Heteroclinic Orbits Research Articles

Influence of optical property contrast on the critical distribution of electrostatic torques in double-beam torsional Casimir actuators: Non-linear actuation toward chaotic motion.

Here, we discuss how to achieve the stable actuation of a double beam torsional micro-actuator over the largest possible displacement of the moving component under the influence of Casimir and electrostatic torques, when the rotating component is constructed from different materials. The main part of this study is devoted to finding the optimal distribution of the electrostatic torque between the left and right sides of the micro-actuator to reach the maximum stable operation of the device. The latter is manifested by switching from homoclinic to heteroclinic orbits in the phase portraits. Indeed, the bifurcation curves and the phase portraits have been employed to show the sensitivity of the critical distribution of the electrostatic torque, beyond which the device does show stable performance, on the contrast of the optical properties of the moving component and the applied voltage in a conservative autonomous system. Moreover, for driven systems, the Melnikov function approach and the Poincaré portraits are used to study the presence of chaotic motion, which eventually leads to stiction. It is shown that the application of the optimal distribution of the electrostatic torque can significantly decrease the possibility of chaotic motion, and at this optimal level, the threshold curves reveal less difference between systems with different optical contrast.

Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition

Dynamical systems containing heteroclinic cycles and networks can be invoked as models of intransitive competition between three or more species. When populations are assumed to be well-mixed, a system of ordinary differential equations (ODEs) describes the interaction model. Spatially extending these equations with diffusion terms creates a system of partial differential equations which captures both the spatial distribution and mobility of species. In one spatial dimension, travelling wave solutions can be observed, which correspond to periodic orbits in ODEs that describe the system in a steady-state travelling frame of reference. These new ODEs also contain a heteroclinic structure. For three species in cyclic competition, the topology of the heteroclinic cycle in the well-mixed model is preserved in the steady-state travelling frame of reference. We demonstrate that with four species, the heteroclinic cycle which exists in the well-mixed system becomes a heteroclinic network in the travelling frame of reference, with additional heteroclinic orbits connecting equilibria not connected in the original cycle. We find new types of travelling waves which are created in symmetry-breaking bifurcations and destroyed in an orbit flip bifurcation with a cycle between only two species. These new cycles explain the existence of ‘defensive alliances’ observed in previous numerical experiments. We further describe the structure of the heteroclinic network for any number of species, and we conjecture how these results may generalise to systems of any arbitrary number of species in cyclic competition.

Invariant Tori and Heteroclinic Invariant Ellipsoids of a Generalized Hopf–Langford System

In this paper, the bounded invariant surfaces of a generalized Langford system are discussed. Firstly, by the first integrals of systems restricted in the Poincaré sections of a periodic orbit, the accurate expressions of a heteroclinic orbit, a family of invariant tori and a heteroclinic invariant ellipsoid are given near a periodic orbit. Then, applying the successor functions to compute the periods of periodic orbits for the systems in the Poincaré sections, we present the parameter conditions for the existence of periodic orbits with any periods on these invariant tori. Finally, using the averaging theory and the theory of the Poincaré bifurcation and by determining the monotonicity of the ratio of two Abelian integrals, we give the conditions respectively such that the system has a unique invariant torus and a unique heteroclinic invariant ellipsoid near a zero-Hopf equilibrium.

Periodic Perturbations of Codimension-Two Bifurcations with a Double Zero Eigenvalue in Symmetrical Dynamical Systems

We study bifurcation behavior in periodic perturbations of two-dimensional symmetric systems exhibiting codimension-two bifurcations with a double zero eigenvalue when the frequencies of the perturbation terms are small. We transform the periodically perturbed systems to simpler ones which are periodic perturbations of the normal forms for the codimension-two bifurcations, and apply the subharmonic and homoclinic Melnikov methods to analyze bifurcations occurring there. In particular, we show that there exist transverse homoclinic or heteroclinic orbits, which yield chaotic dynamics, in wide parameter regions. These results can be applied to three or higher-dimensional systems and even to infinite-dimensional systems with the assistance of center manifold reduction and the invariant manifold theory. We illustrate our theory for a pendulum subjected to position and velocity feedback control when the desired position is periodic in time. We also give numerical computations by the computer tool AUTO to demonstrate the theoretical results.

Orbits and chaos of a two‐fluid magnetized plasma oscillator subjected to parametric excitation with square delayed feedback

AbstractIn this manuscript, a two‐fluid magnetized plasma oscillator subjected to parametric excitation with square delayed feedback is studied both analytically and numerically. The Hamilton systems of triple‐well and narrow single‐well are discussed in detail. The scenarios of phase portraits and equilibria are given. Homoclinic and heteroclinic orbits are strictly derived. With the Melnikov method, the critical value of chaos arising from homoclinic or heteroclinic intersections is derived analytically. We have discovered some interesting dynamic phenomena, such as uncontrollable time delays with which chaos always occurs for this system. The influence of time‐delay on the chaotic property is also studied rigorously. On the basis of theoretical analysis, some numerical simulations including time histories, phase portraits, bifurcation diagrams, Poincaré sections, Lyapunov exponential spectrums and attractor basins are given. Numerical simulations are consistent with theoretical results.

Stochastic Hopf bifurcation and random chaos of the ship rolling dynamic system in random longitudinal wave induced by GWN

In this paper, the dynamic system of ship rolling in random longitudinal waves induced by Gaussian white noise (GWN) is studied analytically and numerically. The dynamic equation of this model which is a quintic extended Duffing system with cubic damping as well as parametric and random excitations is obtained with the Li’s modeling method. The basic dynamic properties including equilibria, potential wells, and classical solutions are derived using analytical methods and their parameter bifurcations are discussed. Stochastic Hopf bifurcation is investigated by using the three-indexes method and the stochastic dynamic theory. It is found that there is a security threshold and a risk factor affecting the traveling of the ship in random waves. Random chaos is studied by random Melnikov method. The chaotic thresholds of some applicable homoclinic orbits and heteroclinic orbits are given. Numerical simulations including phase portraits, time histories, basin of attraction, Poincaré maps, Lyapunov spectra and bifurcation diagrams are given, which verify the analytical results. Utilizing numerical methods, it is also found that the system can experience random chaotic states through random Hopf bifurcation.

Design of Sturm global attractors 1: Meanders with three noses, and reversibility.

We systematically explore a simple class of global attractors, called Sturm due to nodal properties, for the semilinear scalar parabolic partial differential equation (PDE) ut=uxx+f(x,u,ux) on the unit interval 0<x<1, under Neumann boundary conditions. This models the interplay of reaction, advection, and diffusion. Our classification is based on the Sturm meanders, which arise from a shooting approach to the ordinary differential equation boundary value problem of equilibrium solutions ut=0. Specifically, we address meanders with only three "noses," each of which is innermost to a nested family of upper or lower meander arcs. The Chafee-Infante paradigm, with cubic nonlinearity f=f(u), features just two noses. Our results on the gradient-like global PDE dynamics include a precise description of the connection graphs. The edges denote PDE heteroclinic orbits v1⇝v2 between equilibrium vertices v1,v2 of adjacent Morse index. The global attractor turns out to be a ball of dimension d, given as the closure of the unstable manifold Wu(O) of the unique equilibrium with maximal Morse index d. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graph indicates time reversibility on the (d-1)-sphere boundary of the global attractor.

Spectral Stability of Shock-fronted Travelling Waves Under Viscous Relaxation

Reaction-nonlinear diffusion partial differential equations can exhibit shock-fronted travelling wave solutions. Prior work by Li et al. (Physica D 423:132916, 2021) has demonstrated the existence of such waves for two classes of regularizations, including viscous relaxation (see Li et al. in Physica D 423:132916, 2021). Their analysis uses geometric singular perturbation theory: for sufficiently small values of a parameter ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon > 0$$\\end{document} characterizing the ‘strength’ of the regularization, the waves are constructed as perturbations of a singular heteroclinic orbit. Here we show rigorously that these waves are spectrally stable for the case of viscous relaxation. Our approach is to show that for sufficiently small ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon >0$$\\end{document}, the ‘full’ eigenvalue problem of the regularized system is controlled by a reduced slow eigenvalue problem defined for ε=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon = 0$$\\end{document}. In the course of our proof, we examine the ways in which this geometric construction complements and differs from constructions of other reduced eigenvalue problems that are known in the wave stability literature.

Complex dynamics of a sub-quadratic Lorenz-like system

Abstract Motivated by the generic dynamical property of most quadratic Lorenz-type systems that the unstable manifolds of the origin tending to the stable manifold of nontrivial symmetrical equilibria forms a pair of heteroclinic orbits, this technical note reports a new 3D sub-quadratic Lorenz-like system: x ˙ = a ( y − x ) \dot{x}=a(y-x) , y ˙ = c x 3 + d y − x 3 z \dot{y}=c\sqrt[3]{x}+{\rm{d}}y-\sqrt[3]{x}z and z ˙ = − b z + x 3 y \dot{z}=-bz+\sqrt[3]{x}y . Instead, the unstable manifolds of nontrivial symmetrical equilibria tending to the stable manifold of the origin creates a pair of heteroclinic orbits. This drives one to further investigate it and reveal its other hidden dynamics: Hopf bifurcation, invariant algebraic surfaces, ultimate bound sets, globally exponentially attractive sets, existence of homoclinic and heteroclinic orbits, singularly degenerate heteroclinic cycles, and so on. The main contributions of this work are summarized as follows: First, the ultimate boundedness of that system yields the globally exponentially attractive sets of it. Second, the existence of another heteroclinic orbits is also proved by utilizing two different Lyapunov functions. Finally, on the invariant algebraic surface z = 3 4 a x 4 3 z=\frac{3}{4a}\sqrt[3]{{x}^{4}} , the existence of a pair of homoclinic orbits to the origin, and two pairs of heteroclinic orbits to two pairs of nontrivial symmetrical equilibria is also proved by utilizing a Hamiltonian function. In addition, the correctness of the theoretical results is illustrated via numerical examples.

Instability of waves in deep water — A discrete Hamiltonian approach

The stability of waves in deep water has classically been approached via linear stability analysis, with various model equations, such as the nonlinear Schrödinger equation, serving as points of departure. Some of the most well-studied instabilities involve the interaction of four waves – so called Type I instabilities – or five waves – Type II instabilities. A unified description of four and five wave interaction can be provided by the reduced Hamiltonian derived by Krasitskii (1994). Exploiting additional conservation laws, the discretised Hamiltonian may be used to shed light on these four and five wave instabilities without restrictions on spectral bandwidth. We derive equivalent autonomous, planar dynamical systems which allow for straightforward insight into the emergence of instability and the long time dynamics. They also yield new steady-state solutions, as well as discrete breathers associated with heteroclinic orbits in the phase space.

Persistence of traveling waves to the time fractional Keller‐Segel system with a small parameter

This paper aims to investigate the time fractional Keller‐Segel system with a small parameter. After the fractional order traveling wave transformation, the heteroclinic orbit of the degenerate time fractional Keller‐Segel system is demonstrated through the method of constructing a suitable invariant region. Moreover, the persistence of traveling waves in the system with a small parameter can be further illustrated. The results are mainly reliance on the application of geometric singular perturbation theory and Fredholm theorem, which are fundamental theoretical frameworks for dealing with problems of complexity and high dimensionality. Eventually, the asymptotic behavior is depicted by the asymptotic theory to illustrate the rate of decay for traveling waves.

Dynamics of coexisting rotating waves in unidirectional rings of bistable Duffing oscillators

We study the dynamics of multistable coexisting rotating waves that propagate along a unidirectional ring consisting of coupled double-well Duffing oscillators with different numbers of oscillators. By employing time series analysis, phase portraits, bifurcation diagrams, and basins of attraction, we provide evidence of multistability on the route from coexisting stable equilibria to hyperchaos via a sequence of bifurcations, including the Hopf bifurcation, torus bifurcations, and crisis bifurcations, as the coupling strength is increased. The specific bifurcation route depends on whether the ring comprises an even or odd number of oscillators. In the case of an even number of oscillators, we observe the existence of up to 32 coexisting stable fixed points at relatively weak coupling strengths, while a ring with an odd number of oscillators exhibits 20 coexisting stable equilibria. As the coupling strength increases, a hidden amplitude death attractor is born in an inverse supercritical pitchfork bifurcation in the ring with an even number of oscillators, coexisting with various homoclinic and heteroclinic orbits. Additionally, for stronger coupling, amplitude death coexists with chaos. Notably, the rotating wave speed of all coexisting limit cycles remains approximately constant and undergoes an exponential decrease as the coupling strength is increased. At the same time, the wave frequency varies among different coexisting orbits, exhibiting an almost linear growth with the coupling strength. It is worth mentioning that orbits originating from stronger coupling strengths possess higher frequencies.

Rate-induced tipping to metastable Zombie fires

Zombie firesin peatlands disappear from the surface, smoulder underground during the winter, and ‘come back to life’ in the spring. They can release hundreds of megatonnes of carbon into the atmosphere per year and are believed to be caused by surface wildfires. Here, we propose rate-induced tipping (R-tipping) to a subsurface hot metastable state in bioactive peat soils as a main cause of Zombie fires. Our hypothesis is based on a conceptual soil-carbon model subjected to realistic changes in weather and climate patterns, including global warming scenarios and summer heatwaves. Mathematically speaking, R-tipping to the hot metastable state is a genuine nonautonomous instability, due to crossing an elusive quasi-threshold, in a multiple-timescale dynamical system. To explain this instability, we provide a framework combining a special compactification technique with concepts from geometric singular perturbation theory. This framework allows us to reduce an R-tipping problem due to crossing a quasi-threshold to a heteroclinic orbit problem in a singular limit. We identify generic cases of tracking–tipping transitions via: (i) unfolding of a codimension-twoheteroclinic folded-saddle-node type-I singularityfor global warming and (ii) analysis of a codimension-onesaddle-to-saddle hetroclinic orbitfor summer heatwaves, in turn revealing new types of excitability quasi-thresholds.

Spreading and Extinction of Solutions to the Logarithmic Diffusion Equation with a Logistic Reaction

We investigate the behavior of positive solutions of \begin{equation*} \left\{ \ \begin {aligned} &\partial_t u = \partial_x^2 (\log u) + u(1-u), \qquad \quad x\in \mathbb {R}, \\ & \lim_{x\to -\infty } \partial_x (\log u) = \alpha, \quad \lim_{x\to \infty } \partial_x (\log u) = - \beta, \end {aligned} \right. \nonumber \end{equation*} where are given nonnegative constants. This equation has a traveling pulse solution if and only if , and any solution vanishes in a finite time when . For the case of , we successfully classified the behavior of solutions into three categories: spreading, boundedness of total mass, and vanishing. Then, we describe in detail the asymptotic behavior of solutions in each category. First, the transition layer of the spreading solution propagates with the speed of a traveling front. Second, the mass-bounded solution converges to a traveling pulse solution. Third, a rescaled profile of vanishing solution converges to some traveling pulse in an asymptotically self-similar manner. In addition, the heteroclinic orbits that connect the traveling pulse to are constructed.

Heteroclinic switching between chimeras in a ring of six oscillator populations.

In a network of coupled oscillators, a symmetry-broken dynamical state characterized by the coexistence of coherent and incoherent parts can spontaneously form. It is known as a chimera state. We study chimera states in a network consisting of six populations of identical Kuramoto-Sakaguchi phase oscillators. The populations are arranged in a ring, and oscillators belonging to one population are uniformly coupled to all oscillators within the same population and to those in the two neighboring populations. This topology supports the existence of different configurations of coherent and incoherent populations along the ring, but all of them are linearly unstable in most of the parameter space. Yet, chimera dynamics is observed from random initial conditions in a wide parameter range, characterized by one incoherent and five synchronized populations. These observable states are connected to the formation of a heteroclinic cycle between symmetric variants of saddle chimeras, which gives rise to a switching dynamics. We analyze the dynamical and spectral properties of the chimeras in the thermodynamic limit using the Ott-Antonsen ansatz and in finite-sized systems employing Watanabe-Strogatz reduction. For a heterogeneous frequency distribution, a small heterogeneity renders a heteroclinic switching dynamics asymptotically attracting. However, for a large heterogeneity, the heteroclinic orbit does not survive; instead, it is replaced by a variety of attracting chimera states.

Existence of traveling wave solutions to reaction-diffusion-ODE systems with hysteresis

This paper establishes the existence of traveling wave solutions to a reaction-diffusion equation coupled with a singularly perturbed first order ordinary differential equation with a small parameter ϵ>0. The system is a toy model for biological pattern formation. Traveling wave solutions correspond to heteroclinic orbits of a fast-slow system. Under some conditions, the reduced problem (with ϵ=0) has a heteroclinic orbit with jump discontinuity, while the layer problem (i.e., the fast subsystem obtained as another limit of ϵ→0) has an orbit filling the gap. We thus construct a singular orbit by piecing together these two orbits. The traveling wave solution is obtained in the neighborhood of the singular orbit. However, unlike the classical FitzHugh-Nagumo equations, the singular orbit contains a fold point where the normal hyperbolicity breaks down and the standard Fenichel theory is not applicable. To circumvent this difficulty we employ the directional blowup method for geometric desingularization around the fold point.

Modeling, dynamical analysis and numerical simulation of a new 3D cubic Lorenz-like system

Little seems to be considered about the globally exponentially asymptotical stability of parabolic type equilibria and the existence of heteroclinic orbits in the Lorenz-like system with high-order nonlinear terms. To achieve this target, by adding the nonlinear terms yz and x^{2}y to the second equation of the system, this paper introduces the new 3D cubic Lorenz-like system: dot{x}=a(y - x), dot{y}=b_{1}y+b_{2}yz+b_{3}xz+b_{4}x^{2}y, dot{z}= -cz + y^{2}, which does not belong to the generalized Lorenz systems family. In addition to giving rise to generic and degenerate pitchfork bifurcation, Hopf bifurcation, hidden Lorenz-like attractors, singularly degenerate heteroclinic cycles with nearby chaotic attractors, etc., one still rigorously proves that not only the parabolic type equilibria S_{x} = {(x, x, frac{x^{2}}{c})|xin mathbb {R}, cne 0} are globally exponentially asymptotically stable, but also there exists a pair of symmetrical heteroclinic orbits with respect to the z-axis, as most other Lorenz-like systems. This study may offer new insights into revealing some other novel dynamic characteristics of the Lorenz-like system family.

Fear Effect on a Predator–Prey Model with Non-Differential Fractional Functional Response

In this paper, we study the factor of the fear effect in a predator–prey model with prey refuge and a non-differentiable fractional functional response due to the group defense. Since the functional response is non-differentiable, the dynamics of this system are considerably different from the dynamics of a classical predator–prey system. The persistence, the stability and the existence of the steady states are investigated. We examine the Hopf bifurcation at the unique positive equilibrium. Direct Hopf bifurcation is studied via the central manifold theorem. When the value of the fear factor decreases and is less than a threshold κH, the limit cycle appears, and it disappears through a loop of heteroclinic orbits when the value of the fear factor is equal to a value κhet.

Stability and chaotic dynamic analysis of Li-doped fullerene-IRMOF composite materials for hydrogen storage

Isoreticular metal organic framework (IRMOF) material is a type of ideal hydrogen storage material for its porosity and large surface area. Lithium doping and fullerene impregnation can enhance the hydrogen storage capacity of IRMOF efficiently. In this paper, the nonlinear dynamic characteristics of Li-doped fullerene-IRMOF composite material are qualitatively discussed in theory, which determine its thermal stability. A type of Li-doped fullerene-IRMOF composite material is proposed for hydrogen storage, and a new constitutive model is proposed to describe the hysteretic property of IRMOF. The system's nonlinear dynamic model is developed, and the fast-slow dynamic analysis method is introduced to interpret not only the system's jumping phenomena between the multi-pulse homoclinic and heteroclinic orbits but also its chaotic dynamics. The nonlinear dynamic responses of the system under deterministic and stochastic excitation are obtained respectively, and symmetrical and asymmetric orbital fusion phenomena have been discovered. The results of theoretical analysis and numerical simulation both show that the system has novel nonlinear dynamic characteristics such as double periodic motion, multi-pulse jumping orbits, and chaos; multi-pulse jumping phenomena are caused by the stiffness difference between fullerene and IRMOF, which form a fast-slow system; symmetrical fusion and asymmetrical fusion of system's orbits occur under different parameters. Safe, efficient and rapid hydrogen storage and release can be achieved through controlling the system's parameters to adjust the dynamic behavior of the system, which will be a development direction in the application of hydrogen energy.

Global dynamics of an SIS compartment model with resource constraints.

This paper formulates a mathematical framework to describe the dynamics of SIS-type infectious diseases with resource constraints. We first define the basic reproduction number that determines disease prevalence and analyze the existence and local stability of the equilibria. Subsequently, we analyze the global dynamics of the model, excluding periodic solutions and heteroclinic orbits, using the compound matrix approach. The analysis implies that the model can undergo forward and backward bifurcations depending on critical parameters. In the former scenario, the disease persists when the basic reproduction number under resource constraints exceeds one. In the latter scenario, the backward bifurcation creates bistability dynamics in which the disease may persist or become extinct depending on the initial level of infected individuals and the resource abundance.