Existence of a new three-dimensional chaotic attractor

In this paper, one heteroclinic orbit of a new three-dimensional continuous autonomous chaotic system, whose chaotic attractor belongs to the conjugate Lü attractor, is found. The series expression of the heteroclinic orbit of Šhil’nikov type is derived by using the undetermined coefficient method. The uniform convergence of the precise series expansions of this heteroclinic orbits is proved. According to the Šhil’nikov theorem, this system clearly has Smale horseshoes and the horseshoe chaos.     

Heteroclinic orbits in Chen circuit with time delay

Existence of Shil’nikov type of heteroclinic orbit in Chen circuit with direct time delay feedback is proved using the undetermined coefficient method. As a result, Shil’nikov criterion guarantees that the circuit has Smale horseshoes and the circuit demonstrates chaos in a rigorous analytical sense. The geometric structure of the generated chaos is determined by the heteroclinic orbits. Both the simulation and the experimental results show that chaos is indeed generated in the non-chaotic Chen circuit with the direct time delay feedback.     

Comment on “Heteroclinic orbits in Chen circuit with time delay” [Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 3058–3066]

The authors claim in the abstract of the referenced paper that Existence of Shil’nikov type of heteroclinic orbit in Chen circuit with direct time delay feedback is proved using the undetermined coefficient method. As a result, Shil’nikov criterion guarantees that the circuit has Smale horseshoes and the circuit demonstrates chaos in a rigorous analytical sense. Unfortunately, we show in this comment that their demonstration is incorrect.     

A discussion on the coexistence of heteroclinic orbit and saddle foci for third-order systems

The coexistence of heteroclinic orbits and saddle foci is concerned with the basic assumption in Shil?nikov heteroclinic theorem. Two aspects of this discussion are conducted in the paper. Firstly, many third-order systems, which possess exact heteroclinic orbits expressed by pure hyperbolic functions or the combination of hyperbolic and triangle functions and so on, have been constructed. At the same time, the existence of saddle foci is tested and some problems are proposed. Secondly and more importantly, the existence of heteroclinic orbits to saddle foci is studied. The necessary condition for the coexistence of heteroclinic orbits and saddle foci is obtained. Finally, an example is given to show the effectiveness of the results, and some conclusions and problems are presented.     

Partial symmetries and dynamical systems

We start recalling the characterizing property of the ‘partial symmetries’ of a differential problem, that is, the property of transforming solutions into solutions only in a proper subset of the full solution set. This paper is devoted to analyze the role of partial symmetries in the special context of dynamical systems and also to compare this notion with other notions of ‘weak’ symmetries, namely, the λ‐symmetries and the orbital symmetries. Particular attention is addressed to discuss the relevance of partial symmetries in dynamical systems admitting homoclinic (or heteroclinic) manifolds, which can be ‘broken’ by periodic perturbations, thus giving rise, according to the (suitably rewritten) Mel'nikov theorem, to the appearance of a chaotic behavior of Smale‐horseshoes type. Many examples illustrate all the various aspects and situations. Copyright © 2016 John Wiley & Sons, Ltd.     

Constructing piecewise linear chaotic system based on the heteroclinic Shil’nikov theorem

In this paper, a kind of piecewise linear chaotic system is constructed based on the Shil’nikov theorem. These systems have the same Jacobian in each equilibrium, and the piecewise linear functions in them are discontinuous, piecewise constants. The condition for the existence of the heteroclinic orbits in this kind of system is discussed. According to the separating plane and the position of the equilibriums, four different chaotic systems are given. Computer simulations confirm that the proposed method can be used to construct arbitrary chaotic attractors with multi-scrolls.     

Complex dynamics of a simple 3D autonomous chaotic system with four-wing

The present paper revisits a three dimensional (3D) autonomous chaotic system with four-wing occurring in the known literature [Nonlinear Dyn (2010) 60(3): 443--457] with the entitle ``A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems'' and is devoted to discussing its complex dynamical behaviors, mainly for its non-isolated equilibria, Hopf bifurcation, heteroclinic orbit and singularly degenerate heteroclinic cycles, etc. Firstly, the detailed distribution of its equilibrium points is formulated. Secondly, the local behaviors of its equilibria, especially the Hopf bifurcation, are studied. Thirdly, its such singular orbits as the heteroclinic orbits and singularly degenerate heteroclinic cycles are exploited. In particular, numerical simulations demonstrate that this system not only has four heteroclinic orbits to the origin and other four symmetry equilibria, but also two different kinds of infinitely many singularly degenerate heteroclinic cycles with the corresponding two-wing and four-wing chaotic attractors nearby.     

The number of public keys in the McEliece-Sidel’nikov cryptosystem

The McEliece-Sidel’nikov cryptosystem is a modification of the McEliece cryptosystem, which is one of the oldest public-key cryptosystems. It was proposed by V.M. Sidel’nikov in 1994 and is based on the u-fold application of Reed-Muller codes RM(r, m). The lower bound is obtained for the power of the set of public keys of the McEliece-Sidel’nikov cryptosystem using an arbitrary number of blocks (u).     

Using Lin's method to solve Bykov's problems

We consider nonwandering dynamics near heteroclinic cycles between two hyperbolic equilibria. The constituting heteroclinic connections are assumed to be such that one of them is transverse and isolated. Such heteroclinic cycles are associated with the termination of a branch of homoclinic solutions, and called T-points   in this context. We study codimension-two T-points and their unfoldings in Rⁿ${\mathbb{R}}^n$. In our consideration we distinguish between cases with real and complex leading eigenvalues of the equilibria. In doing so we establish Lin's method as a unified approach to (re)gain and extend results of Bykov's seminal studies and related works. To a large extent our approach reduces the study to the discussion of intersections of lines and spirals in the plane.     

Homoclinic Bifurcations in n Dimensions

Bifurcations near homoclinic orbits in n dimensions are described. Depending on the eigenvalues of the Jacobian at the fixed point whose real parts are closest to zero, a strange invariant set of periodic and aperiodic orbits can be produced, which can be described by a Bernoulli shift on a finite set of symbols. These results generalize earlier ones of Shil'nikov, Gaspard, Tresser, Glendinning, and Sparrow, amongst others.     

Truncation Analysis for the Derivative Schrödinger Equation

The truncation equation for the derivative nonlinear Schrödinger equation has been discussed in this paper. The existence of a special heteroclinic orbit has been found by using geometrical singular perturbation theory together with Melnikov's technique.     

MODELING THE EFFECT OF RARITY VALUE ON THE EXPLOITATION OF A WILDLIFE SPECIES SUBJECTED TO THE ALLEE EFFECT

The overexploitation of wildlife species is a serious problem in the field of biodiversity conservation. The species subjected to natural Allee effects are even more threatened by exploitation. Moreover, for many wildlife species, their rarity can fuel their exploitation by making them disproportionately desirable and consequently increasing their market price. In this paper, a mathematical model is proposed and analyzed to study how the value that consumers place on rarity can threaten the survival of a species subjected to natural Allee effects. It is assumed that the value of a species increases as its density declines. The analysis of model shows that the increase in the consumers' response to rarity can drive the system to admit Hopf‐bifurcation and heteroclinic bifurcation. The occurrence of the heteroclinic cycle indicates that the increase in consumers' response to rarity can cause the extinction of the species. It is found that an increase in the Allee threshold causes a decrease in the threshold value of consumers' response below which extinction is inevitable.     

Periodic orbit of the pendulum with a small nonlinear damping

We study the pendulum with a small nonlinear damping, which can be expressed by a Hamiltonian system with a small perturbation. We prove that a unique periodic orbit exists for any initial position between the equilibrium point and the heteroclinic orbit of the unperturbed system, depending on the choice of the bifurcation parameter in the damping. The main tools are bifurcation theory and Abelian integral technique, as well as the Zhang''s uniqueness theorem on Li\''enard equations.     

Detonatlve travelling waves for combustions

The existence of travelling wave solution to equations of a viscous heat conducting combustible fluid is proved. The reactions are assumed to be one step exothermic reactions with a natural discontinuous reaction rate function. The problem is studied for a general gas. Instead of assuming the ideal gas conditions we consider a general thermodynamics which is described by a fairly mild set of hypotheses. Travelling waves for detonations reduce to specific heteroclinic orbits of a discontinuous system of ODE's. The existence proof for heteroclinic orbits corresponding to weak and strong detonation waves is carried out by some general topological arguments in ODE. The uniqueness and nonuniqueness of these waves are also considered.     

Correlation of the two-prime Sidel��nikov sequence

Motivated by the concepts of Sidel??nikov sequences and two-prime generator (or Jacobi sequences) we introduce and analyze some new binary sequences called two-prime Sidel??nikov sequences. In the cases of twin primes and cousin primes equivalent 3 modulo 4 we show that these sequences are balanced. In the general case, besides balancedness we also study the autocorrelation, the correlation measure of order k and the linear complexity profile of these sequences showing that they have many nice pseudorandom features.     

General soliton and (semi-)rational solutions to the nonlocal Mel'nikov equation on the periodic background

In this paper, the Hirota's bilinear method and Kadomtsev-Petviashvili hierarchy reduction method are applied to construct soliton, line breather and (semi-)rational solutions to the nonlocal Mel'nikov equation with nonzero boundary conditions. These solutions are expressed as Gram-type determinants. When N is even, soliton, line breather and (semi-)rational solutions on the constant background are derived while these solutions are located on the periodic background for odd N. Regularity of these solutions and their connections with the local Mel'nikov equation are analyzed for proper choices of parameters that appear in the solutions. The dynamics of the solutions are discussed in detail. All possible configurations of soliton and lump solutions are found for . Several interesting dynamical behaviors of semi-rational solutions are observed. It is shown that certain lumps may exhibit fusion and fission phenomena during their interactions with solitons while some lump may change its direction of movement after it collides with solitons.     

Heteroclinic Ratchets in Networks of Coupled Oscillators

We analyze an example system of four coupled phase oscillators and discover a novel phenomenon that we call a “heteroclinic ratchet”; a particular type of robust heteroclinic network on a torus where connections wind in only one direction. The coupling structure has only one symmetry, but there are a number of invariant subspaces and degenerate bifurcations forced by the coupling structure, and we investigate these. We show that the system can have a robust attracting heteroclinic network that responds to a specific detuning Δ between certain pairs of oscillators by a breaking of phase locking for arbitrary Δ>0 but not for Δ≤0. Similarly, arbitrary small noise results in asymmetric desynchronization of certain pairs of oscillators, where particular oscillators have always larger frequency after the loss of synchronization. We call this heteroclinic network a heteroclinic ratchet because of its resemblance to a mechanical ratchet in terms of its dynamical consequences. We show that the existence of heteroclinic ratchets does not depend on symmetry or number of oscillators but depends on the specific connection structure of the coupled system.     

Asymptotic Stability of Pseudo-simple Heteroclinic Cycles in $${\mathbb R}^4$$

Robust heteroclinic cycles in equivariant dynamical systems in \({\mathbb R}^4\) have been a subject of intense scientific investigation because, unlike heteroclinic cycles in \({\mathbb R}^3\), they can have an intricate geometric structure and complex asymptotic stability properties that are not yet completely understood. In a recent work, we have compiled an exhaustive list of finite subgroups of O(4) admitting the so-called simple heteroclinic cycles, and have identified a new class which we have called pseudo-simple heteroclinic cycles. By contrast with simple heteroclinic cycles, a pseudo-simple one has at least one equilibrium with an unstable manifold which has dimension 2 due to a symmetry. Here, we analyze the dynamics of nearby trajectories and asymptotic stability of pseudo-simple heteroclinic cycles in \({\mathbb R}^4\).     

Codimension 3 Non-resonant Bifurcations of Rough Heteroclinic Loops with One Orbit Flip

Heteroclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near a rough heteroclinic loop. This heteroclinic loop has a principal heteroclinic orbit and a non-principal heteroclinic orbit that takes orbit flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold or three-fold 1-periodic orbit is also obtained.     

Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit

We analyze homoclinic orbits near codimension-1 and -2 heteroclinic cycles between an equilibrium and a periodic orbit for ordinary differential equations in three or higher dimensions. The main motivation for this study is a self-organized periodic replication process of travelling pulses which has been observed in reaction-diffusion equations. We establish conditions for existence and uniqueness of countably infinite families of curve segments of 1-homoclinic orbits which accumulate at codimension-1 or -2 heteroclinic cycles. The main result shows the bifurcation of a number of curves of 1-homoclinic orbits from such codimension-2 heteroclinic cycles which depends on a winding number of the transverse set of heteroclinic points. In addition, a leading order expansion of the associated curves in parameter space is derived. Its coefficients are periodic with one frequency from the imaginary part of the leading stable Floquet exponents of the periodic orbit and one from the winding number.