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.     

New nonlinear mechanisms of midlatitude atmospheric low-frequency variability

This paper studies the dynamical mechanisms potentially involved in the so-called atmospheric low-frequency variability, occurring at midlatitudes in the Northern Hemisphere. This phenomenon is characterised by recurrent non-propagating and temporally persistent flow patterns, with typical spatial and temporal scales of 6000-10 000 km and 10-50 days, respectively.We study a low-order model derived from the 2-layer shallow-water equations on a β-plane channel. The main ingredients of the low-order model are a zonal flow, a planetary scale wave, orography, and a baroclinic-like forcing.A systematic analysis of the dynamics of the low-order model is performed using techniques and concepts from dynamical systems theory. Orography height (h₀) and magnitude of zonal wind forcing (U₀) are used as control parameters to study the bifurcations of equilibria and periodic orbits. Along two curves of Hopf bifurcations an equilibrium loses stability () and gives birth to two distinct families of periodic orbits. These periodic orbits bifurcate into strange attractors along three routes to chaos: period doubling cascades, breakdown of 2-tori by homo- and heteroclinic bifurcations, or intermittency ( and ).The observed attractors exhibit spatial and temporal low-frequency patterns comparing well with those observed in the atmosphere. For the periodic orbits have a period of about 10 days and patterns in the vorticity field propagate eastward. For , the period is longer (30-60 days) and patterns in the vorticity field are non-propagating. The dynamics on the strange attractors are associated with low-frequency variability: the vorticity fields show weakening and strengthening of non-propagating planetary waves on time scales of 10-200 days. The spatio-temporal characteristics are “inherited” (by intermittency) from the two families of periodic orbits and are detected in a relatively large region of the parameter plane. This scenario provides a characterisation of low-frequency variability in terms of intermittency due to bifurcations of waves.     

Complex oscillations and waves of calcium in pancreatic acinar cells

We perform a bifurcation analysis of a model of Ca²⁺ wave propagation in the basal region of pancreatic acinar cells. The model we consider was first presented in Sneyd et al. [J. Sneyd, K. Tsaneva-Atanasova, J.I.E. Bruce, S.V. Straub, D.R. Giovannucci, D.I. Yule, A model of calcium waves in pancreatic and parotid acinar cells, Biophys. J. 85 (2003) 1392–1405], where a partial bifurcation analysis was given of the model in the absence of diffusion. We obtain more complete information about bifurcations of the diffusionless model via numerical studies, then analyse the spatially extended model by numerical investigation of the travelling wave equations and direct numerical solution of the model equations. We find solitary waves in the model equations arising from homoclinic bifurcations in the travelling wave equations. The solitary waves exist and appear to be stable for a significant interval of the primary bifurcation parameter (i.e., the concentration of inositol trisphosphate) but are eventually replaced by irregular spatio-temporal behaviour. The homoclinic bifurcations are related to a number of complicated mathematical structures in the travelling wave equations, including an anomalous homoclinic-Hopf bifurcation, heteroclinic bifurcations between an equilibrium and a periodic orbit, and homoclinic bifurcations of periodic orbits.     

Existence and Dynamics of Bounded Traveling Wave Solutions to Getmanou Equation

In this paper, we study the existence and dynamics of bounded traveling wave solutions to Getmanou equations by using the qualitative theory of differential equations and the bifurcation method of dynamical systems. We show that the corresponding traveling wave system is a singular planar dynamical system with two singular straight lines, and obtain the bifurcations of phase portraits of the system under different parameters conditions. Through phase portraits, we show the existence and dynamics of several types of bounded traveling wave solutions including solitary wave solutions, periodic wave solutions, compactons, kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions are given. Additionally, we confirm abundant dynamical behaviors of the traveling wave s olutions to the equation, which are summarized as follows: i) We confirm that two types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system. ii) We confirm that two types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, and the homoclinic orbit of associated system, which is tangent to the singular line at the singular point of associated system.     

Drift of slow variables in slow-fast Hamiltonian systems

We study the drift of slow variables in a slow-fast Hamiltonian system with several fast and slow degrees of freedom. Keeping the slow variables frozen, for any periodic trajectory of the fast subsystem we define an action. For a family of periodic orbits, the action is a scalar function of the slow variables and can be considered as a Hamiltonian function which generates some slow dynamics. These dynamics depend on the family of periodic orbits.Assuming that for the frozen slow variables the fast system has a pair of hyperbolic periodic orbits connected by two transversal heteroclinic trajectories, we prove that for any path composed of a finite sequence of slow trajectories generated by action Hamiltonians, there is a trajectory of the full system whose slow component shadows the path.     

New bifurcations of basin boundaries involving Wada and a smooth Wada basin boundary

This paper demonstrates and analyses double heteroclinic tangency in a three-well potential model, which can produce three new types of bifurcations of basin boundaries including from smooth to Wada basin boundaries, from fractal to Wada basin boundaries in which no changes of accessible periodic orbits happen, and from Wada to Wada basin boundaries. In a model of mechanical oscillator, it shows that a Wada basin boundary can be smooth.     

Unstable Manifolds of Relative Periodic Orbits in the Symmetry-Reduced State Space of the Kuramoto–Sivashinsky System

Systems such as fluid flows in channels and pipes or the complex Ginzburg–Landau system, defined over periodic domains, exhibit both continuous symmetries, translational and rotational, as well as discrete symmetries under spatial reflections or complex conjugation. The simplest, and very common symmetry of this type is the equivariance of the defining equations under the orthogonal group O(2). We formulate a novel symmetry reduction scheme for such systems by combining the method of slices with invariant polynomial methods, and show how it works by applying it to the Kuramoto–Sivashinsky system in one spatial dimension. As an example, we track a relative periodic orbit through a sequence of bifurcations to the onset of chaos. Within the symmetry-reduced state space we are able to compute and visualize the unstable manifolds of relative periodic orbits, their torus bifurcations, a transition to chaos via torus breakdown, and heteroclinic connections between various relative periodic orbits. It would be very hard to carry through such analysis in the full state space, without a symmetry reduction such as the one we present here.     

New bifurcations of basin boundaries involving Wada and a smooth Wada basin boundary

This paper demonstrates and analyses double heteroclinic tangency in a three-well potential model, which can produce three new types of bifurcations of basin boundaries including from smooth to Wada basin boundaries, from fractal to Wada basin boundaries in which no changes of accessible periodic orbits happen, and from Wada to Wada basin boundaries. In a model of mechanical oscillator, it shows that a Wada basin boundary can be smooth.     

Dynamics of a single ion in a perturbed Penning trap: octupolar perturbation

Imperfections in the design or implementation of Penning traps may give rise to electrostatic perturbations that introduce nonlinearities in the dynamics. In this paper we investigate, from the point of view of classical mechanics, the dynamics of a single ion trapped in a Penning trap perturbed by an octupolar perturbation. Because of the axial symmetry of the problem, the system has two degrees of freedom. Hence, this model is ideal to be managed by numerical techniques like continuation of families of periodic orbits and Poincaré surfaces of section. We find that, through the variation of the two parameters controlling the dynamics, several periodic orbits emanate from two fundamental periodic orbits. This process produces important changes (bifurcations) in the phase space structure leading to chaotic behavior.     

Chaos in the second—order autonomous Birkhoff system with a heteroclinic circle

Chaotic behaviour in a second-order autonomous Birkhoff system with a heteroclinic circle under weakly periodic perturbation is studied using the Melnikov method.The equations of heteroclinic orbits and the criteria for chaos are given.One example is also presented to illustrate the application of the results.     

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.     

一类相对转动非线性动力系统的混沌运动

研究一类具有同宿轨道、异宿轨道的相对转动非线性动力系统的混沌运动. 建立具有非线性刚度、非线性阻尼和外扰激励作用的一类两质量相对转动非线性动力系统的动力学方程. 利用Melnikov方法讨论了系统的全局分岔和系统进入混沌状态的可能途径,给出了系统发生混沌的必要条件,并利用最大Lyapunov指数图,分岔图,Poincare截面图和相轨迹图进一步分析了系统的混沌行为. 关键词： 相对转动 非线性动力系统 混沌 Melnikov方法     

Bifurcations and chaotic threshold for a nonlinear system with an irrational restoring force

Nonlinear dynamical systems with an irrational restoring force often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational nonlinearity avoiding the conventional Taylor's expansion to retain the natural characteristics of the system. A series of transformations are proposed to convert the homoclinic orbits of the unperturbed system to the heteroclinic orbits in the new coordinate, which can be transformed back to the analytical expressions of the homoclinic orbits. Melnikov's method is employed to obtain the criteria for chaotic motion, which implies that the existence of homoclinic orbits to chaos arose from the breaking of homoclinic orbits under the perturbation of damping and external forcing. The efficiency of the criteria for chaotic motion obtained in this paper is verified via bifurcation diagrams, Lyapunov exponents, and numerical simulations. It is worthwhile noting that our study is an attempt to make a step toward the solution of the problem proposed by Cao Q J et al. (Cao Q J, Wiercigroch M, Pavlovskaia E E, Thompson J M T and Grebogi C 2008 Phil. Trans. R. Soc. A 366 635).     

Heteroclinic Connections Between Periodic Orbits in Planar Restricted Circular Three Body Problem. Part II

We present a method for proving the existence of symmetric periodic, heteroclinic or homoclinic orbits in dynamical systems with the reversing symmetry. As an application we show that the Planar Restricted Circular Three Body Problem (PCR3BP) corresponding to the Sun-Jupiter-Oterma system possesses an infinite number of symmetric periodic orbits and homoclinic orbits to the Lyapunov orbits. Moreover, we show the existence of symbolic dynamics on six symbols for PCR3BP and the possibility of resonance transitions of the comet. This extends earlier results by Wilczak and Zgliczynski [12]. Electronic Supplementary Material: Supplementary material is available in the online version of this article at An erratum to this article is available at .     

Faraday waves, streaming flow, and relaxation oscillations in nearly circular containers

Faraday waves near onset in an elliptical container are described by a third-order system of ordinary differential equations with characteristic slow-fast structure. These equations describe the interaction of standing waves with a weakly damped streaming flow driven by Reynolds stresses in boundary layers at the free surface and the rigid walls, and capture the proliferation with decreasing damping of periodic and nonperiodic relaxation oscillations observed near onset in previous simulations. These structures are the result of slow drift through symmetry-related Hopf bifurcations.     

Dislocations in an Anisotropic Swift-Hohenberg Equation

We study the existence of dislocations in an anisotropic Swift-Hohenberg equation. We find dislocations as traveling or standing waves connecting roll patterns with different wavenumbers in an infinite strip. The proof is based on a bifurcation analysis. Spatial dynamics and center-manifold reduction yield a reduced, coupled-mode system of differential equations. Existence of traveling dislocations is then established by showing that this reduced system possesses robust heteroclinic orbits.     

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.     

Heteroclinic orbits in a model for Langmuir circulations

Numerical studies of a 5-mode model for Langmuir circulations show the existence if two fundamentally different types of heteroclinic orbits and period-doubling bifurcations. The first is common to other models of competing instabilities; the second is quite new and appears to be related to the quintic Duffing equation.     

Langevin Dynamics with a Tilted Periodic Potential

We study a Langevin equation for a particle moving in a periodic potential in the presence of viscosity γ and subject to a further external field α. For a suitable choice of the parameters α and γ the related deterministic dynamics yields heteroclinic orbits. In such a regime, in absence of stochastic noise both confined and unbounded orbits coexist. We prove that, with the inclusion of an arbitrarly small noise only the confined orbits survive in a sub-exponential time scale.