Nikov Theorem Research Articles

Generating Chaos with Saddle-Focus Homoclinic Orbit

This paper develops an anticontrol approach to design a 3D continuous-time autonomous chaotic system with saddle-focus homoclinic orbit, based on two chaotification criterions for all orbits to be globally bounded with positive Lyapunov exponents. By using the Shil’nikov theorem, a Poincaré return map near the origin is found in the designed controlled system, confirming the existence of chaos in sense of the Smale horseshoe.

Colorings of complements of line graphs

AbstractOur purpose is to show that complements of line graphs (of graphs) enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number to be equal to a natural upper bound. A consequence of this latter condition is a complete characterization of all induced subgraphs of the Kneser graph that have a chromatic number equal to its chromatic number, namely . In addition to the upper bound, a lower bound is provided by Dol'nikov's theorem, a classical result of the topological method in graph theory. We prove the NP‐hardness of deciding the equality between the chromatic number and any of these bounds. The topological method is especially suitable for the study of coloring properties of complements of line graphs of hypergraphs. Nevertheless, all proofs in this article are elementary and we also provide a short discussion on the ability for the topological methods to cover some of our results.

Analysis, Stabilization, and DSP-Based Implementation of a Chaotic System with Nonhyperbolic Equilibrium

This paper reports an autonomous dynamical system, and it finds that one nonhyperbolic zero equilibrium and two hyperbolic nonzero equilibria coexist in this system. Thus, it is difficult to demonstrate the existence of chaos by Šil’nikov theorem. Consequently, the topological horseshoe theory is adopted to rigorously prove the chaotic behaviors of the system in the phase space of Poincaré map. Then, a single control scheme is designed to stabilize the dynamical system to its zero-equilibrium point. Besides, to verify the theoretical analyses physically, the attractor and stabilization scheme are further realized via DSP-based technique.

Dissipative periodic and chaotic patterns to the KdV–Burgers and Gardner equations

We investigate the KdV-Burgers and Gardner equations with dissipation and external perturbation terms by the approach of dynamical systems and Shil'nikov's analysis. The stability of the equilibrium point is considered, and Hopf bifurcations are investigated after a certain scaling that reduces the parameter space of a three-mode dynamical system which now depends only on two parameters. The Hopf curve divides the two-dimensional space into two regions. On the left region the equilibrium point is stable leading to dissapative periodic orbits. While changing the bifurcation parameter given by the velocity of the traveling waves, the equilibrium point becomes unstable and a unique stable limit cycle bifurcates from the origin. This limit cycle is the result of a supercritical Hopf bifurcation which is proved using the Lyapunov coefficient together with the Routh-Hurwitz criterion. On the right side of the Hopf curve, in the case of the KdV-Burgers, we find homoclinic chaos by using Shil'nikov's theorem which requires the construction of a homoclinic orbit, while for the Gardner equation the supercritical Hopf bifurcation leads only to a stable periodic orbit.

Hidden extreme multistability generated from a fractional-order chaotic system

A new 4D fractional-order chaotic system with no equilibrium is proposed in this paper. One salient feature of this system is that it has no equilibrium point, which leads to the lack of homo-clinic or hetero-clinic trajectory. Thus, Shil’nikov theorem is not suitable to prove the existence of chaos in this system. However, the system can exhibit complex dynamic behaviors when its system parameter and fractional derivative order are varied. More interesting, the dynamics depending on initial values are also investigated and a more complex phenomenon of hidden extreme multistability is revealed. Furthermore, by selecting the appropriate system parameters, the state transition behaviors can be also found in this system. Finally, in order to verify the feasibility of the proposed fractional-order chaotic system, the corresponding physical circuit is designed by using the modular design method. The hardware experimental results are in good agreement with numerical analysis.

A Novel Four-Dimensional No-Equilibrium Hyper-Chaotic System With Grid Multiwing Hyper-Chaotic Hidden Attractors

By using a simple state feedback control technique and introducing two new nonlinear functions into a modified Sprott B system, a novel four-dimensional (4D) no-equilibrium hyper-chaotic system with grid multiwing hyper-chaotic hidden attractors is proposed in this paper. One remarkable feature of the new presented system is that it has no equilibrium points and therefore, Shil'nikov theorem is not suitable to demonstrate the existence of chaos for lacking of hetero-clinic or homo-clinic trajectory. But grid multiwing hyper-chaotic hidden attractors can be obtained from this new system. The complex hidden dynamic behaviors of this system are analyzed by phase portraits, the time domain waveform, Lyapunov exponent spectra, and the Kaplan–York dimension. In particular, the Lyapunov exponent spectra are investigated in detail. Interestingly, when changing the newly introduced nonlinear functions of the new hyper-chaotic system, the number of wings increases. And with the number of wings increasing, the region of the hyper-chaos is getting larger, which proves that this novel proposed hyper-chaotic system has very rich and complicated hidden dynamic properties. Furthermore, a corresponding improved module-based electronic circuit is designed and simulated via multisim software. Finally, the obtained experimental results are presented, which are in agreement with the numerical simulations of the same system on the matlab platform.

Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability.

By using a simple state feedback controller in a three-dimensional chaotic system, a novel 4D chaotic system is derived in this paper. The system state equations are composed of nine terms including only one constant term. Depending on the different values of the constant term, this new proposed system has a line of equilibrium points or no equilibrium points. Compared with other similar chaotic systems, the newly presented system owns more abundant and complicated dynamic properties. What interests us is the observation that if the value of the constant term of the system is nonzero, it has no equilibria, and therefore, the Shil'nikov theorem is not suitable to verify the existence of chaos for the lack of heteroclinic or homoclinic trajectory. However, one-wing, two-wing, three-wing, and four-wing hidden attractors can be obtained from this new system. In addition, various coexisting hidden attractors are obtained and the complex transient transition behaviors are also observed. More interestingly, the unusual and striking dynamic behavior of the coexistence of infinitely many hidden attractors is revealed by selecting the different initial values of the system, which means that extreme multistability arises. The rich and complex hidden dynamic characteristics of this system are investigated by phase portraits, bifurcation diagrams, Lyapunov exponents, and so on. Finally, the new system is implemented by an electronic circuit. A very good agreement is observed between the experimental results and the numerical simulations of the same system on the Matlab platform.

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.

Generation Method of Multipiecewise Linear Chaotic Systems Based on the Heteroclinic Shil’nikov Theorem and Switching Control

Based on the heteroclinic Shil’nikov theorem and switching control, a kind of multipiecewise linear chaotic system is constructed in this paper. Firstly, two fundamental linear systems are constructed via linearization of a chaotic system at its two equilibrium points. Secondly, a two-piecewise linear chaotic system which satisfies the Shil’nikov theorem is generated by constructing heteroclinic loop between equilibrium points of the two fundamental systems by switching control. Finally, another multipiecewise linear chaotic system that also satisfies the Shil’nikov theorem is obtained via alternate translation of the two fundamental linear systems and heteroclinic loop construction of adjacent equilibria for the multipiecewise linear system. Some basic dynamical characteristics, including divergence, Lyapunov exponents, and bifurcation diagrams of the constructed systems, are analyzed. Meanwhile, computer simulation and circuit design are used for the proposed chaotic systems, and they are demonstrated to be effective for the method of chaos anticontrol.

Decomposition of pseudo-radioactive chemical products with a mathematical approach

The aim of this paper is to study the decomposition of pseudo–radioactive products that follow a dynamics determined by a trigonometric factor. In particular for maps of the form \(e^{\cos (\pi t)}\) is proved that an asymptotic sampling recomposition property, generalizing the classical Shannon–Whittaker–Kotel’nikov Theorem, works.

Analysis of a New Quadratic 3D Chaotic Attractor

A new three-dimensional chaotic system is introduced. Basic properties of this system show that its corresponding attractor is topologically different from some well-known systems. Next, detailed information on dynamic of this system is obtained numerically by means of Lyapunov exponents spectrum, bifurcation diagrams, and 0-1 chaos indicator test. We finally prove existence of this chaotic attractor theoretically using Shil’nikov theorem and undetermined coefficient method.

Generating Grid Multiwing Chaotic Attractors by Constructing Heteroclinic Loops Into Switching Systems

Over the last two decades, multiwing chaos generation has seen promising advances and becomes an active research field today. It is well known that there is a gap between theoretical design and engineering applications in multiwing chaos generation. That is, most theoretical designs of multiwing chaotic attractors with mathematical proofs or numerical verification have rather complex expressions; however, most engineering applications of multiwing chaotic attractors without theoretical supports have simple expressions. To bridge the gap between theoretical design and engineering applications in multiwing chaos generation, this paper introduces a novel practical approach for generating grid multiwing butterfly chaotic attractors from the multipiecewise Lu system by constructing heteroclinic loops. It should be particularly pointed out that the designed multiwing chaotic attractors exhibit typical heteroclinic chaos from the heteroclinic Shil'nikov theorem and also have clear potential engineering applications. The proposed method can be easily extended to the generalized Lorenz system family.

Algorithm for estimating the state vector of controlled plants based on the Kotel’nikov theorem

An algorithm for estimating the state vector of controlled plants in the class of band-limited functions with a cutoff regularizing function is proposed which is more effective and accurate than available algorithms. The effect of errors in assigning samples of the differentiated functions on the estimate quality is evaluated, and the methodical error of the algorithm is analyzed.

Bifurcations and chaotic dynamics in a 4-dimensional competitive Lotka–Volterra system

In this paper, we study the Hopf bifurcation for the four-dimensional competitive Lotka–Volterra system, and give an example which can display chaotic dynamics apparent like Rossler’s folded band attractor. This demonstrates that Smale’s conclusions in (J. Math. Biol., 3:5–7, 1976) are true even for the simplest competitive Lotka–Volterra systems when the dimension n is four. We explore the mechanism of occurrence of chaotic behavior for the four-dimensional competitive Lotka–Volterra system. The numerical study indicates that a periodic solution by Hopf bifurcation can undergo successive period-doubling cascades, and a homoclinic orbit can undergo homoclinic bifurcation by Shil’nikov theorem.

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.

Calculation of Fresnel diffraction patterns via the Kotel’nikov theorem

The efficiency of calculation of Fresnel diffraction patterns via the Kotel’nikov theorem is considered. A computer model based on an algorithm proposed earlier is built. It is shown that this algorithm provides satisfactory agreement with the results of calculations performed with the use of well-known, but more laborious, methods.

The dual distance of a CRC and bounds on the probability of undetected error, the weight distribution, and the covering radius

Dual codes play an important role in the field of error detecting codes on a binary symmetric channel. Via the MacWilliams Identities they can be used to calculate the original code's weight distribution and its probability of undetected error. Moreover, knowledge of the minimum distance of the dual code provides insight in the properties of the weights of a code. In this paper firstly the order of growth of the dual distance of a CRC as a function of the block length n is investigated, and a new lower bound is proven. Then this bound is used to derive a weaker version of the 2-r-bound on the probability of undetected error, and the relationship of this bound to the 2-r-bound is discussed. Estimates of the range of binomiality and the covering radius are given, depending only on the code rate R and the degree r of the generating polynomial of the CRC. In the case of a CRC, two results of Tietavainen are improved. Furthermore, wit is shown that there is binomial behavior of the weight distribution, if only n is large enough. Then, by means of an estimate of the tail of the binomial, another bound on the probability of undetected error is verified. Finally a new version of Sidel'nikov's theorem on the normality of the cumulative distribution function of the weights of a code is presented, where the dual distance is replaced by an expression depending on n and the degree r. In this way the conclusions of the present paper may attribute a new meaning to some well known results about codes with known dual distance and give some new insight in this kind of problems.

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.

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.

Sturm–Liouville problems with parameter dependent potential and boundary conditions

This paper deals with the computation of the eigenvalues of Sturm–Liouville problems with parameter dependent potential and boundary conditions. We shall extend the domain of application of the method based on sampling theory to the case where the classical Whittaker–Shannon–Kotel’nikov theorem is not applicable. A few numerical examples will be presented.