Nikov Type Research Articles

Design of multi-wing 3D chaotic systems with only stable equilibria or no equilibrium point using rotation symmetry

Some dissipative dynamical systems only have stable equilibria or have no equilibrium point but are able to generate chaos of non-Sil’nikov type. This category of chaotic attractors belong to hidden attractor, which is difficult to be located in the phase space. In this work, we aim at designing multi-wing 3D chaotic systems with hidden attractors. We first proposes a quadratic system with only two stable node-foci, which can generate double-wing chaotic hidden attractor. Using rotation symmetry, we construct multi-fold cover of the quadratic system and obtain multiple symmetric stable equilibria. Two examples of 2-fold and 3-fold covers of the chaotic system are presented. Furthermore, the rotation symmetry is applied to no-equilibrium system to generate multi-wing chaotic hidden attractors. By means of bifurcation diagram, finite-time Lyapunov exponents, phase portraits, and attraction basins, it is shown that the multi-wing chaotic attractors always coexist with stable point attractors. The local initial condition areas and complex boundaries for different coexisting attractors are addressed. Finally, the designed systems are implemented by analog circuit and microcontroller to verify the multi-wing hidden attractors.

Hopf Bifurcation and Hopf-Pitchfork Bifurcation in an Integro-Differential Reaction-Diffusion System

We study the bifurcations of small amplitude time-periodic solutions and chaotic solutions of a two-component integro-differential reaction-diffusion system in one spatial dimension. The system has doubly degenerate points and triply degenerate points. The following results are obtained. (I) Around the doubly degenerate points, a reduced two-dimensional dynamical system on the center manifold is obtained. We find that the small amplitude stable time-periodic solutions can bifurcate from the non-uniform stationary solutions through the Hopf bifurcations for all $n$. (II) Around the triply degenerate point, a three-dimensional dynamical system on the center manifold is obtained. The reduced system can be transformed into normal form for the Hopf-Pitchfork bifurcation. The truncated normal form can possess the invariant tori and the heteroclinic loop. Furthermore, the system under the non $S^{1}$-symmetric perturbation may possess the Shil'nikov type homoclinic orbit. Numerical results for the integro-differential reaction-diffusion system are presented and found to be convincing.

Fourier–Dunkl system of the second kind and Euler–Dunkl polynomials

We prove a partial fraction decomposition of a quotient of two functions Eα(itx) and Iα(it) which are defined in terms of the Bessel functions Jα and Jα+1 of the first kind. This expansion leads naturally to the introduction of an orthonormal system with respect to the measure |x|2α+1dx2α+1Γ(α+1) in [−1,1], which we call the Fourier–Dunkl system of the second kind. Euler–Dunkl polynomials En,α(x) of degree n are defined by considering Eα(tx)∕Iα(t) as a generating function. It is shown that the sum ∑m=1∞1∕jm,α2k, where jm,α are the positive zeros of Jα, is equal (up to an explicit factor) to E2k−1,α(1). For α=1∕2 this leads to classical results of Euler since the function E1∕2(x) is the exponential function and En,1∕2(x) are (essentially) the Euler polynomials. In the second part of the paper a sampling theorem of Whittaker–Shannon–Kotel’nikov type is established which is strongly related to the above-mentioned partial decomposition and which holds for all functions in the Payley–Wiener space defined by the Dunkl transform in [−1,1].

Horseshoes in 4-dimensional piecewise affine systems with bifocal heteroclinic cycles.

By studying the Poincaré map in a neighborhood of the bifocal heteroclinic cycle (the corresponding subsystems only have conjugate complex eigenvalues), this paper provides a result on the existence of chaotic invariant sets for the two-zone 4-dimensional piecewise affine systems with bifocal heteroclinic cycles that cross the switching manifold transversally at two points. Different from Shil'nikov type theorems, the existence of chaotic invariant sets near the heteroclinic cycles depends not only on the eigenvalue conditions but also on the way of intersections of the stable manifolds and unstable manifolds of the subsystems.

Chaotic Dynamics in Four-Dimensional Piecewise Affine Systems with Bifocal Heteroclinic Cycles

The well-known Shil’nikov type theory provides an approach to proving the existence of chaotic invariant sets for some classes of smooth dynamical systems with homoclinic orbits or heteroclinic cycles. However, it cannot be applied to nonsmooth systems directly. Based on the similar ideas, this paper studies the existence of chaotic invariant sets for a class of two-zone four-dimensional piecewise affine systems with bifocal heteroclinic cycles that cross the switching manifold transversally at two points. It turns out that there exist countable infinite chaotic invariant sets in a neighborhood of the bifocal heteroclinic cycle under some eigenvalue conditions. Moreover, the horseshoes of the corresponding Poincaré map are topologically semi-conjugated to a full shift on four symbols.

Chaos in the Takens–Bogdanov bifurcation with O(2) symmetry

ABSTRACTThe Takens–Bogdanov bifurcation is a codimension-two bifurcation that provides a key to the presence of complex dynamics in many systems of physical interest. When the system is translation-invariant in one spatial dimension with no left-right preference the imposition of periodic boundary conditions leads to the Takens–Bogdanov bifurcation with O(2) symmetry. This bifurcation, analyzed by G. Dangelmayr and E. Knobloch, Phil. Trans. R. Soc. London A 322, 243 (1987), describes the interaction between steady states and travelling and standing waves in the nonlinear regime and predicts the presence of modulated travelling waves as well. The analysis reveals the presence of several global bifurcations near which the averaging method (used in the original analysis) fails. We show here, using a combination of numerical continuation and the construction of appropriate return maps, that near the global bifurcation that terminates the branch of modulated travelling waves, the normal form for the Takens–Bogdanov bifurcation admits cascades of period-doubling bifurcations as well as chaotic dynamics of Shil'nikov type. Thus chaos is present arbitrarily close to the codimension-two point.

On the Position of Chaotic Trajectories

The main purpose of this paper is to locate trajectories of a perturbed system, which is known to behave chaotically. The unperturbed system is assumed to have the origin as a hyperbolic fixed point, and to admit a trajectory homoclinic to the origin. This homocline is assumed to lie in a prescribed region having the origin in its border. Using a Mel’nikov type approach, we introduce natural conditions ensuring that all the chaotic trajectories of the perturbed system, given by classical results, lie in the same region. The applicability of our results is illustrated in two examples. In the first one, we find positive radial solutions for a class of P.D.E.’s, obtaining new results in the case of critical equations ruled by Laplacian with Hardy potentials. In the other one, we show that under certain conditions one of two weakly coupled pendula moves in one direction only.

Chaos generator design with piecewise affine systems

This paper considers design of chaos generator with a class of three-dimensional two-zone piecewise affine systems. A criterion is established to guarantee the existence of a heteroclinic cycle connecting two saddle-focus equilibrium points located in two zones, respectively, by which the existence of chaotic attractors in a neighborhood of the heteroclinic cycle can be easily obtained by Shil’nikov type theory. Based on the mathematical criteria proposed, we present a methodology for design of chaos generator.

Shilnikov Homoclinic Bifurcation of Mixed-Mode Oscillations

The Koper model is a three-dimensional vector field that was developed to study complex electrochemical oscillations arising in a diffusion process. Koper and Gaspard described paradoxical dynamics in the model: they discovered complicated, chaotic behavior consistent with a homoclinic orbit of Shil'nikov type, but were unable to locate the orbit itself. The Koper model has since served as a prototype to study the emergence of mixed-mode oscillations (MMOs) in slow-fast systems, but only in this paper is the existence of these elusive homoclinic orbits established. They are found first in a larger family that has been used to study singular Hopf bifurcation in multiple time scale systems with two slow variables and one fast variable. A curve of parameters with homoclinic orbits in this larger family is obtained by continuation and shown to cross the submanifold of the Koper system. The strategy used to compute the homoclinic orbits is based upon systematic investigation of intersections of invariant manifolds in this system with multiple time scales. Both canards and folded nodes are multiple time scale phenomena encountered in the analysis. Suitably chosen cross-sections and return maps illustrate the complexity of the resulting MMOs and yield a modified geometric model from the one Shil'nikov used to study spiraling homoclinic bifurcations.

Analysis of Dynamical Behaviors in a Continuous Chaotic System Without Šil'nikov Orbits

This paper discusses a simple yet interesting three-dimensional autonomous quadratic chaotic system, which has two fixed points of saddle-focus type but does not belong to the class of Šil'nikov type since there are neither homoclinic nor heteroclinic orbits. Dynamical behaviors of this chaotic system are investigated in detail through analyzing bifurcation routes, Lyapunov exponents and Poincaré mappings. These investigations partially reveal the mechanism of how chaos in non-Šil'nikov type of systems is generated.

Mel’nikov Methods and Homoclinic Orbits in Discontinuous Systems

We consider a discontinuous system exhibiting a, possibly non-smooth, homoclinic trajectory. We assume that the critical point lies on the discontinuity level. We study the persistence of such a trajectory when the system is subject to a smooth non-autonomous perturbation. We use a Mel’nikov type approach and we introduce conditions which enable us to reformulate the problem in the setting of smooth systems so that we can follow the outline of the classical theory.

Chaotification in the stretch-twist-fold (STF) flow

Chaotic behavior and detailed parameter analysis of stretch-twist-fold (STF) flow are investigated. STF flow is associated with fluid particle motion which naturally arises in the dynamo theory. It proposes a mechanism, by which a celestial bodies, such as earth and sun, can maintain and amplify the magnetic field continuously. Parameter analysis is performed using linearization theory for different choices of parameters. The existence of Heteroclinic trajectory of Sil’nikov type is proved using an undetermined coefficient method. It connects two non trivial equilibrium points. As a consequence, the Sil’nikov criterion guarantees that STF flow has Smale horseshoes chaos.

QUADRATIC VECTOR FIELDS EQUIVARIANT UNDER THE D2 SYMMETRY GROUP

Symmetry often plays an important role in the formation of complicated structures in the dynamics of vector fields. Here, we study a specific family of systems defined on ℝ3, which are invariant under the D2 symmetry group. Under the assumption that they are polynomial of degree at most two, they belong to a two-parameter family of vector fields, called the D2 model. We describe the global behavior of the system, for most parameter values, and locate a region of parameter space where complicated structures occur. The existence of heteroclinic and homoclinic orbits is shown, as well as of heteroclinic cycles (for other parameter values), implying the presence of (different types of) Shil'nikov type of chaos in the D2 systems. We then employ Poincaré maps to illustrate the bifurcations leading to this behavior. The global bifurcations exhibited by its strange attractors are explained as an effect of symmetry. We conclude by describing the behavior of the system at infinity.

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.

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.

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.

Sil'nikov chaos of the Liu system

A new chaotic attractor is discovered for the Liu system. The homoclinic and heteroclinic orbits in the Liu system have been found by using the undetermined coefficient method. It analytically demonstrates that there exists one heteroclinic orbit of the Sil'nikov type that connects two nontrivial equilibrium points, and therefore Smale horseshoes and the horseshoe chaos occur for this system via the Sil'nikov criterion. In addition, there also exists one homoclinic orbit joined to the origin. The convergence of the series expansions of these two types of orbits is proved.

SHIL'NIKOV HETEROCLINIC ORBITS IN A CHAOTIC SYSTEM

In this paper, a chaotic system which exhibits a chaotic attractor with only three equilibria for some parameters is considered. The existence of heteroclinic orbits of the Shil'nikov type in a chaotic system has been proved using the undetermined coefficient method. As a result, the Shil'nikov criterion guarantees that the system has Smale horseshoes. Moreover, the geometric structures of the attractor are determined by these heteroclinic orbits.

Construction and Verification of a Simple Smooth ChaoticSystem

This article introduces a new chaotic system of three-dimensionalquadratic autonomous ordinary differential equations, which candisplay different attractors with two unstable equilibrium pointsand four unstable equilibrium points respectively. Dynamicalproperties of this system are then studied. Furthermore, by applyingthe undetermined coefficient method, heteroclinic orbit of Shil'nikov's type in this system is found and the convergence of theseries expansions of this heteroclinic orbit are proved in thisarticle. The Shil'nikov's theorem guarantees that this system hasSmale horseshoes and the horseshoe chaos.

Shil’nikov chaos control using homoclinic orbits and the Newhouse region

A method of controlling Shil'nikov's type chaos using windows that appear in the 1 dimensional bifurcation diagram when perturbations are applied, and using existence of stable homoclinic orbits near the unstable one is presented and applied to the electronic Chua's circuit. A demonstration of the chaos control in the electronic circuit experiments and their simulations and bifurcation analyses are given.