Branching of 2D tori off an equilibrium of a cosymmetric system (codimension-1 bifurcation)

The standard object for vector fields with a nontrivial cosymmetry is a continuous one-parameter family of equilibria. Characteristically, the stability spectrum of equilibrium varies along such a family, though the spectrum always contains a zero point. Consequently, in the general position a family consists of stable and unstable arcs separated by boundary equilibria, which are neutrally stable in the linear approximation. In the present paper the central manifold method and the Lyapunov-Schmidt method are used to investigate the branching bifurcation of invariant two-dimensional tori in cosymmetric systems off a boundary equilibrium whose spectrum contains, besides the requisite point 0, two pairs of purely imaginary eigenvalues. A number of new effects, as compared with the classic case of an isolated equilibrium, are found: the bifurcation studied has codimension 1 (2 for an isolated equilibrium); it is accompanied by a branching bifurcation of a normal limit cycle; and, a stable arc can be created on an unstable arc. (c) 2001 American Institute of Physics.     

Bifurcations and selection of equilibria in a simple cosymmetric model of filtrational convection

A three-dimensional set of ordinary differential equations that constitutes a simple abstract model of Darcy convection is investigated. The model reproduces a number of effects that are typical for dynamic systems with nontrivial cosymmetry. Nontrivial cosymmetry can give rise to a continuous family of equilibria where, in this case, the equilibrium stability spectrum varies along the family. The family of equilibria and its stability are examined analytically, and special bifurcations that occur in the system are investigated. It is shown that discrete and continual symmetries, called "flash symmetries," can be present in the system for certain parameter values. Computer experiments on the selection of equilibria in the symmetric and cosymmetric cases have been carried out. They showed that, for initial points that are far enough from a cycle of equilibria, the neighborhood of a single equilibrium is established in the case of cosymmetry, but all the equilibria are equivalent in the case of symmetry. The authors hope that these results, as well as the formulation of the problems and the approach to their solution, will serve as a sample in the investigation of more complex systems in mathematical physics. (c) 1999 American Institute of Physics.     

Bifurcation of the branching of a cycle in n-parameter family of dynamic systems with cosymmetry

A study is reported of the bifurcation of the branching of a cycle (Poincare-Andronov-Hopf bifurcation) from a smooth one-dimensional submanifold of equilibria of a dynamical system that depends on a vector parameter and admits cosymmetry. The paper reports a topological classification of local phase portraits near a known equilibrium, when the system parameter is close to its critical value that corresponds to an oscillatory instability. New phenomena that are not observed in the classical case of an isolated equilibrium include a delay of cycle creation with respect to the system parameter, loss of stability by the family of equilibria without loss of attraction, and the possibility of unstable supercritical self-oscillations. (c) 1997 American Institute of Physics.     

Nonlinear analysis of a maglev system with time-delayed feedback control

This paper undertakes a nonlinear analysis of a model for a maglev system with time-delayed feedback. Using linear analysis, we determine constraints on the feedback control gains and the time delay which ensure stability of the maglev system. We then show that a Hopf bifurcation occurs at the linear stability boundary. To gain insight into the periodic motion which arises from the Hopf bifurcation, we use the method of multiple scales on the nonlinear model. This analysis shows that for practical operating ranges, the maglev system undergoes both subcritical and supercritical bifurcations, which give rise to unstable and stable limit cycles respectively. Numerical simulations confirm the theoretical results and indicate that unstable limit cycles may coexist with the stable equilibrium state. This means that large enough perturbations may cause instability in the system even if the feedback gains are such that the linear theory predicts that the equilibrium state is stable.     

Bifurcation and Stability Analysis of the Equilibrium States in Thermodynamic Systems in a Small Vicinity of the Equilibrium Values of Parameters

The dynamic behavior of thermodynamic system, described by one order parameter and one control parameter, in a small neighborhood of ordinary and bifurcation equilibrium values of the system parameters is studied. Using the general methods of investigating the branching (bifurcations) of solutions for nonlinear equations, we performed an exhaustive analysis of the order parameter dependences on the control parameter in a small vicinity of the equilibrium values of parameters, including the stability analysis of the equilibrium states, and the asymptotic behavior of the order parameter dependences on the control parameter (bifurcation diagrams). The peculiarities of the transition to an unstable state of the system are discussed, and the estimates of the transition time to the unstable state in the neighborhood of ordinary and bifurcation equilibrium values of parameters are given. The influence of an external field on the dynamic behavior of thermodynamic system is analyzed, and the peculiarities of the system dynamic behavior are discussed near the ordinary and bifurcation equilibrium values of parameters in the presence of external field. The dynamic process of magnetization of a ferromagnet is discussed by using the general methods of bifurcation and stability analysis presented in the paper.     

Heteroclinic bifurcations and chaotic transport in the two-harmonic standard map

We study a two-parameter family of standard maps: the so-called two-harmonic family. In particular, we study the areas of lobes formed by the stable and unstable manifolds. Variational methods are used to find heteroclinic orbits and their action. A specific pair of heteroclinic orbits is used to define a difference in action function and to study bifurcations in the stable and unstable manifolds. Using this idea, two phenomena are studied: the change of orientation of lobes and tangential intersections of stable and unstable manifolds.     

Generators of quasiperiodic oscillations with three-dimensional phase space

Considering a family of three-dimensional oscillators originating in the field of radio-engineering, the paper describes three different mechanisms of torus formation. Particular emphasis is paid to a process in which a saddle-node bifurcation eliminates a stable cycle and leaves the system to find a stationary state between a saddle cycle and a pair of equilibrium points of unstable focus/stable node and unstable node/stable focus type.     

Sequential escapes: onset of slow domino regime via a saddle connection

We explore sequential escape behaviour of coupled bistable systems under the influence of stochastic perturbations. We consider transient escapes from a marginally stable “quiescent” equilibrium to a more stable “active” equilibrium. The presence of coupling introduces dependence between the escape processes: for diffusive coupling there is a strongly coupled limit (fast domino regime) where the escapes are strongly synchronised while for intermediate coupling (slow domino regime) without partially escaped stable states, there is still a delayed effect. These regimes can be associated with bifurcations of equilibria in the low-noise limit. In this paper, we consider a localized form of non-diffusive (i.e. pulse-like) coupling and find similar changes in the distribution of escape times with coupling strength. However, we find transition to a slow domino regime that is not associated with any bifurcations of equilibria. We show that this transition can be understood as a codimension-one saddle connection bifurcation for the low-noise limit. At transition, the most likely escape path from one attractor hits the escape saddle from the basin of another partially escaped attractor. After this bifurcation, we find increasing coefficient of variation of the subsequent escape times.     

Heteroclinic primary intersections and codimension one Melnikov method for volume-preserving maps

We study families of volume preserving diffeomorphisms in R(3) that have a pair of hyperbolic fixed points with intersecting codimension one stable and unstable manifolds. Our goal is to elucidate the topology of the intersections and how it changes with the parameters of the system. We show that the "primary intersection" of the stable and unstable manifolds is generically a neat submanifold of a "fundamental domain." We compute the intersections perturbatively using a codimension one Melnikov function. Numerical experiments show various bifurcations in the homotopy class of the primary intersections. (c) 2000 American Institute of Physics.     

非线性电路通向混沌的演化过程

给出了四阶非线性电路通向复杂性的两种演化模式,指出这两种模式与三个共存的平衡点有关.在第一种模式中,不稳定的平衡点由Hopf分岔导致了稳定的周期运动,经过倍周期分岔通向混沌,其所有的吸引子都保持对称结构;而在第二种模式中,另两个平衡点由Hopf分岔产生相互对称的极限环,并分别导致了两个混沌吸引子,其分岔过程步调一致,而且所有的吸引子都相互对称.随着参数的变化,这两个混沌吸引子相互作用形成一个扩大的混沌吸引子,导致与第一种分岔模式中定性一致的混沌运动.     

Synchronization of multi-frequency noise-induced oscillations

Using a model system of FitzHugh-Nagumo type in the excitable regime, the similarity between synchronization of self-sustained and noise-induced oscillations is studied for the case of more than one main frequency in the spectrum. It is shown that this excitable system undergoes the same frequency lockings as a self-sustained quasiperiodic oscillator. The presence of noise-induced both stable and unstable limit cycles and tori, as well as their tangential bifurcations, are discussed. As the FitzHugh-Nagumo oscillator represents one of the basic neural models, the obtained results are of high importance for neuroscience.     

Zip and velcro bifurcations in competition models in ecology and economics

During the last six years or so, a number of interesting papers discussed systems with line segments of equilibria, planes of equilibria, and with more general equilibrium configurations. This note draws attention to the fact that such equilibria were considered previously by Miklós Farkas (1932–2007), in papers published in 1984–2005. He called zip bifurcations those involving line segments of equilibria, and velcro bifurcations those involving planes of equilibria. We briefly describe prototypical situations involving zip and velcro bifurcations.

     

Noise-induced generation of saw-tooth type transitions between climate attractors and stochastic excitability of paleoclimate

Motivated by important paleoclimate applications we study a three dimensional model ofthe Quaternary climatic variations in the presence of stochastic forcing. It is shown thatthe deterministic system exhibits a limit cycle and two stable system equilibria. Wedemonstrate that the closer paleoclimate system to its bifurcation points (lying either inits monostable or bistable zone) the smaller noise generates small or large amplitudestochastic oscillations, respectively. In the bistable zone with two stable equilibria,noise induces a complex multimodal stochastic regime with intermittency of small and largeamplitude stochastic fluctuations. In the monostable zone, the small amplitude stochasticoscillations localized in the vicinity of unstable equilibrium appear along with the largeamplitude oscillations near the stable limit cycle. For the analysis of thesenoise-induced effects, we develop the stochastic sensitivity technique and use theMahalanobis metric in the three-dimensional case. To approximate the distribution ofrandom trajectories in Poincare sections, we use a method of confidence ellipses. Aspatial configuration of these ellipses is defined by the stochastic sensitivity and noiseintensity. The glaciation/deglaciation transitions going between two polar Earth’s stateswith the warm and cold climate become easier and quicker with increasing the noiseintensity. Our stochastic analysis demonstrates a near 100 ky saw-tooth type climate selffluctuations known from paleoclimate records. In addition, the enhancement of noiseintensity blurs the sharp climate cycles and reduces the glaciation-deglaciation periodsof the Earth’s paleoclimate.     

状态反馈控制声光双稳系统的倍周期分岔和混沌

设计了一种动力学状态反馈(DSF)方法控制非线性混沌系统.介绍了DSF方法的控制原理,并用此方法控制声光双稳(AOB)系统的混沌,以此验证其有效性.仿真模拟显示,通过选择恰当的控制参数,有效地实现了声光双稳(AOB)系统中倍周期分岔的延迟控制和混沌吸引子中原不稳定周期轨道的稳定控制,同时,还可以将系统控制在2np、3mp 和2np×3mp这样其它任意所需的周期轨道上.     

Stability of solitary waves in nonlinear composite media

The system of equations for planar waves in elastic composite media in the presence of anisotropy is considered. In anisotropic case two two-parametric families of solitary waves are found in an explicit form. In case of the absence of anisotropy these two families coalesce into the unique three parametric family. The solitary wave solutions are found to be orbitally stable in a certain range of their phase speeds (range of stability) both in an anisotropic as well as in an isotropic materials. It is also shown that the initial value problem for the governing equations is locally well posed which is needed to prove the stability result. The local well-posedness of the initial value problem along with stability of solitary waves implies global existence result provided the initial data lie in a neighbourhood of a stable solitary wave. This complements the previous results of blow-up for this type of equations.     

Instability of pole solutions for planar propagating flames in sufficiently large domains

It is well known that the partial differential equation (PDE) describing the dynamics of a hydrodynamically unstable planar flame front has exact pole solutions for which the PDE reduces to a set of ordinary differential equations (ODEs). The paradox, however, lies in the fact that the set of ODEs does not permit the appearance of new poles in the complex plane, or the formation of cusps in the physical space, as observed in experiments. The validity of the PDE itself has thus been questioned. We show here that the discrepancy between the PDE and the ODEs is due to the instability of exact pole solutions for the PDE. In previous work, we have reported that most exact pole solutions are indeed unstable for the PDE but, for each interval of relatively small length L, there remains one solution (up to translation symmetry) which is neutrally stable. The latter is a one-peak, coalescent solution for which the poles (whose number is maximal) are steady. The front undergoes bifurcations as the length of the domain considered increases: the one-pole, one-peak coalescent solution is first neutrally stable. As the length of the interval increases, it becomes unstable and the two-pole one-peak coalescent solution is, in turn, neutrally stable. This phenomenon occurs once again: as the two-pole solution becomes unstable, the three-pole solution becomes stable. The contribution of the present work is to show that subsequent bifurcations are of a different nature. As the interval length increases, the steady one-peak, coalescent solutions whose number of poles is maximal are no longer stable and bifurcations to unsteady states occur. In all cases, the appearance of new poles is observed in the unsteady dynamics. We also show analytically that such an instability is not permitted in the ODEs for which all steady one-peak, coalescent solutions are neutrally stable.     

Asymmetric coexisting bifurcations and multi-stability in an asymmetric memristive diode-bridge-based Jerk circuit

An asymmetric memristive diode-bridge (MDB) emulator is raised to imitate the asymmetric volt-ampere characteristic of a physical memristor. Then, an asymmetric MDB-based Jerk circuit is built and its state equation is derived, upon which the theoretical analysis, MATLAB-based numerical simulations, and hardware measurements are executed to reveal the asymmetric coexisting bifurcations and the phenomenon of multi-stability. The memristive Jerk circuit has three equilibrium points of a pair nontrivial equilibrium points of asymmetric unstable saddle-foci and a zero equilibrium point of unstable saddle-focus, which leads to the occurrence of asymmetric coexisting bifurcations and asymmetric local attraction basins. The asymmetrical bifurcations are numerically disclosed by 1-D/2-D bifurcation plots, Lyapunov spectrum, and phase plane trajectories. Multi-stability with asymmetric coexisting attractors under two sets of system parameters are demonstrated as examples by local attraction basins and phase plane trajectories. Thereafter, experimental circuit prototype employing discrete components is manually welded and hardware measurements are executed to validate the numerical simulations.     

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.     

Bifurcations of mixed-mode oscillations in a stellate cell model

Experimental recordings of the membrane potential of stellate cells within the entorhinal cortex show a transition from subthreshold oscillations (STOs) via mixed-mode oscillations (MMOs) to relaxation oscillations under increased injection of depolarizing current. Acker et al. introduced a 7D conductance based model which reproduces many features of the oscillatory patterns observed in these experiments. For the first time, we present a comprehensive bifurcation analysis of this model by using the software package AUTO. In particular, we calculate the stable MMO branches within the bifurcation diagram of this model, as well as other MMO patterns which are unstable. We then use geometric singular perturbation theory to demonstrate how the bifurcations are governed by a 3D reduced model introduced by Rotstein et al. We extend their analysis to explain all observed MMO patterns within the bifurcation diagram. A key role in this bifurcation analysis is played by a novel homoclinic bifurcation structure connecting to a saddle equilibrium on the unstable branch of the corresponding critical manifold. This type of homoclinic connection is possible due to canards of folded node (folded saddle-node) type.