Kaplan-Yorke Dimension Research Articles

A chaos-geometric approach to analysis, modelling and forecasting atmospheric pollutants dynamics for industrial regions

We applied an advanced chaos-geometric approach to analysis, modeling, forecasting and processing the time series of the air pollutants (NO2) concentrations in an atmosphere of the industrial cities (regions). The approach includes such advanced non-linear analysis and a chaos theory methods such as a multifractal approach, correlation integral algorithm, the Lyapunov’s exponents and Kolmogorov entropy analysis, a power spectrum analysis, prediction models with neural networks blocks etc. The dynamical and topological invariants (including the Lyapunov’s exponents spectrum, Kaplan-Yorke dimension, Kolmogorov entropy etc) for the air pollutants (NO2) concentrations time series in an atmosphere of the industrial cities are computed. Our study has shown an existence of a deterministic chaos in the atmospheric pollutants fluctuations dynamics. It is presented an effective prediction model for description of the temporal evolutionary dynamics of the air pollutants concentration in atmosphere of the industrial city.

Dimension of reservoir computers.

A reservoir computer is a complex dynamical system, often created by coupling nonlinear nodes in a network. The nodes are all driven by a common driving signal. In this work, three dimension estimation methods, false nearest neighbor, covariance dimension, and Kaplan-Yorke dimension, are used to estimate the dimension of the reservoir dynamical system. It is shown that the signals in the reservoir system exist on a relatively low dimensional surface. Changing the spectral radius of the reservoir network can increase the fractal dimension of the reservoir signals, leading to an increase in a testing error.

A Three-Dimensional Chaotic System and Its New Proposed Electronic Circuit

In this paper, new electronic circuit was designed with two-equilibrium as an engineering application on a three-dimensional chaotic system. The circuit consists of resistors, capacitors, voltages and operational amplifiers TL032CN. The adopted continuous-time chaotic dynamical system is with quadratic cross-product nonlinear terms and parameters. The basic characteristics of the proposed circuit model were analyzed in detail by equilibrium points, stability analysis, Lyapunov exponents and Kaplan-Yorke dimension. The results were simulated theoretically using MultiSIM 10, and it was well consistent with the results obtained from the Matlab program.

A new family of 9D and 10D hyperchaotic systems from the 8D hyperchaotic Benkouider system, the bifurcation analysis of the 10D hyperchaotic system, circuit design and an application to secure voice communication

This work presents a new 10D polynomial hyperchaotic system with eight positive Lyapunov exponents. We propose a family of new 9D and 10D hyperchaotic systems, which are derived from the 8D hyperchaotic Benkouider system (2020). With its eight positive Lyapunov exponents, the proposed 10D hyperchaotic system generates a very complex behaviour than the existing systems, which makes it very useful in many fields of applications, especially in secure communication. Major properties of the new system are investigated using Lyapunov exponents, bifurcation diagrams, phase portraits, equilibrium points, Kaplan-Yorke dimension and multistability. In addition, an equivalent electronic circuit is implemented using Multisim software to validate the physical feasibility of the constructed 10D hyperchaotic system. Finally, a new secure voice communication scheme is developed based on the chaotic masking approach and using all the complex hyperchaotic signals generated by the new 10D hyperchaotic system.

Characterization of spatiotemporal chaos in arrays of nonlinear plasmonic nanoparticles

We investigate spatiotemporal chaos dynamics in a finite nanoparticles array with Kerr-type nonlinear response, excited by an incident plane wave of varying intensity and tunable frequency close to the localized plasmon resonance of a single particle. Considering dipole-dipole coupling between the nanoparticles described by their polarizability, we compute the temporal evolution of the dipoles and numerically extract the Lyapunov spectra, allowing us to characterize different dynamical behaviors. Furthermore, we estimate the Kaplan-Yorke dimension that provides a measure of the strange attractor complexity. We show that time-modulated solutions which are generated at the onset of modulational instability experience secondary instabilities leading to a complex nonlinear dynamic. It is also shown that in the highly nonlinear regime, the spatiotemporal chaos is robust and exists in a large range of parameters that we have determined numerically.

Determining accurate Lyapunov exponents of a multiscroll chaotic attractor based on SNFS

In n-scroll attractors design using active devices, the behavior of the nonlinear part is, often, modeled by using a piecewise linear (PWL) approach. Although little effort is required to build a PWL model, the main performance parameters associated with the active device to be used in the physical design are not included. Moreover, the main metrics used for quantifying chaos are: Lyapunov exponent (LE), Kolmogorov–Sinai entropy, Kaplan–Yorke dimension, fractal power spectra and fractal dimension. Among them, the former not only is used as indicator of chaotic motion when a positive LE is found, but if its value positively increases, then the unpredictability grade of the chaotic system also increases. All reported works until today about the computed LEs on chaotic systems are derived using a PWL approach. This is a serious drawback since PWL models cannot model a dynamic behavior, and hence, introducing a level of error during the numerical analysis becomes evident when experimental and numerical results are examined at high frequency. Under this scheme, new results on the accurate computing of LEs for a chaotic oscillator based on a nonlinear function called saturated nonlinear function series (SNFS) are introduced. A mathematical model is used for characterizing the dynamical behavior of the SNFS that includes the main performance parameters of Op Amps such as slew rate, DC gain, gain bandwidth product and dynamic range, showing a much better accuracy than the traditional PWL approach. This behavioral modeling demonstrates that the positive LE increases in relation to the number of scrolls and even is also depending on the frequency of the system, overthrowing the results that have been handled in previous works, where a constant value of the LE was obtained when the number of scrolls increased.

A New 4-D Hyperchaotic System with Four-Scroll Hidden Attractor, Its Properties and Bifurcation Analysis

This paper announces a new four-dimensional hyperchaotic system with a four-scroll attractor and discusses its dynamic properties such as Lyapunov exponents, phase portraits, Kaplan-Yorke dimension and equilibrium points. Our calculations show that the new hyperchaotic system has no equilibrium point and hence it exhibits hidden attractor. Our new hyperchaotic system has three nonlinearities in total. A detailed bifurcation analysis has been presented for the new hyperchaotic system with four-scroll hidden attractor. Specifically, we discussed bifurcation analysis such as route to four-scroll hyperchaos, coexisting bifurcation, multistability, two parameter Lyapunov exponents and antimonotonicity.

A new 3-D chaotic system with four quadratic nonlinear terms, its global chaos control via passive control method and circuit design

This paper reports the finding of a new three-dimensional chaotic system with four quadratic nonlinear terms. The paper starts with a detailed dynamic analysis of the properties of the system such as phase plots, Lyapunov exponents, Kaplan-Yorke dimension and equilibrium points. Our new chaotic system is obtained by modifying the dynamics of the Zhu chaotic system (2010), and it has complex chaotic properties. As an engineering application, passive control method is applied for the global chaos control of the new chaotic system. Finally, an electronic circuit implementation of the new chaotic system is designed and implemented in MultiSIM. A good qualitative agreement has been shown between the MATLAB simulations of the new chaotic system and the MultiSIM results.

A new 4-D hyperchaotic hyperjerk system with a single equilibrium, its dynamic properties and circuit design

This paper announces a new four-dimensional hyperchaotic hyperjerk system with a single equilibrium and discusses its dynamic properties such as Lyapunov exponents, phase portraits, Kaplan-Yorke dimension and equilibrium points. Hyperjerk systems have a nice triangular structure in their dynamics and they have many engineering applications. Our new hyperjerk system has three nonlinearities in total. New synchronization results based on active backstepping control are also derived for the new hyperjerk system. In addition, an electronic circuit implementation of the new hyperjerk system is designed carefully and examined well in MultiSIM. A good qualitative agreement has been shown between the MATLAB simulations of the theoretical hyperchaotic hyperjerk model and the MultiSIM results.

A new two-scroll chaotic system with two nonlinearities: dynamical analysis and circuit simulation

Chaos theory has several applications in science and engineering. In this work, we announce a new two-scroll chaotic system with two nonlinearities. The dynamical properties of the system such as dissipativity, equilibrium points, Lyapunov exponents, Kaplan-Yorke dimension and bifurcation diagram are explored in detail. The presence of coexisting chaotic attractors, coexisting chaotic and periodic attractors in the system is also investigated. In addition, the offset boosting of a variable in the new chaotic system is achieved by adding a single controlled constant. It is shown that the new chaotic system has rotation symmetry about the z-axis. An electronic circuit simulation of the new two-scroll chaotic system is built using Multisim to check the feasibility of the theoretical model.

Dynamic Analysis of a Snap Oscillator Based on a Unique Diode Nonlinearity Effect, Offset Boosting Control and Sliding Mode Control Design for Global Chaos Synchronization

This paper focuses on the behavior of a specific class of oscillator (snap) under a unique diode effect by checking the complex dynamic of the proposed oscillator. From linear and nonlinear analysis methods, the numerical integration of the system such as, fixed-point and stability analysis, bifurcation diagrams, Kaplan–Yorke dimension, Lyapunov exponent spectrum, frequency spectra, Poincare section and cross section of basins of attraction reveals that oscillator is chaotic and the chaotic robustness of the system depends on parameters changing. To the best knowledge of the authors, the system is the simplest in its category but the study revealed some interesting phenomena: asymmetric coexisting attractors, antimonotonicity phenomenon and even period-doubling bifurcation. These phenomena confer oscillating the quality of multistable or resistance to attacks in engineering application of data encryption. Furthermore, an offset boosting operation of a variable is used to control attractors and a robust sliding mode control for control engineering application of the system is designed to achieve global chaos synchronization based on Lyapunov stability theory. An appropriate electronic circuit or analog simulator is designed; PSpice simulations demonstrate feasible the proposed snap.

On maximizing the positive Lyapunov exponent of chaotic oscillators applying DE and PSO

Lyapunov exponents are related to the exponentially fast divergence or convergence of nearby orbits in phase space, and they can be used to evaluate the Kaplan–Yorke dimension. In this manner, since the existence of a positive Lyapunov exponent (LE+) is taken as an indication that chaotic behavior exists, and due to the huge search spaces of the design variables of chaotic oscillators, we show the application of differential evolution (DE) and particle swarm optimization (PSO) algorithms to maximize LE+. Four chaotic oscillators are optimized herein, for which we detail the evaluation of their equilibrium points and their eigenvalues that are used to estimate the step-size h to perform appropriate numerical simulation. Both DE and PSO are calibrated to perform different number of generations with three different sizes of individuals in the populations, and with search spaces around the values already published for the four chaotic oscillators. As a result, we show that both DE and PSO algorithms provide higher values of LE+ and Kaplan–Yorke dimension compared to the ones already published in the literature.

Lyapunov exponents of the Kuramoto--Sivashinsky PDE

The Kuramoto–Sivashinsky equation is a prototypical chaotic nonlinear partial differential equation (PDE) in which the size of the spatial domain plays the role of a bifurcation parameter. We investigate the changing dynamics of the Kuramoto–Sivashinsky PDE by calculating the Lyapunov spectra over a large range of domain sizes. Our comprehensive computation and analysis of the Lyapunov exponents and the associated Kaplan–Yorke dimension provides new insights into the chaotic dynamics of the Kuramoto–Sivashinsky PDE, and the transition to its one-dimensional turbulence.
 
 
 doi:10.1017/S1446181119000105

A New Chaotic System with Line of Equilibria: Dynamics, Passive Control and Circuit Design

<p>A new chaotic system with line equilibrium is introduced in this paper. This system consists of five terms with two transcendental nonlinearities and two quadratic nonlinearities. Various tools of dynamical system such as phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and Poincarè map are used. It is interesting that this system has a line of fixed points and can display chaotic attractors. Next, this paper discusses control using passive control method. One example is given to insure the theoretical analysis. Finally, for the new chaotic system, An electronic circuit for realizing the chaotic system has been implemented. The numerical simulation by using MATLAB 2010 and implementation of circuit simulations by using MultiSIM 10.0 have been performed in this study.</p>

New Chaotic Systems with Two Closed Curve Equilibrium Passing the Same Point: Chaotic Behavior, Bifurcations, and Synchronization

In this work, we introduce a chaotic system with infinitely many equilibrium points laying on two closed curves passing the same point. The proposed system belongs to a class of systems with hidden attractors. The dynamical properties of the new system were investigated by means of phase portraits, equilibrium points, Poincaré section, bifurcation diagram, Kaplan–Yorke dimension, and Maximal Lyapunov exponents. The anti-synchronization of systems was obtained using the active control. This study broadens the current knowledge of systems with infinite equilibria.

Chaotic system with bondorbital attractors

Two kinds of chaotic attractors with nontrivial topologies are found in a 4D autonomous continuous dynamical system. Since the equilibria of the system are located on both sides of the basic structures of these attractors, and the basic structures of these attractors are bond orbits, we call them bondorbital attractors. They have fractional dimensions and their Kaplan–Yorke dimensions are greater than 3. The generation mechanisms of the two types of attractors are explored and analyzed based on the Shilnikov’s theorems. The type I attractors generated by system with parameter $$P_{1}$$ possess coexistence features, and the type II attractors generated by system with parameter $$P_{2}$$ have the ability to realize consecutive bond orbits. Furthermore, the type I attractors have continuous attracting basins with diagonal distribution and can be caught by means of a method of shorting capacitors in hardware experiments, whereas the type II attractors possess discrete basins of attraction and are difficult to be captured in hardware experiments. The difference between the two types of bondorbital attractors and traditional self-excited attractors in generation method is analyzed, and we also verify that they are not hidden attractors. Based on the step function sequence f(x, M, N), the type II attractors with at most ( $$N+M+1$$ )-fold structures can be generated in the system with parameter $$P_{2}$$ . Two sets of symmetric specific initial conditions are used to verify that the system with parameter $$P_{2}$$ can generate bondorbital attractors with fourfold and fivefold basic structures based on f(x, 2, 2). Some characteristics of the two classes of bondorbital attractors are listed in tabular form.

A hyperchaotic hyperjerk system with four nonlinearities, its dynamical analysis and circuit realization

A new four-dimensional hyperchaotic hyperjerk system with four nonlinearities is proposed in this paper. The dynamical properties of the new hyperjerk system are described in terms of phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, etc. Also, a detailed dynamical analysis of the new hyperjerk system has been carried out with bifurcation diagram and Lyapunov exponents. As an engineering application, an electronic circuit realization of the new hyperchaotic hyperjerk system is designed via MultiSIM to confirm the feasibility of the theoretical hyperchaotic hyperjerk model.

A Fish Biology Chaotic System and its Circuit Design

Applications of dynamical systems in chaos theory arising in several areas are investigated and circuital implementations of dynamical systems are studied by many researchers. In this work, we consider the fish-biology dynamical system introduced by Foutayeni and Khaladi (2016). By taking a different set of parameter values, we observe chaos in the Foutayeni-Khaladi fish-biology model and derive a new fish-biology chaotic system. Properties of the new fish-biology chaotic system are analyzed by means of phase portraits, Lyapunov exponents, and Kaplan-Yorke dimension. An electronic circuit realization is shown to validate the chaotic behavior of the new 3-D fish-biology chaotic system. The circuit experimental results of the 3-D fish-biology chaotic system show agreement with numerical simulations.

A Hyperchaotic System with Three Quadratic Nonlinearities, its Dynamical Analysis and Circuit Realization

A new four-dimensional hyperchaotic system with three quadratic nonlinearities is proposed in this paper. The dynamical properties of the new hyperchaotic system are explored in terms of phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, etc. Also, a detailed dynamical analysis of the new hyperchaotic system has been carried out with bifurcation diagram and Lyapunov exponents. As an engineering application, an electronic circuit realization of the new hyperchaotic system is designed via MultiSIM to confirm the feasibility of the theoretical hyperchaotic model.

A Chaotic Jerk System with Three Cubic Nonlinearities, Dynamical Analysis, Adaptive Chaos Synchronization and Circuit Simulation

A 3-D new chaotic jerk system with three cubic nonlinearities is proposed in this paper. The dynamical properties of the new jerk system are described in terms of phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, symmetry, dissipativity, etc. Also, a detailed dynamical analysis of the jerk system has been carried out with bifurcation diagram and Lyapunov exponents. As an engineering application, adaptive synchronization of the new chaotic jerk system with itself is designed via backstepping control method. Furthermore, an electronic circuit realization of the new chaotic jerk system is presented in detail to confirm the feasibility of the theoretical chaotic jerk model.