a third parameter a. The transition "mechanism," from an area type to another one, is given with a three-dimensional representation describing the sheet configuration of the parameter plane.

Codimension-three bifurcation point in a laser system with an injected signal.

In single-mode laser systems with an injected signal, we find a codimension-three bifurcation point at \ensuremath{\Delta}=-\ensuremath{\theta}=(2C${\mathrm{\ensuremath{-}}1)}^{1/2}$ when C>(1/2. In the vicinity of the bifurcation point, the initial five-dimensional coupled differential equations are reduced to two-dimensional ones by adiabatic elimination. The dynamics about the critical point is analyzed. Based on the revealed critical point, a parameter region where a full hysteresis loop exists can be predicted. This optical-bistability system can be operated by a rather small signal.

On codimension-three bifurcations in the motion of articulated tubes conveying a fluid

We consider the planar motion of two articulated tubes conveying a fluid near a double degeneracy in the linearized dynamical equations. In particular, the interaction between Hopf and pitchfork (symmetric saddle-node) bifurcations, and the effects of asymmetries on that interaction, are studied in detail. The fourth order system of nonlinear equations is reduced to a third order system using center manifold theory. The method of averaging provides the normal form which can be considered as a planar vector field with three important parameters, one each for the Hopf bifurcation, the pitchfork bifurcation and the asymmetry. This second order system is shown to possess twenty-three open sets in the parameter space, each with qualitatively different phase portraits. Some of these portraits are presented, along with bifurcation analysis, and the physical implications of the results are discussed.