Torus destruction via global bifurcations in a piecewise-smooth, continuous map with square-root nonlinearity

It has been shown recently that torus formation in piecewise-smooth maps can occur through a special type of border collision bifurcation in which a pair of complex conjugate Floquet multipliers “jump” from the inside to the outside of the unit circle. It has also been shown that a large class of impacting mechanical systems yield piecewise-smooth maps with square-root singularity. In this Letter we investigate the dynamics of a two-dimensional piecewise-smooth map with square-root type nonlinearity, and describe two new routes to chaos through the destruction of two-frequency torus. In the first scenario, we identify the transition to chaos through the destruction of a loop torus via homoclinic bifurcation. In the other scenario, a change of structure in the torus occurs via heteroclinic saddle connections. Further parameter changes lead to a homoclinic bifurcation resulting in the creation of a chaotic attractor. However, this scenario is much more complex, with the appearance of a sequence of heteroclinic and homoclinic bifurcations.     

Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems

This paper presents a unified framework for performing local analysis of grazing bifurcations in n-dimensional piecewise-smooth systems of ODEs. These occur when a periodic orbit has a point of tangency with a smooth (n−1)-dimensional boundary dividing distinct regions in phase space where the vector field is smooth. It is shown under quite general circumstances that this leads to a normal-form map that contains to lowest order either a square-root or a (3/2)-type singularity according to whether the vector field is discontinuous or not at the grazing point. In particular, contrary to what has been reported in the literature, piecewise-linear local maps do not occur generically. First, the concept of a grazing bifurcation is carefully defined using appropriate non-degeneracy conditions. Next, complete expressions are derived for calculating the leading-order term in the normal form Poincaré map at a grazing bifurcation point in arbitrary systems, using the concept of a discontinuity mapping. Finally, the theory is compared with numerical examples including bilinear oscillators, a relay feedback controller and general third-order systems.     

Birth of bilayered torus and torus breakdown in a piecewise-smooth dynamical system

Border-collision bifurcations arise when the periodic trajectory of a piecewise-smooth system under variation of a parameter crosses into a region with different dynamics. Considering a three-dimensional map describing the behavior of a DC/DC power converter, the Letter discusses a new type of border-collision bifurcation that leads to the birth of a “bilayered torus”. This torus consists of the union of two saddle cycles, their unstable manifolds, and a stable focus cycle. When changing the parameters, the bilayered torus transforms through a border-collision bifurcation into a resonance torus containing the stable cycle and a saddle. The Letter also presents scenarios for torus destruction through homoclinic and heteroclinic tangencies.     

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

The dynamics near a Hopf saddle-node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study: a two-parameter model map G is constructed, such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1. To investigate the effect of resonances, the complex eigenvalues are selected to have a 1:5 resonance. It is shown that, near the origin of the parameter space, the family G has two secondary Hopf saddle-node bifurcations of period five points. A cone-like structure exists in the neighbourhood, formed by two surfaces of saddle-node and a surface of Hopf bifurcations. Quasi-periodic bifurcations of an invariant circle, forming a frayed boundary, are numerically shown to occur in model G. Along such Cantor-like boundary, an intricate bifurcation structure is detected near a 1:5 resonance gap. Subordinate quasi-periodic bifurcations are found nearby, suggesting the occurrence of a cascade of quasi-periodic bifurcations.     

Fixed equilibria of point vortices in symmetric multiply connected domains

The paper provides symmetric fixed configurations of point vortices in multiply connected domains in the unit circle with many circular obstacles. When the circular domain is invariant with respect to rotation around the origin by a degree of 2π/M, a regular M-polygonal ring configuration of identical point vortices becomes a fixed equilibrium. On the other hand, when we assume a special symmetry, called the folding symmetry, on the circular domain, we find a fixed equilibrium in which M point vortices with the positive unit strength and M point vortices with the negative unit strength are arranged alternately at the vertices of a 2M-polygon. We also investigate the stability of these fixed equilibria and their bifurcation for a special circular domain with the rotational symmetry as well as the folding symmetry. Furthermore, we discuss fixed equilibria in non-circular multiply connected domains with the same symmetries. We give sufficient conditions for the conformal mappings, by which fixed equilibria in the circular domains are mapped to those in the general multiply connected domains. Some examples of such conformal mappings are also provided.     

A novel four-wing non-equilibrium chaotic system and its circuit implementation

In this paper, we construct a novel, 4D smooth autonomous system. Compared to the existing chaotic systems, the most attractive point is that this system does not display any equilibria, but can still exhibit four-wing chaotic attractors. The proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, bifurcation diagram, and Poincaré maps. There is little difference between this chaotic system without equilibria and other chaotic systems with equilibria shown by phase portraits and Lyapunov exponents. But the bifurcation diagram shows that the chaotic systems without equilibria do not have characteristics such as pitchfork bifurcation, Hopf bifurcation etc. which are common to the normal chaotic systems. The Poincaré maps show that this system is a four-wing chaotic system with more complicated dynamics. Moreover, the physical existence of the four-wing chaotic attractor without equilibria is verified by an electronic circuit.     

Wave train selection behind invasion fronts in reaction-diffusion predator-prey models

Wave trains, or periodic travelling waves, can evolve behind invasion fronts in oscillatory reaction-diffusion models for predator-prey systems. Although there is a one-parameter family of possible wave train solutions, in a particular predator invasion a single member of this family is selected. Sherratt (1998) [13] has predicted this wave train selection, using a λ-ω system that is a valid approximation near a supercritical Hopf bifurcation in the corresponding kinetics and when the predator and prey diffusion coefficients are nearly equal. Away from a Hopf bifurcation, or if the diffusion coefficients differ somewhat, these predictions lose accuracy. We develop a more general wave train selection prediction for a two-component reaction-diffusion predator-prey system that depends on linearizations at the unstable homogeneous steady states involved in the invasion front. This prediction retains accuracy farther away from a Hopf bifurcation, and can also be applied when the predator and prey diffusion coefficients are unequal. We illustrate the selection prediction with its application to three models of predator invasions.     

Direct transition from a stable equilibrium to quasiperiodicity in non-smooth systems

The purpose of this Letter is to show how a border-collision bifurcation in a piecewise-smooth dynamical system can produce a direct transition from a stable equilibrium point to a two-dimensional invariant torus. Considering a system of nonautonomous differential equations describing the behavior of a power electronic DC/DC converter, we first determine the chart of dynamical modes and show that there is a region of parameter space in which the system has a single stable equilibrium point. Under variation of the parameters, this equilibrium may collide with a discontinuity boundary between two smooth regions in phase space. When this happens, one can observe a number of different bifurcation scenarios. One scenario is the continuous transformation of the stable equilibrium into a stable period-1 cycle. Another is the transformation of the stable equilibrium into an unstable period-1 cycle with complex conjugate multipliers, and the associated formation of a two-dimensional (ergodic or resonant) torus.     

Tunneling without barriers

The evolution of quantum fields in theories with relatively flat potentials is considered. It is shown that bubble nucleation, a quantum mechanical tunneling process, may occur even if V(φ) has no barrier. Some implications for inflationary universe scenarios are discussed.     

Entanglement and quantum phase transition in a mixed-spin Heisenberg chain with single-ion anisotropy

We study the ground-state and thermal entanglement in the mixed-spin (S,s)=(1,1/2) Heisenberg chain with single-ion anisotropy D using exact diagonalization of small clusters. In this system, a quantum phase transition is revealed to occur at the value D=0, which is the bifurcation point for the global ground state; that is, when the single-ion anisotropy energy is positive, the ground state is unique, whereas when it is negative, the ground state becomes doubly degenerate and the system has the ferrimagnetic long-range order. Using the negativity as a measure of entanglement, we find that a pronounced dip in this quantity, taking place just at the bifurcation point, serves to signal the quantum phase transition. Moreover, we show that the single-ion anisotropy helps to improve the characteristic temperatures above which the quantum behavior disappears.     

铂族金属氧化过程中的簇发振荡及其诱发机理

铂族金属表面氧化过程是典型的多相催化反应之一, 具有广泛的应用背景及丰富的振荡行为, 因此深入研究铂族金属的氧化中的物理及化学过程具有重要的理论意义及工程应用前景. 通过对铂族金属CO的氧化过程中实测数据的回归分析, 建立了不同尺度耦合解析动力学理论模型. 通过对平衡态的稳定性分析, 指出在一定条件下稳态解会由鞍-结同宿轨道分岔导致周期振荡. 当快子系统产生Hopf分岔时, 该周期振荡会进一步演化为两尺度耦合的周期簇发振荡, 即N^k振荡, 并由加周期分岔使得系统处于激发态的时间显著增加.在此基础上, 利用分岔理论进一步分析了周期簇发及加周期分岔的产生机理, 揭示了周期簇发中沉寂态和激发态相互转化时的不同分岔模式.     

Packet switching start-stop bits generation based on bifurcation behavior of light in a micro ring resonator

This paper proposes the interesting nonlinear behavior of light known as bifurcation, where the use of such behavior in a micro ring resonator to form the secure digital codes for optical packet switching application is demonstrated. A new concept of the stop-start bits in an optical packet switching protocol is formed by using the bifurcation codes. Bifurcation is introduced when light is input into a nonlinear micro ring device, where the refractive index of an InGaAsP/InP is one of device parameters. The other parameters of the device are coupling coefficient (K) and the ring radius (R), where the ring radii used are ranged from 5 to 10 μm. Simulation results obtained have shown that the packet switching data can be secured by using the generated start-stop bits as the secured codes.     

Estimating ultimate bound and finding topological horseshoe for a new chaotic system

This paper presents a new three-dimensional autonomous chaotic system with only one positive term. Basic dynamical properties of the new attractor are demonstrated in terms of phase portraits, equilibria, Lyapunov exponents, Poincare mapping, bifurcation diagram. Furthermore, we derive a three-dimensional spheriform ultimate bound and positively invariant set for all the positive values of its parameters a, b, c. At last, the horseshoe chaos in this system is investigated based on the topological theory.     

Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators

We study the bifurcation and dynamical behaviour of the system of N globally coupled identical phase oscillators introduced by Hansel, Mato and Meunier, in the cases N=3 and N=4. This model has been found to exhibit robust ‘slow switching’ oscillations that are caused by the presence of robust heteroclinic attractors. This paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors. We consider bifurcations that occur in a system of identical oscillators on varying the parameters in the coupling function. These bifurcations preserve the permutation symmetry of the system. We then investigate the implications of these bifurcations for the sensitivity to detuning (i.e. the size of the smallest perturbations that give rise to loss of frequency locking).For N=3 we find three types of heteroclinic bifurcation that are codimension-one with symmetry. On varying two parameters in the coupling function we find three curves giving (a) an S₃-transcritical homoclinic bifurcation, (b) a saddle-node/heteroclinic bifurcation and (c) a Z₃-heteroclinic bifurcation. We also identify several global bifurcations with symmetry that organize the bifurcation diagram; these are codimension-two with symmetry.For N=4 oscillators we determine many (but not all) codimension-one bifurcations with symmetry, including those that lead to a robust heteroclinic cycle. A robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region. Furthermore, we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system, while for other parts of the phase plane it can coexist with stable limit cycles. We finish with a discussion of bifurcations that appear for this coupling function and general N, as well as for more general coupling functions.     

FREE VIBRATIONS OF MULTILAYERED PLATES BASED ON A MIXED VARIATIONAL APPROACH IN CONJUNCTION WITH GLOBAL PIECEWISE-SMOOTH FUNCTIONS

This paper presents a two-dimensional model for the analysis of freely vibrating laminated plates. The governing differential equations, associated boundary conditions and constitutive equations are derived from Reissner's mixed variational theorem. Both the governing differential equations and the related boundary conditions are presented in terms of resultant stresses and displacements. The model is able to provide the results for the corresponding three-dimensional theory. Such a performance is guaranteed from an appropriate expansion of relevant kinetic and stress quantities through the thickness of the multilayered plate. The expansion is realized by using a novel selection of global piecewise-smooth functions (GPSFs). The number of GPSFs can be arbitrarily increased to achieve a two-dimensional plate theory which is, at least, as accurate as that of a full layerwise theory. It is also shown that GPSFs permit to deal a multilayered plate as if it was virtually made of a single layer. Indeed, the theory need not explicitly introduce continuity conditions for both displacements and relevant stresses. The performance of the present two-dimensional model in conjunction with the global piecewise-smooth functions is tested and discussed by comparing its resulting eigen-parameters, for a class of simply supported plates, with those of other two-dimensional models and with those existing of the exact three-dimensional theory.     

Degenerate Hopf bifurcation in the presence of symmetry

We investigate the bifurcation of time-periodic states from a stationary state destabilized by the undamping of a set of modes associated with a degenerate pair of complex-conjugate frequencies. This problem is of particular interest for bifurcations in driven systems with symmetry whose order-parameter dimension n is even and n ≥ 4. For this case of a degenerate Hopf bifurcation a star of symmetry-equivalent limit cycles bifurcates in analogy to the star of symmetry-related domains arising at a symmetry-breaking phase transition in equilibrium systems. We illustrate this fact by analyzing a concrete example with n = 4. Within the framework of an amplitude expansion, we explicitly construct the time-periodic states and discuss their stability. In particular, it is shown that fairly general conclusions for the bifurcation behaviour can be drawn on the sole basis of the knowledge of the order-parameter symmetry.     

Relative equilibria of D2H + and H2D+

Relative equilibria of molecules are classical trajectories corresponding to steady rotations about stationary axes during which the shape of the molecule does not change. They can be used to explain and predict features of quantum spectra at high values of the total angular momentum J in much the same way that absolute equilibria are used at low J. This paper gives a classification of the symmetry types of relative equilibria of AB₂ molecules and computes the relative equilibria bifurcation diagrams and normal mode frequencies for D₂H⁺ and H₂D⁺. These are then fed into a harmonic quantization procedure to produce a number of predictions concerning the structures of energy level clusters and their rearrangements as J increases. In particular the formation of doublet pairs is predicted for H₂D⁺ from J ≈ 26.     

Coexisting tori and torus bubbling in non-smooth systems

Pulse modulated power electronic converters represent an important class of piecewise-smooth dynamical systems with a broad range of applications in modern power supply systems. The paper presents a detailed investigation of a number of unusual bifurcation phenomena that can occur in power converters with multilevel control. In the first example a closed invariant curve arises in a border-collision bifurcation as a period-6 saddle cycle collides with a stable fixed point of focus type and transforms it into an unstable focus point. The second example involves the formation of a structure of coexisting tori through the interplay between border-collision and global bifurcations. We examine the behavior of the system in the presence of two coexisting stable resonance tori and finally show how an existing torus can develop heteroclinic bubbles that connect the points of a stable resonance cycle with an external pair of saddle and focus cycles. The appearance of these structures is explained in terms of a sequence torus-birth bifurcations with pairs of stable and unstable tori folding one over the other.