Existence Of Chaotic Attractors Research Articles

Multi-wing chaotic system based on smooth function and its predefined time synchronization

In this paper, the construction of multi-wing chaotic system and predefined time synchronization are investigated. First, a new 4D chaotic system is constructed, and the existence of chaotic attractors is proved by dissipativity, boundedness, and Lyapunov exponents. Then the existence of a saddle focus of index-2 is proved by equilibrium point analysis, and a multi-winged hyperchaotic system is generated by extending the saddle focus with the constructed smooth function, which is found to be more complex and more conducive to synchronization applications by complexity analysis. Then the simulation of the circuit also proves the chaotic properties of the system in a physical sense. Finally, a new robust controller is designed using the fast terminal sliding mode algorithm to achieve predefined time synchronization between heterogeneous chaotic systems, and the numerical simulation results indicate that this synchronization method has a rapid time of convergence and achieves the predefined time effect.

Bifurcation of Novel Seven-Dimension Hyper Chaotic System

In this paper, introduced a novel seven dimensions (7D) nonlinear hyperchaotic system in third-order. A chaotic behavior that has twelve positive parameters for novel 7D hyperchaotic is analyzed through calculating the Lyapunov exponent, attracter of the system, fractional dimension, influence parameters, dissipative, bifurcation path, and phase portraits. It is well known that one of the chaotic definitions is the novel 7D chaotic if it satisfies positive Lyapunov exponent at each point on its domain (eventually periodic). The results from the numerical analysis of Lyapunov exponents, bifurcation have shown that there are periodic dynamic behaviors, quasi-periodic, the existence of chaotic attractors, and hyperchaotic for our analyzed to the proposed system. Also, some complex dynamic behaviors are discussed, such as equilibrium stability, Besides, discuss when a parameter changes a properties phase portrait change also. The dynamics of the proposed novel 7D hyperchaotic simulated and implemented using the Mathematica program provided qualitatively and it illustrated phase portraits.

Generation of Extremely Multistable Systems Based on Lurie Systems

Chaotic signals and systems are widely used in image encryption, secure communications, weak signal detection, and radar systems. Many researchers have focused in recent years on the development of systems with an infinite number of coexisting chaotic attractors. We propose some approaches in this work to the generation of self-reproducing systems with an infinite number of coexisting self-excited or hidden chaotic attractors with the same Lyapunov exponents based on mathematical models of Lurie systems. The proposed approach makes it possible to generate extremely multistable systems using numerous known examples of the existence of chaotic attractors in Lurie systems without resorting to an exhaustive computer search. We illustrate the proposed methods by constructing extremely multistable systems with 1-D and 2-D grids of hidden chaotic attractors using the generalized Chua system, in which hidden attractors were first discovered by G.A. Leonov and N.V. Kuznetsov.

Nonlinear dynamics and chaos in a simplified memristor-based fractional-order neural network with discontinuous memductance function

In this paper, a simplified memristor-based fractional-order neural network (MFNN) with discontinuous memductance function is proposed. It is essentially a switched system with irregular switching laws and consists of eight fractional-order neural network (FNN) subsystems. The nonlinear dynamics of the simplified MFNN including equilibrium points and their stability, bifurcation and chaos is investigated analytically and numerically. In particular, the effect of the switching jump on the dynamics of the simplified MFNN is explored for the first time. Taking the fractional order, the memristive connection weight or the switching jump as the bifurcation parameter, the dynamics such as chaotic motion, tangent bifurcation and intermittent chaos is identified over a wide range of some specified parameter. It can be seen that the incorporation of the memristors greatly improves and enriches the dynamics of the corresponding FNN. Different from the period-doubling route to chaos, this paper reveals that the mechanism behind the emergence of chaos for the simplified MFNN is the intermittency route to chaos. In particular, for some typical parameter, the existence of chaotic attractors is verified with the phase portraits, bifurcation diagrams, Poincare sections and maximum Lyapunov exponents, respectively. This paper not only provides a way of designing chaotic MFNN with discontinuous memductance function but also suggests a possible method of generating more complicated chaotic attractors, such as multi-scroll or multi-wing attractors.

On novel conditions of chaotic attractors existence in autonomous polynomial dynamical systems

The novel sufficient conditions for the existence of chaotic dynamics in n-dimensional autonomous polynomial dynamical systems are found. These conditions allowed to extend collections of known systems, for which the existence of chaotic attractors is rigorously proved and to complement these collections by new chaotic systems. As an example, a new chaotic attractor generated by the known Rabinovich–Fabrikant system is shown. In addition, new discrete maps generating chaos in Rabinovich–Fabrikant system are also represented. Besides, an application of got results for research of cubic dynamical systems is demonstrated.

Cross-Like Surface Waves Between Finite Cylindrical Shells

We construct a new mathematical model of the interaction of two resonant surface waves in a volume between two cylindrical shells of finite length. For the first time, we establish the existence of chaotic attractors for a system satisfying the resonance conditions for cross-like and forced waves. We also study the regular modes in the system and describe their phase portraits and frequency spectra.

Multistability and complex basins in a nonlinear duopoly with price competition and relative profit delegation.

In this article, we investigate the local and global dynamics of a nonlinear duopoly model with price-setting firms and managerial delegation contracts (relative profits). Our study aims at clarifying the effects of the interaction between the degree of product differentiation and the weight of manager's bonus on long-term outcomes in two different states: managers behave more aggressively with the rival (competition) under product complementarity and less aggressively with the rival (cooperation) under product substitutability. We combine analytical tools and numerical techniques to reach interesting results such as synchronisation and on-off intermittency of the state variables (in the case of homogeneous attitude of managers) and the existence of chaotic attractors, complex basins of attraction, and multistability (in the case of heterogeneous attitudes of managers). We also give policy insights.

Localization of invariant compacts of a phase-lock system

The localization method, which makes it possible to find regions in the phase space that contain all attractors of the system is used to analyze a phase system. Systems of inequalities describing such sets have been obtained. Phase-lock systems of the fourth and third order, which allow existence of chaotic attractors of various types, have been investigated.

Shilnikov chaos and Hopf bifurcation in three-dimensional differential system

In this paper, a three-dimensional differential system with only one equilibrium point is proposed. The existence of chaotic attractor in the system is verified via the homoclinic Silnikov theorem. By using the undetermined coefficient method, the homoclinic orbit of the system is determined. The Hopf bifurcation of the system is also investigated by analyzing the characteristic equation at equilibrium point and computing the first Lyapunov coefficient. Numerical simulations support the correctness of the theoretical results.

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.

The Generation of a Series of Multiwing Chaotic Attractors Using Integer and Fractional Order Differential Equation Systems

In this article, we present a systematic approach to design chaos generators using integer order and fractional order differential equation systems. A series of multiwing chaotic attractors and grid multiwing chaotic attractors are obtained using linear integer order differential equation systems with switching controls. The existence of chaotic attractors in the corresponding fractional order differential equation systems is also investigated. We show that, using the nonlinear fractional order differential equation system, or linear fractional order differential equation systems with switching controls, a series of multiwing chaotic attractors can be obtained.

The dynamic of plankton–nutrient interaction with delay

In this paper we consider the plankton–nutrient interaction model with the help of delay differential equations. Firstly the elementary dynamical properties of the plankton–nutrient system in the absence of time delay is discussed. Then we establish the existence of local Hopf-bifurcation as the time delay crosses a threshold value. Explicit results are derived for stability and direction of the bifurcating periodic solution by using normal form theory and center manifold arguments. Finally, outcomes of the system are validated through numerical simulations and complex dynamic of the system is explored with the existence of chaotic attractors.

Pattern Formation in Spatially Extended Tritrophic Food Chain Model Systems: Generalist versus Specialist Top Predator

The complex dynamics of two types of tritrophic food chain model systems when the species undergo spatial movements, modeling two real situations of marine ecosystem, are investigated in this study analytically and using numerical simulations. The study has been carried out with the objective to explore and compare the competitive effects of fish and molluscs species being the top predators, when phytoplankton and zooplankton species are undergoing spatial movements in the subsurface water. Reaction diffusion systems have been used to represent temporal evolution and spatial interaction among the species. The two model systems differ in an essential way that the top predators are generalist and specialist, respectively, in two models. “Wave of Chaos” mechanism is found to be the responsible factor for the pattern (non-Turing) formation in one dimension seen in the food chain ending with top generalist predator. In the present work we have reported WOC phenomenon, for the first time in the literature, in a three-species spatially extended food chain model system. The numerical simulation leads to spontaneous and interesting pattern formation in two dimensions. Constraints on different parameters under which Turing and non-Turing patterns may be observed are obtained analytically. Diffusion-driven analysis is carried out, and the effect of diffusion on the chaotic dynamics of the model systems is studied. The existence of chaotic attractor and long-term chaotic behavior demonstrate the effect of diffusion on the dynamics of the model systems. It is observed from numerical study that food chain model system with top predator as generalist has very rich dynamics and shows very interesting patterns. An ecosystem having top predator as specialist leads to the stability of the system.

Nonlinear Dynamics and Chaos in Fractional-Order Hopfield Neural Networks with Delay

A fractional-order two-neuron Hopfield neural network with delay is proposed based on the classic well-known Hopfield neural networks, and further, the complex dynamical behaviors of such a network are investigated. A great variety of interesting dynamical phenomena, including single-periodic, multiple-periodic, and chaotic motions, are found to exist. The existence of chaotic attractors is verified by the bifurcation diagram and phase portraits as well.

Chaotic dynamics in quadratic systems with singular linear part

New existence conditions are found for homoclinic and heteroclinic orbits for systems of quadratic ordinary differential equations with singular linear part. Implementing these conditions together with Shilnikov's theorems guarantees the existence of chaotic attractors in 3-D autonomous quadratic systems. The examples of the chaotic attractors are given.

Chaos Analysis and Control of Permanent Magnet Synchronous Motors

As a nonlinear dynamic system, the permanent magnet synchronous motor (PMSM) can exhibit prominent chaotic characteristics under some choices of system parameters. The existence of chaotic attractor of the PMSM is verified through the centre manifold theory and Poincaré section. Chaotic phenomenon affects the normal operation of motor. In this paper, it makes the PMSM in a stable state to control chaos of the PMSM with a control strategy of delay feedback, which can eliminate chaos well.

Chaos Characteristics and Control of Permanent Magnet Synchronous Motors

The permanent magnet synchronous motor (PMSM), a nonlinear dynamic system, can exhibit prominent chaotic characteristics under some choices of system parameters and external inputs. Based on a mathematical model of the permanent magnet synchronous motor, the existence of chaotic attractor is verified by the phase trajectory, Lyapunov exponent map and the bifurcation diagram. Chaotic phenomenon, such as a strong oscillation of speed and torque, unstable operating performance, affects the normal operation of motor. It makes the PMSM in a stable state to control chaos of the PMSM with a control strategy of infinitesimal geometry, which can eliminate chaos well.

Shilnikov sense chaos in a simple three-dimensional system

The Shilnikov sense Smale horseshoe chaos in a simple 3D nonlinear system is studied. The proportional integral derivative (PID) controller is improved by introducing the quadratic and cubic nonlinearities into the governing equations. For the discussion of chaos, the bifurcate parameter value is selected in a reasonable regime at the requirement of the Shilnikov theorem. The analytic expression of the Shilnikov type homoclinic orbit is accomplished. It depends on the series form of the manifolds surrounding the saddle-focus equilibrium. Then the methodology is extended to research the dynamical behaviours of the simplified solar-wind-driven-magnetosphere-ionosphere system. As is illustrated, the Lyapunov characteristic exponent spectra of the two systems indicate the existence of chaotic attractor under some specific parameter conditions.

A novel chaotic attractor with constant Lyapunov exponent spectrum and its circuit implementation

Based on the further evolvement of the improved chaotic system with constant Lyapunov exponent spectrum, by introducing an absolute term in the dynamic equation, a novel chaotic attractor is found in this paper. Firsty, the existence of chaotic attractor is verified by simulation of phase portrait, Poincaré mapping, and Lyapunov exponent spectrum. Secondly, the basic dynamical behaviour of the new system is investigated and expounded. Simulation of Lyapunov exponent spectrum, bifurcation diagram and numerical analysis on amplitude evolvement of state variables show that the state variables of the chaotic system can be modified linearly by a global linear amplitude adjuster while the Lyapunov exponent spectrum keeps on stable and the chaotic attractor displays the same phase portrait. Finally, an analog circuit is designed to implement the new system, the chaotic attractor is observed and the action of global linear amplitude adjuster is verified, all of which show a good agreement between numerical simulation and experimental results.

Chaos in the fractional-order conjugate Chen system and its circuit emulation

This paper studies the chaotic behaviors of the fractional-order conjugate Chen system. The necessary condition for the existence of chaotic attractors in the fractional-order conjugate Chen chaotic system is obtained. Furthermore, a new circuit unit is proposed to realize the fractional-order chaotic system. The results between numerical emulation and circuit experimental simulation are in agreement with each other and prove that chaos actually exits in the fractional-order conjugate Chen system.