Symmetry-increasing bifurcation of chaotic attractors

Bifurcation in symmetric is typically associated with spontaneous symmetry breaking. That is, bifurcation is associated with new solution having less symmetry.In this paper we show that symmetry-increasing bifurcation in the discrete dynamics of symmetric mappings is possible (and is perhaps generic). The reason for these bifurcations may be understood as follows. The existence of one attractor in a system with symmetry gives rise to a family of conjugate attractors all related by symmetry. Typically, in computer experiments, what we see is a sequence of symmetry-breaking bifurcations leading to the existence of conjugate chaotic attractors. As the bifurcation parameter is varied these attractors grow in size and merge leading to a single attractor having greater symmetry.We prove a theorem suggesting why this new attractor should have greater symmetry and present a number of striking examples of the symmetric patterns that can be formed by iterating the simplest mappings on the plane with the symmetry of the regular m-gon. In the last section we discuss period-doubling in the presence of symmetry.     

Blowout bifurcations of codimension two

We consider examples of loss of stability of chaotic attractors in invariant subspaces (blowouts) that occur on varying two parameters, i.e. codimension-two blowout bifurcations. Such bifurcations act as organising centres for nearby codimension-one behaviour, analogous to the case for codimension-two bifurcations of equilibria. We consider examples of blowout bifurcations showing change of criticality, blowouts that occur into two different invariant subspaces and interact, blowouts that occur with onset of hyperchaos, interaction of blowout and symmetry increasing bifurcations and collision of blowout bifurcations. As in the case of bifurcation of equilibria, there are many cases in which one can infer the presence and form of secondary bifurcations and associated branches of attractors. There is presently no generic theory of such higher codimension blowouts (there is not even such a theory for codimension-one blowouts). We want to present a number of examples that would need to be covered by such a theory.     

A discrete model exhibiting successive bifurcations leading to the onset of turbulence

A simple discrete model which consists ofN limit-cycle oscillators interacting with a linear coupling is numerically investigated in order to study the sequence of oscillatory states leading to the onset of turbulence. The systems withN=2 and 3 are studied. The system ofN=2 does not exhibit a nonperiodic motion, whereas the system ofN=3 does exhibit a nonperiodic motion. It is shown that, as an external parameter changes, the system ofN=3 undergoes a sequence of bifurcations, exhibiting the singly periodic, doubly periodic and nonperiodic motions, successively. This is similar to the bifurcation scheme for the onset of turbulence proposed by Ruelle and Takens and experimentally shown by Gollub and Swinny in a rotating Couette flow. The successive bifurcations are investigated in details and new features are reported.     

Observing a codimension-two heteroclinic bifurcation

This paper reports experimental observations of codimension-two heteroclinic bifurcations in an autonomous third-order electrical circuit. The paper also reports confirmations by computer simulations. In the laboratory experiments, a pair of programmable resistors are used in order to adjust two bifurcation parameters. In the associated two-parameter space, several codimension-one bifurcation sets are experimentally measured to capture codimension-two bifurcation structures. All of these bifurcation sets are numerically confirmed by exact bifurcation equations which are derived from piecewise-linear circuit dynamics.     

Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry

We analyze unfoldings of a codimension two, steady-state/steady-state modal interaction possessing O(2) symmetry. At the degenerate bifurcation point there are two zero eigenvalues, each of multiplicity two. The spatial wavenumbers of the critical modes k_i are assumed to satisfy k₂ = 2k₁. We base our analysis on a detailed study of the third order truncation of the resulting equivariant normal form, which is a four-dimensional vector field. We find that heteroclinic cycles and modulated travelling waves exist for open sets of parameter values near the codimension two bifurcation point. We provide conditions on parameters which guarantee existence and uniqueness of such solutions and we investigate their stability types. We argue that such motions will be prevalent in continuum systems having the symmetry of translation and reflection with respect to one (or more) spatial directions.     

Analysis of the T-point-Hopf bifurcation

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-2 heteroclinic cycle occurs. If the parameter space is three-dimensional, such a bifurcation is located generically on a curve. A more degenerate scenario appears when this curve reaches a surface of Hopf bifurcations of one of the equilibria involved in the heteroclinic cycle. We are interested in the analysis of this codimension-3 bifurcation, which we call T-point-Hopf. In this work we propose a model, based on the construction of a Poincaré map, that describes the global behavior close to a T-point-Hopf bifurcation. The existence of certain kinds of homoclinic and heteroclinic connections between equilibria and/or periodic orbits is proved. The predictions deduced from this model strongly agree with the numerical results obtained in a modified van der Pol-Duffing electronic oscillator.     

Symmetry chaotic attractors and bursting dynamics of semiconductor lasers subjected to optical injection

This paper presents the nonlinear dynamics and bifurcations of optically injected semiconductor lasers in the frame of relative high injection strength. The behavior of the system is explored by means of bifurcation diagrams; however, the exact nature of the involved dynamics is well described by a detailed study of the dynamics evolutions as a function of the effective gain coefficient. As results, we notice the different types of symmetry chaotic attractors with the riddled basins, supercritical pitchfork and Hopf bifurcations, crisis of attractors, instability of chaos, symmetry breaking and restoring bifurcations, and the phenomena of the bursting behavior as well as two connected parts of the same chaotic attractor which merge in a periodic orbit.     

Conductivity of single-walled carbon nanotubes

In a system of N interacting single-level quantum dots (QDs), we study the relaxation dynamics and the current–voltage characteristics determined by symmetry properties of the QD arrangement. Different numbers of dots, initial charge configurations, and various coupling regimes to reservoirs are considered. We reveal that effective charge trapping occurs for particular regimes of coupling to the reservoir when more than two dots form a ring structure with the C_N spatial symmetry. We reveal that the effective charge trapping caused by the C_N spatial symmetry of N coupled QDs depends on the number of dots and the way of coupling to the reservoirs. We demonstrate that the charge trapping effect is directly connected with the formation of dark states, which are not coupled to reservoirs due to the system spatial symmetry C_N. We also reveal the symmetry blockade of the tunneling current caused by the presence of dark states.     

Period-doubling/symmetry-breaking mode interactions in iterated maps

We consider iterated maps with a reflectional symmetry. Possible bifurcations in such systems include period-doubling bifurcations (within the symmetric subspace) and symmetry-breaking bifurcations. By using a second parameter, these bifurcations can be made to coincide at a mode interaction. By reformulating the period-doubling bifurcation as a symmetry-breaking bifurcation, two bifurcation equations with Z₂×Z₂ symmetry are derived. A local analysis of solutions is then considered, including the derivation of conditions for a tertiary Hopf bifurcation. Applications to symmetrically coupled maps and to two coupled, vertically forced pendulums are described.     

Symmetry breaking bifurcations in a circular chain of N coupled logistic maps

A circular chain of N cells with logistic dynamics, coupled together with symmetric nearest neighbor coupling and periodic boundary conditions is investigated. For certain coupling parameters we observe bifurcation of a stable state into two types of period two solutions. By using the symmetry of this Coupled Map Lattice model, we show that the bifurcated system only can have periodic solutions with symmetry group corresponding to certain subgroups of the full symmetry group of the system.     

Global analysis of periodic orbit bifurcations in coupled Morse oscillator systems: time-reversal symmetry,permutational representations and codimension-2 collisions

In this paper we study periodic orbit bifurcation sequences in a system of two coupled Morse oscillators. Time-reversal symmetry is exploited to determine periodic orbits by iteration of symmetry lines. The permutational representation of Tsuchiya and Jaffe is employed to analyze periodic orbit configurations on the symmetry lines. Local pruning rules are formulated, and a global analysis of possible bifurcation sequences of symmetric periodic orbits is made. Analysis of periodic orbit bifurcations on symmetry lines determines bifurcation sequences, together with periodic orbit periodicities and stabilities. The correlation between certain bifurcations is explained. The passage from an integrable limit to nointegrability is marked by the appearance of tangent bifurcations; our global analysis reveals the origin of these ubiquitous tangencies. For period-1 orbits, tangencies appear by a simple disconnection mechanism. For higher period orbits, a different mechanism involving 2-parameter collisions of bifurcations is found. (c) 1999 American Institute of Physics.     

Primary resonant optimal control for nested homoclinic and heteroclinic bifurcations in single-dof nonlinear oscillators

A unified control theorem is presented in this paper, whose aim is to suppress the transversal intersections of stable and unstable manifolds of homoclinic and heteroclinic orbits in the Poincarè map embedding in system dynamics. Based on the control theorem, a primary resonant optimal control technique (PROCT for short) is applied to a general single-dof nonlinear oscillator. The novelty of this technique is able to obtain the unified analytical expressions of the control gain and the control parameters for suppressing the homoclinic and heteroclinic bifurcations, where the control gain can guarantee that the control region where the homoclinic and heteroclinic bifurcations do not occur can be enlarged as much as possible at least cost. The technique is applied to a nonlinear oscillator with a pair of nested homoclinic and heteroclinic orbits. By the PROCT, the transversal intersections of homoclinic and heteroclinic orbits can be suppressed, respectively. The hopping phenomenon that there coexist two kinds of chaotic attractors of Duffing-type and pendulum-type can be suppressed. On the contrary, if the first amplitude coefficient is greater than the critical heteroclinic bifurcation value, then another degenerate hopping behavior of chaos will take place again. Therefore, the phenomenon of hopping is the dominant type of chaos in this oscillator, whose suppressing or inducing is admissible from the points of practical and theoretical view.     

Path integral approach to the full Dicke model

The full Dicke model describes a system of N identical two level-atoms coupled to a single mode quantized bosonic field. The model considers rotating and counter-rotating coupling terms between the atoms and the bosonic field, with coupling constants g₁ and g₂, for each one of the coupling terms, respectively. We study finite temperature properties of the model using the path integral approach and functional methods. In the thermodynamic limit, N→∞, the system exhibits phase transition from normal to superradiant phase, at some critical values of temperature and coupling constants. We distinguish between three particular cases, the first one corresponds to the case of rotating wave approximation, where g₁≠0 and g₂=0, the second one corresponds to the case of g₁=0 and g₂≠0, in these two cases the model has a continuous symmetry. The last one, corresponds to the case of g₁≠0 and g₂≠0, where the model has a discrete symmetry. The phase transition in each case is related to the spontaneous breaking of its respective symmetry. For each one of these three particular cases, we find the asymptotic behaviour of the partition function in the thermodynamic limit, and the collective spectrum of the system in the normal and the superradiant phase. For the case of rotating wave approximation, and also the case of g₁=0 and g₂≠0, in the superradiant phase, the collective spectrum has a zero energy value, corresponding to the Goldstone mode associated to the continuous symmetry breaking of the model. Our analysis and results are valid in the limit of zero temperature, β→∞, in which, the model exhibits a quantum phase transition.     

Exploiting the Hamiltonian structure of a neural field model

We study the unexpected disappearance of stable homoclinic orbits in regions of parameter space in a neural field model with one spatial dimension. The usual approach of using numerical continuation techniques and local bifurcation theory is insufficient to explain the qualitative change in the model’s behaviour. The lack of robustness of the model to small perturbations in parameters is surprising, and the phenomenon may be of broader significance than just our model. By exploiting the Hamiltonian structure of the time-independent system, we develop a numerical technique with which we discover that a small, separate solution curve exists for a range of parameter values. As the firing rate function steepens, the small curve causes the main curve to break and stable homoclinic orbits are destroyed in a region of parameter space. Numerically, we use level set analysis to find that a codimension-one heteroclinic bifurcation occurs at the terminating ends of the solution curves. By replacing the firing rate function with a step function, we show analytically that the bifurcation is related to the value of the firing threshold. We also show the existence of heteroclinic orbits at the breakpoints using a travelling front analysis in the time-dependent system.     

Autonomous coupled oscillators with hyperbolic strange attractors

We propose several examples of smooth low-order autonomous dynamical systems which have apparently uniformly hyperbolic attractors. The general idea is based on the use of coupled self-sustained oscillators where, due to certain amplitude nonlinearities, successive epochs of damped and excited oscillations alternate. Because of additional, phase sensitive coupling terms in the equations, the transfer of excitation from one oscillator to another is accompanied by a phase transformation corresponding to some chaotic map (in particular, an expanding circle map or Anosov map of a torus). The first example we construct is a minimal model possessing an attractor of the Smale-Williams type. It is a four-dimensional system composed of two oscillators. The underlying amplitude equations are similar to those of the predator-pray model. The other three examples are systems of three coupled oscillators with a heteroclinic cycle. This scheme presents more variability for the phase manipulations: in the six-dimensional system not only the Smale-Williams attractor, but also an attractor with Arnold cat map dynamics near a two-dimensional toral surface, and a hyperchaotic attractor with two positive Lyapunov exponents, are realized.     

Synchronization and basin bifurcations in mutually coupled oscillators

Synchronization behaviour of two mutually coupled double-well Duffing oscillators exhibiting cross-well chaos is examined. Synchronization of the subsystems was observed for coupling strength k > 0.4. It is found that when the oscillators are operated in the regime for which two attractors coexist in phase space, basin bifurcation sequences occur leading to n + 1, n ≥ 2 basins as the coupling is varied — a signature of Wada structure and final-state sensitivity. However, in the region of complete synchronization, the basins structure is identical with that of the single oscillators and retains its essential features including fractal basin boundaries.      

Complex chaos and bifurcations of semiconductor lasers subjected to optical injection

This paper presents the nonlinear dynamics and bifurcations of optically injection semiconductor lasers in the frame of relative weak injection strength. We consider the new modified rate equations model established recently and the behavior of the system is explored by means of bifurcation diagrams. However, the exact nature of the involved dynamics is well described by a detailed study of the changes of dynamics as a function of the effective gain coefficient. As results, we notice symmetry spectra of intensity, the sudden transition between chaos and stable limit cycle, double scroll attractors together with the phenomenon of a sequence of period-doubling route of chaos, strict crisis between the two basins attraction and the boundary crisis as well as the effects of frequency detuning and linewidth enhancement factor on the nonlinear behaviors.     

Comparison of the influences of nodal dynamics on network synchronized regions

A new piecewise linear unified chaotic (PLUC) system is firstly presented, and then its fundamental dynamical behaviors are analyzed. This modified chaotic system, as well as the unified chaotic (UC) one, is taken as network nodal oscillators for investigating the difference of influences of nodal dynamics on the bifurcation of network synchronized regions. It is found that beyond the greatly similar bifurcation modes between PLUC and UC networks, the synchronized regions in PLUC networks are far narrower at almost each parameter a than those in UC networks for most of inner coupling matrices, indicating the PLUC node makes the network more difficult to synchronization. Our numerical investigations show that this phenomenon is closely related with nodal dynamical properties, such as the boundary of attractors, the largest Lyapunov exponent and Lyapunov dimension.     

Phase structure and critical behavior of multi-Higgs U(1) lattice gauge theory in three dimensions

We study the three-dimensional (3D) compact U(1) lattice gauge theory coupled with N-flavor Higgs fields by means of the Monte Carlo simulations. This model is relevant to multi-component superconductors, antiferromagnetic spin systems in easy plane, inflational cosmology, etc. It is known that there is no phase transition in the N = 1 model. For N = 2, we found that the system has a second-order phase transition line in the c₂ (gauge coupling)-c₁ (Higgs coupling) plane, which separates the confinement phase and the Higgs phase. Numerical results suggest that the phase transition belongs to the universality class of the 3D XY model as the previous works by Babaev et al. and Smiseth et al. suggested. For N = 3, we found that there exists a critical line similar to that in the N = 2 model, but the critical line is separated into two parts; one for c₂<c_2tc=2.4±0.1 with first-order transitions, and the other for c_2tc<c₂ with second-order transitions, indicating the existence of a tricritical point. We verified that similar phase diagram appears for the N = 4 and N = 5 systems. We also studied the case of anistropic Higgs coupling in the N = 3 model and found that there appear two second-order phase transitions or a single second-order transition and a crossover depending on the values of the anisotropic Higgs couplings. This result indicates that an “enhancement” of phase transition occurs when multiple phase transitions coincide at a certain point in the parameter space.     

Nonlinear dynamos: A complex generalization of the Lorenz equations

Plane nonlinear dynamo waves can be described by a sixth order system of nonlinear ordinary differential equations which is a complex generalization of the Lorenz system. In the regime of interest for modelling magnetic activity in stars there is a sequence of bifurcations, ending in chaos, as a stability parameter D (the dynamo number) is increased. We show that solutions undergo three successive Hopf bifurcations, followed by a transition to chaos. The system possesses a symmetry and can therefore be reduced to a fifth order system, with trajectories that lie on a 2-torus after the third bifurcation. As D is then increased, frequency locking occurs, followed by a sequence of period-doubling bifurcations that leads to chaos. This behaviour is probably caused by the Shil'nikov mechanism, with a (conjectured) homoclinic orbit when D is infinite.