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The stochastic dynamical ρ4 equation is utilized as a robust framework for modeling the behavior of complex systems characterized by randomness and nonlinearity, with applications spanning various scientific fields. The aim of this paper is to employ an analytical method to identify stochastic traveling wave solutions of the dynamical ρ4 equation. Novel hyperbolic and rational functions are investigated through this method. A Galilean transformation is applied to reformulate the model into a planar dynamical system, which enables a comprehensive qualitative analysis. Additionally, the emergence of chaotic and quasi-periodic patterns following the introduction of a perturbation term is addressed. Simulation results indicate that significant changes in the systems’ dynamic behavior are caused by adjusting the amplitude and frequency parameters. Our findings indicate the impact of the method on system dynamics and its efficacy in analyzing solitons and phase behavior in nonlinear models. These discoveries provide fresh perspectives on how the suggested method can lead to notable shifts in the systems’ dynamic behavior. The effectiveness and practicality of the proposed methodology in scrutinizing soliton solutions and phase visualizations across diverse nonlinear models are underscored by these revelations. [ABSTRACT FROM AUTHOR]

This research explores the incorporation of variable order (VO) fractional calculus into the hyperchaotic Lorenz system and studies various chaotic features and attractors. Initially, we propose a variable fractional order hyperchaotic Lorenz system and numerically solve it. The solutions are obtained for multiple choices of control parameters, and these results serve as reference solutions for exploring chaos with the artificial intelligence tool radial basis function network (RBFN). We rebuild phase spaces and trajectories of system states to exhibit chaotic behavior at various levels. To further assess the sensitivity of chaotic attractors, Lyapunov exponents are calculated. The efficacy of the designed computational RBFN is validated through the RMSE and extensive error analysis. The proposed research on AI capabilities aims to introduce an innovative methodology for modeling and analyzing hyperchaotic dynamical systems with variable orders. [ABSTRACT FROM AUTHOR]

Using the sawtooth map as the basis of a coupled map lattice enables simple analytic results to be obtained for the global Lyapunov spectra of a number of standard lattice networks. The results presented can be used to enrich a course on chaos or dynamical systems by providing tractable examples of higher dimensional maps and links to a number of standard results in matrices. A number of suggestions for classroom use are given. [ABSTRACT FROM AUTHOR]

The fact that very small changes cause unpredictable big changes in nature has led to the chaos theory. Many researchers focusing on chaos have improved this theory in several directions. One of the prominent areas in chaos theory is controlled chaos. Due to the usefulness of controlled chaos, it has been used in many working areas as the iteration methods of fixed point theory. In this study, we have developed a chaos controlling mechanism using the Ishikawa iteration method. Firstly, Ishikawa fixed point type controlling mechanism is defined and the theorems determining the control parameters are proved. Afterward, the results of Ishikawa fixed point type controlling mechanism were obtained by taking the logistic map as an example. In addition, by using the MATLAB software program, the convergence, periodical behavior and chaotic regions occurring outside the range of the parameters forming the controlling mechanism have been investigated and numerical simulations studied. Subsequently, the effectiveness of the stability intervals obtained for the fixed point and periodic fixed points of the Ishikawa fixed point controlling mechanism by applying the Lyapunov exponent is demonstrated. Finally, we have given an application as control chaos of Ishikawa traffic model. [ABSTRACT FROM AUTHOR]

Further exploration into the influence of a memristor on the behavior of chaotic systems deserves attention. When constructing memristor chaotic systems, it is commonly believed that increasing the number of memristors will lead to better system performance. This paper proposes a class of chaotic maps with different discrete memristors, achieved through internal perturbation based on the Sine map. The I-V curve of the discrete memristor has a symmetrical structure. The dynamic characteristics of the designed system are analyzed using the chaotic attractor phase diagram, Lyapunov exponent (LE) spectrum, and bifurcation diagram. Numerical simulations demonstrate that internal perturbations of discrete memristors enhance the Sine map's chaotic characteristics, expand the chaos range, and improve the ergodicity and LE value. Moreover, the type of discrete memristors has a significant impact on the dynamic characteristics of the system, while the number of discrete memristors has little influence. Therefore, in this paper, a direction for the design of a discrete memristor chaotic system is provided. Finally, a discrete memristor chaotic map with a simple structure and better performance is selected. Based on this, a pseudo-random sequence generator is designed, and the generated sequence passes the National Institute of Standards and Technology (NIST) test. [ABSTRACT FROM AUTHOR]

The coexistence of different attractors with fixed parameters affords versatility in the performance of the dynamical system. As a result, recent research has focused on defining nonlinear oscillators with infinite coexisting attractors, the vast majority of which are non-autonomous systems. In this study, we present an infinite number of equilibriums achieved with the simplest autonomous megastable oscillator. To begin, we explore the symmetry property of distinct coexisting attractors, such as periodic, quasi-periodic, and chaotic attractors. The dynamical properties of the proposed system are further investigated using phase portraits, Lyapunov exponents spectrum, and bifurcation diagram using the local maxima of the state variables. The detailed investigation reveals that the proposed system has a wide range of dynamical properties ranging from periodic oscillations to hyperchaos. Linear quadratic regulator (LQR) control-based controllers are designed to control the chaos in the proposed system and also achieve synchronization between proposed systems that have different initial conditions. The effectiveness of the controller designed based on LQR is presented by simulation results. It is shown that the proposed system converges asymptotically to the desired equilibrium point, and the synchronization between the slave and the master systems is achieved by converging the error to zero after controllers are activated. [ABSTRACT FROM AUTHOR]

In this research, the ship power system is studied with a fractional-order approach. A 2-D model of a two-generator parallel-connected is considered. A chaotic attractor is observed for particular parameter values. The fractional-order form is calculated with the Adam–Bashforth–Moulton method. The chaotic response is identified even for the order 0.99. Phase portrait is generated using the Caputo derivative approach. Wolf's algorithm is used to calculate Lyapunov exponents. For the considered values of parameters, one positive Lyapunov exponent confirms the existence of chaos. Bifurcation diagrams are presented to analyze the various dynamical behaviors and bifurcation points. Interestingly, the considered system is multistable. Also, antimonotonicity, period-doubling, and period halving are observed in the bifurcation diagram. As the last step, a fractional-order controller is designed to remove chaotic dynamics. Time plots are simulated to show the effectiveness of the controller. [ABSTRACT FROM AUTHOR]

Background: Conservative chaos is one of the most interesting objects in modern nonlinear dynamics and chaos theory. Purpose: In this paper, we are focused on a simple symmetric 2–4–6 polynomial oscillatory potential under external periodic excitation. Methods: The Poincaré sections, bifurcation diagrams, and the Lyapunov characteristic exponents for such a dynamical system are obtained and analyzed together with the fixed points. Results: Our numerical results show that the motion of a particle in this potential has rich nonlinear dynamics. Because of two local instabilities, such a system exhibits conservative chaos and can be considered as a simple and efficient source of the chaotic signal. Conclusion: The obtained results show that this system is not only of fundamental interest for the theory of nonlinear oscillations but also can be considered as a promising source of chaos in the field of secure communications and encryption. [ABSTRACT FROM AUTHOR]

This work identifies the influence of chaos theory on fractional calculus by providing a theorem for the existence and stability of solution in fractional-order gyrostat model with the help of a fixed-point theorem. We modified an integer order gyrostat model consisting of three rotors into fractional order by attaching rotatory fuel-filled tank and provided an iterative scheme for our proposed model as a working rule of obtained analytical results. Moreover, this iterative scheme is injected into algorithms for a system of integer order dynamical systems to observe Lyapunov exponents and a bifurcation diagram for our proposed fractional-order dynamical model. Furthermore, we obtained five equilibrium points, including four unstable spirals and one saddle node, using local dynamical analysis which acted as self-exciting attractors and a separatrix in a global domain. [ABSTRACT FROM AUTHOR]

The Lyapunov exponent is the most-well-known measure for quantifying chaos in a dynamical system. However, its computation for any time series without information regarding a dynamical system is challenging because the Jacobian matrix of the map generating the dynamical system is required. The entropic chaos degree measures the chaos of a dynamical system as an information quantity in the framework of Information Dynamics and can be directly computed for any time series even if the dynamical system is unknown. A recent study introduced the extended entropic chaos degree, which attained the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. Moreover, an improved calculation formula for the extended entropic chaos degree was recently proposed to obtain appropriate numerical computation results for multidimensional chaotic maps. This study shows that all Lyapunov exponents of a chaotic map can be estimated to calculate the extended entropic chaos degree and proposes a computational algorithm for the extended entropic chaos degree; furthermore, this computational algorithm was applied to one and two-dimensional chaotic maps. The results indicate that the extended entropic chaos degree may be a viable alternative to the Lyapunov exponent for both one and two-dimensional chaotic dynamics. [ABSTRACT FROM AUTHOR]

In the applied research of nonlinear system, the low degree of chaos in the dynamical system leads to the limitation of using the chaos method to solve some practical problems. In this paper, we use the product trigonometric function and ternary polynomial to build a dynamical system, which has strong chaotic characteristics. The dynamical system is constructed by two product trigonometric functions and a ternary linear equation, and its chaotic properties are verified by bifurcation diagrams, Lyapunov exponents, fractal dimensions, etc. The system has many parameters and large parameter intervals and is not prone to cycles. The conditions for the non-divergence of this system are given by mathematical derivation, and it is found that the linear part of the system can be replaced by an arbitrary ternary polynomial system and still not diverge, and the bifurcation diagram is drawn to verify it. Finally, the chaotic sequence is distributed more uniformly in the value domain space by adding the modulo operation. Then, the bit matrix of multiple images is directly permuted by the above system, and the experiment confirms that the histogram, information entropy, and pixel correlation of its encrypted images are satisfactory, as well as a very large key space. [ABSTRACT FROM AUTHOR]

Memristor, as a typical nonlinear element, is able to produce chaotic signals in chaotic systems easily. Chaotic systems have potential applications in secure communications, information encryption, and other fields. Therefore, it is of importance to generate abundant dynamic behaviors in a single chaotic system. In this paper, a novel memristor-based chaotic system without equilibrium points is proposed. One of the essential features is the absence of symmetry in this system, which increases the complexity of the new system. Then, the nonlinear dynamic behaviors of the system are analyzed in terms of chaos diagrams, bifurcation diagrams, Poincaré maps, Lyapunov exponent spectra, the sum of Lyapunov exponents, phase portraits, 0–1 test, recurrence analysis and instantaneous phase. The results of the sum of Lyapunov exponents show that the given system is a quasi-Hamiltonian system with certain initial conditions (IC) and parameters. Next, other critical phenomena, such as hidden multi-scroll attractors, abundant coexistence characteristics, are found characterized through basins of attraction and others. Especially, it reveals some rare phenomena in other systems that multiple hidden hyperchaotic attractors coexist. Finally, the circuit implementation based on Micro Control Unit (MCU) confirms theoretical analysis and the numerical simulation. [ABSTRACT FROM AUTHOR]

Intensive and repetitive simulations are required to study static and dynamic behaviours of systems. Particular phenomena such as bifurcation and chaos require long simulation times and analysis. To check the existence of bifurcations and chaos in a dynamic system, a fine-tuning procedure of a bifurcation parameter is to be carried out. This increases considerably the computing time, and a great amount of patience is needed to obtain adequate results. Because of the high switching frequency of a boost inverter, the integration process of the dynamic model used to describe it uses an integration step that is in general less than one microsecond. This makes the integration process time consuming even for a short simulation. Thus, a fast, but accurate, method is suitable to analyse the dynamic behaviour of the converter. This work contains two topics. First, we develop a like-discrete integration process that permits precise results in a very fast manner. For one switching period, we compute only two or a maximum of three breaking points depending on whether we treat a continuous conduction mode (CCM) or a discontinuous conduction mode (DCM) of the inductor current. Furthermore, with each segment of the dynamic trajectory, an exact analytic formula is associated. The second goal is to use this result to develop a discrete iterative map formulated as in standard discrete time series models. The Jacobian matrix of the found iterative map is defined and used to compute Lyapunov exponents to prove existence of chaos. Performance of the developed study is positively evaluated by using classical simulations and fine-tuning a bifurcation parameter to detect chaos. This parameter is the desired reference of the inductor current peak. Results show that the proposed scheme is very fast and accurate. The study can be easily extended to other switching topologies of DC-DC inverters. [ABSTRACT FROM AUTHOR]

Many biological ecosystems exhibit chaotic behavior, demonstrated either analytically using parameter choices in an associated dynamical systems model or empirically through analysis of experimental data. In this paper, we use existing software tools (COPASI, R) to explore dynamical systems and uncover regions with positive Lyapunov exponents where thus chaos exists. We evaluate the ability of the software's optimization algorithms to find these positive values with several dynamical systems used to model biological populations. The algorithms have been able to identify parameter sets which lead to positive Lyapunov exponents, even when those exponents lie in regions with small support. For one of the examined systems, we observed that positive Lyapunov exponents were not uncovered when executing a search over the parameter space with small spacings between values of the independent variables. [ABSTRACT FROM AUTHOR]

The purpose of this article is to build an introductory approach to chaos theory from the Chua' oscillator for students of mathematics, physics and engineering. The Chua' circuit is one of the simplest dynamic systems that produce irregular behavior. Despite its simplicity, this oscillating circuit is a very useful device for studying basic principles of chaos theory. This study begins with the definition of equilibrium points, this is done in a graphic and analytical way. The analysis of the stationary solution of differential equations system allows to show the occurrence of pitchfork bifurcation. The eigenvalues are calculated and the trajectories of the Chua' oscillator are analyzed, then the occurrence of Hopf bifurcation is demonstrated. Period-doubling cascade are observed using phase portrait and orbit diagram. Finally, sensitivity to initial conditions is studied by calculating Lyapunov exponents. [ABSTRACT FROM AUTHOR]

Copyright of Al-Mustansiriyah Journal of Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Harbin Institute of Technology. Social Sciences Edition / Haerbin Gongye Daxue Xuebao. Shehui Kexue Ban is the property of Harbin Institute of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper presents a novel memristor-based dynamical system with circuit implementation, which has a 2 × 3 -wing, 2 × 2 -wing, and 2 × 1 -wing non-Shilnikov type of chaotic attractors. The system has two index-2 saddle-focus equilibria, symmetrical with respect to the x -axis. The system is analyzed with bifurcation diagrams and Lyapunov exponents, demonstrating its complex dynamical behaviors: the system reaches the chaotic state from the periodic state through alternating period-doubling bifurcations and then from the chaotic state back to the periodic state through inverse bifurcations, as one parameter changes. It shows two interesting phenomena: a jump-switching periodic state and jump-switching chaotic state. Also, the system can sustain chaos with a constant Lyapunov spectrum in some initial conditions and a parameter set. In addition, a class of symmetric periodic bursting phenomena is surprisingly observed under a particular set of parameters, and its generation mechanism is revealed through bifurcation analysis. Finally, the circuit implementation verifies the theoretical analysis and the jump-switching numerical simulation results. [ABSTRACT FROM AUTHOR]

In this work we introduce a square chaotic attractor based on the collision of two heteroclinic orbits. Before the collision, the system presents the coexistence of two double scroll attractors that are generated via piecewise linear (PWL) systems that deal with two saddle-foci equilibria of different classes, i.e. the dimensions of the unstable and stable manifolds of one of the equilibrium points are one and two, respectively, and vice versa for the unstable and stable manifolds associated with the second equilibrium point. The new class is constructed based on the existence of a heteroclinic loop by linear affine systems with two saddle-focus equilibrium points of different types. The chaotic behavior of the proposed system is tested by the maximum Lyapunov exponent and the 0–1 chaos test. This new class of PWL dynamical systems exhibits different behaviors and allows the generation of different basins of attraction for the coexistence of two or more attractors. Therefore, a mechanism to establish bistability is provided and how it can be controlled (i.e. bistability annihilation) via varying system's parameters is demonstrated. [ABSTRACT FROM AUTHOR]

A simplified hyperchaotic canonical Chua's oscillator (referred as SHCCO hereafter) made of only seven terms and one nonlinear function of type hyperbolic sine is analyzed. The system is found to be self-excited, and bifurcation tools associated with the spectrum of Lyapunov exponents reveal the rich dynamical behaviors of the system including hyperchaos, torus, period-doubling route to chaos, and hysteresis when turning the system control parameters. Wide ranges of hyperchaotic dynamics are highlighted in various two-parameter spaces based on two-parameter Lyapunov diagrams. The analysis of the hysteretic window using a basin of attraction as argument reveals that the SHCCO exhibits three coexisting attractors. Laboratory measurements further confirm the performed numerical investigations and henceforth validate the mathematical model. Of most/particular interest, multistability observed in the SHCCO is further controlled based on a linear augmentation scheme. Numerical results show the effectiveness of the control strategy through annihilation of the asymmetric pair of coexisting attractors. For higher values of the coupling strength, only a unique symmetric periodic attractor survives. [ABSTRACT FROM AUTHOR]

We investigate the ability of simple diagnostics based on Lagrangian descriptor (LD) computations of initially nearby orbits to detect chaos in conservative dynamical systems with phase space dimensionality higher than two. In particular, we consider the recently introduced methods of the difference and the ratio of the LDs of neighboring orbits, as well as a quantity related to the finite-difference second spatial derivative of the LDs, and use them to determine the chaotic or regular nature of ensembles of orbits of a prototypical area-preserving map model, the 4-dimensional symplectic standard map. We compare the characterization obtained by these LDs-based diagnostics against that achieved by the Smaller Alignment Index method of chaos detection, by recording the percentage agreement P A between the two classifications. We also study the influence of the final number of orbit iterations T , the order n of the indices, as well as the distance σ of neighboring orbits on the performance of these methods, and find appropriate T , n and σ values which allow the efficient use of the three indices as short time and computationally cheap chaos diagnostics achieving P A ≳ 90 %. Our findings clearly indicate the capability of LDs to efficiently identify chaos in systems whose phase space is difficult to visualize, due to its high dimensionality, without knowing the variational equations (tangent map) of continuous (discrete) time systems needed by traditional chaos indicators. • Presentation of chaos diagnostics based on computations of Lagrangian descriptors. • Identification of chaos in high dimensional symplectic maps. • Comparison of various efficient chaos detection techniques. • Identifying chaos without using variational equations or tangent maps. [ABSTRACT FROM AUTHOR]

In this research article, we present the concepts of fractional-order dynamical systems and synchronization methodologies of fractional order chaotic dynamical systems using slide mode control techniques. We have analysed the different phase portraits and time-series graphs of fractional order Rabinovich-Fabrikant systems. We have obtained that the lowest dimen- sion of Rabinovich-Fabrikant system is 2.85 through utilization of the fractional calculus and computational simulation. Bifurcation diagrams and Lyapunov exponents of fractional order Rabinovich-Fabrikant system to justify the chaos in the systems. Synchronization of two iden- tical fractional-order chaotic Rabinovich-Fabrikant systems are achieved using sliding mode control methodology. [ABSTRACT FROM AUTHOR]

This paper puts forward the microscopic chaos control in the deterministic dynamics of the chemical reactor system. First, the dynamic behavior of the chemical reactor system is explored for some of the parameters and chaotic behavior is investigated. Phase plane, bifurcation plots and Lyapunov exponents are presented to verify the chaotic behavior. Second, nonlinear active plus proportional integral sliding mode control (NA-PISMC) is proposed to control microscopic chaos in the chemical reactor system. A proportional integral switching surface is proposed to achieve the stability condition of the error dynamics and controller is designed by using the relevant variables of the chemical system. Unlike the open loop and open plus closed loop control techniques, the design of proposed controller does not require the parameter perturbation. The required stability condition is derived based on Lyapunov stability theory. Simulation is done in MATLAB environment. Numerical simulation results are presented in order to show the effective performances of the proposed controller design. Simulation results correspond that the objectives of chaos existence and chaos control are achieved successfully. [ABSTRACT FROM AUTHOR]

Dynamic system "pendulum - source of limited excitation" with taking into account the various factors of delay is considered. Mathematical model of the system is a system of ordinary differential equations with delay. Three approaches are suggested that allow to reduce the mathematical model of the system to systems of differential equations, into which various factors of delay enter as some parameters. Genesis of deterministic chaos is studied in detail. Maps of dynamic regimes, phase-portraits of attractors of systems, phase-parametric characteristics and Lyapunov characteristic exponents are constructed and analyzed. The scenarios of transition from steady-state regular regimes to chaotic ones are identified. It is shown, that in some cases the delay is the main reason of origination of chaos in the system "pendulum - source of limited excitation". [ABSTRACT FROM AUTHOR]

Recently, the notion of hidden extreme multistability and hidden attractors is very attractive in chaos theory and nonlinear dynamics. In this paper, by utilizing a simple state feedback control technique, a novel 4D fractional-order hyperchaotic system is introduced. Of particular interest is that this new system has no equilibrium, which indicates that its attractors are all hidden and thus Shil'nikov method cannot be applied to prove the existence of chaos for lacking hetero-clinic or homo-clinic orbits. Compared with other fractional-order chaotic or hyperchaotic systems, this new system possesses three unique and remarkable features: (i) The amazing and interesting phenomenon of the coexistence of infinitely many hidden attractors with respect to same system parameters and different initial conditions is observed, meaning that hidden extreme multistability arises. (ii) By varying the initial conditions and selecting appropriate system parameters, the striking phenomenon of antimonotonicity is first discovered, especially in such a fractional-order hyperchaotic system without equilibrium. (iii) An attractive special feature of the convenience of offset boosting control of the system is also revealed. The complex and rich hidden dynamic behaviors of this system are investigated by using conventional nonlinear analysis tools, including equilibrium stability, phase portraits, bifurcation diagram, Lyapunov exponents, spectral entropy complexity, and so on. Furthermore, a hardware electronic circuit is designed and implemented. The hardware experimental results and the numerical simulations of the same system on the Matlab platform are well consistent with each other, which demonstrates the feasibility of this new fractional-order hyperchaotic system. [ABSTRACT FROM AUTHOR]

We examine the importance of chaotic attractors when modelling non-metastatic tumour growth using a model in which cells come in three types: host, immune, and tumour. The relationships between these cell populations are derived from the law of mass action, assuming that a conjugate is formed in the interaction between immune and tumour cells. A nonlinearity in the production of immune cells, based on previous analyses, is introduced and explained. Using previously chosen model parameters, the maximal Lyapunov exponent is calculated numerically as 0.0218, demonstrating the existence of chaotic behaviour. Under the variation of one particular nonlinear rate constant, four distinct types of attractor are observed. Of biological importance, chaotic behaviour is shown to lead to a significantly higher maximum tumour size when compared to non-chaotic behaviour. Counterintuitively, increasing the parameter associated with the killing of tumour cells by immune cells is demonstrated to increase the maximum tumour size when this parameter is below the threshold at which the equilibrium is zero tumour cells. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we measure the chaotic behavior for satellite system through the dissipation, equilibrium points, bifurcation diagrams, Poincare section maps, Lyapunov exponents, and Kaplan‐Yorke dimension. We observe the qualitative behavior of satellite systems through these tools to justify the chaos in the system. Synchronization for 2 identical satellite systems using slide mode control is presented. We estimate the equilibrium points of chaotic satellite system. At each equilibrium point, we obtain the eigenvalues of Jacobian matrix of satellite system and verify the unstable region. We calculate Kaplan‐Yorke dimension, ie, DKY=2.1915, which ensures the strange behavior of the system. The qualitative and simulated results are in an excellent agreement. [ABSTRACT FROM AUTHOR]

A new dynamical system based on Thomas' system is described with infinitely many strange attractors on a 3-D spatial lattice. The mechanism for this multistability is associated with the disturbed offset boosting of sinusoidal functions with different spatial periods. Therefore, the initial condition for offset boosting can trigger a bifurcation, and consequently infinitely many attractors emerge simultaneously. One parameter of the sinusoidal nonlinearity can increase the frequency of the second order derivative of the variables rather than the first order and therefore increase the Lyapunov exponents accordingly. We show examples where the lattice is periodic and where it is quasiperiodic, that latter of which has an infinite variety of attractor types. [ABSTRACT FROM AUTHOR]

Modifications of the Mean Exponential Growth factor of Nearby Orbits (MEGNO) linear variational method called MEGNO2 and OMEGNO2 indicators are introduced. The modifications are based on taking into account not only the linear, but also the nonlinear part of the increment of the phase flow in the divergence among nearby trajectories according to the second-order formulas. The new indicators allow one to determine more quickly the nature of the orbits under study in dynamical systems with zero or small Lyapunov exponents in comparison with the first-order variational indicators. They improve the analysis of regular regions and, in particular, periodic orbits as well as prevent the appearance of spurious structures in the resulting mappings. [ABSTRACT FROM AUTHOR]

Recently, systems with chaos and the absence of equilibria have received a great deal of attention. In our work, a simple five-term system and its anti-synchronization are presented. It is special that the system has a hyperbolic sine nonlinearity and no equilibrium. Such a system generates chaotic behaviours, which are verified by phase portraits, positive Lyapunov exponent as well as an electronic circuit. Moreover, the system displays multistable characteristic when changing its initial conditions. By constructing an adaptive control, chaos anti-synchronization of the system with no equilibrium is obtained and illustrated via a numerical example. [ABSTRACT FROM AUTHOR]

Numerical estimation of Lyapunov exponents in non-linear dynamical systems results in a very high computational cost. This is due to the large-scale computational cost of several Runge-Kutta problems that need to be calculated. In this work we introduce a parallel implementation based on MPI (Message Passing Interface) for the calculation of the Lyapunov exponents for a multidimensional dynamical system, considering a weakly coupled algorithm. Since we work on an academic high-latency cluster interconnected with a gigabit switch, the design has to be oriented to reduce the number of messages required. With the design introduced in this work, the computing time is drastically reduced, and the obtained performance leads to close to optimal speed-up ratios. The implemented parallelisation allows us to carry out many experiments for the calculation of several Lyapunov exponents with a low-cost cluster. The numerical experiments showed a high scalability, which we showed with up to 68 cores.e [ABSTRACT FROM AUTHOR]

Copyright of Ingeniería (0121-750X) is the property of Ingenieria and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The connection between the macroscopic description of collective chaos and the underlying microscopic dynamics is thoroughly analysed in mean-field models of one-dimensional oscillators. We investigate to what extent infinitesimal perturbations of the microscopic configurations can provide information also on the stability of the corresponding macroscopic phase. In ensembles of identical one-dimensional dynamical units, it is possible to represent the microscopic configurations so as to make transparent their connection with the macroscopic world. As a result, we find evidence of an intermediate, mesoscopic, range of distances, over which the instability is neither controlled by the microscopic equations nor by the macroscopic ones. We examine a whole series of indicators, ranging from the usual microscopic Lyapunov exponents, to the collective ones, including finite-amplitude exponents. A system of pulse-coupled oscillators is also briefly reviewed as an example of non-identical phase oscillators where collective chaos spontaneously emerges. [ABSTRACT FROM AUTHOR]

The dynamics of two coupled twist maps with weak dissipation is studied. The calculation of Lyapunov exponents is used to analyze the structure of the action plane of the system. The chaotic transient dynamics is revealed for extremely small values of dissipation by calculation of finite-time Lyapunov exponents. The stagger-and-step method is used to obtain the chaotic saddle and it is found that it is similar to the Arnold web. [ABSTRACT FROM AUTHOR]

In this paper, we explore the Parrondo’s paradox when several dynamically simple maps are combined in a periodic way, producing chaotic dynamics. We show that the paradox is not commutative, that is, it depends on the way that the maps are iterated. We also see that the paradox happens more frequently when the number of maps that we iterate increases. [ABSTRACT FROM AUTHOR]

In order to study the behaviour of discrete dynamical systems under adiabatic cyclic variations of their parameters, we consider discrete versions of adiabatically-rotated rotators. Parallelling the studies in continuous systems, we generalize the concept of geometric phase to discrete dynamics and investigate its presence in these rotators. For the rotated sine circle map, we demonstrate an analytical relationship between the geometric phase and the rotation number of the system. For the discrete version of the rotated rotator considered by Berry, the rotated standard map, we further explore this connection as well as the role of the geometric phase at the onset of chaos. Further into the chaotic regime, we show that the geometric phase is also related to the diffusive behaviour of the dynamical variables and the Lyapunov exponent. [ABSTRACT FROM AUTHOR]

In this paper, a generalized scheme is proposed for designing multistable continuous dynamical systems. The scheme is based on the concept of partial synchronization of states and the concept of constants of motion. The most important observation is that by coupling two m-dimensional dynamical systems, multistable nature can be obtained if i number of variables of the two systems are completely synchronized and j number of variables keep a constant difference between them i.e., their differences are constants of motion, where i + j = m and 1 ≤ i, j ≤ m−1. The proposed scheme is illustrated by taking coupled Lorenz systems and coupled chaotic Lorenz-like systems. According to the scheme, two coupled systems reduce to single modified system with some initial condition-dependent parameters. Time evolution plots, phase diagrams, variation of maximum Lyapunov exponent and bifurcation diagrams of the systems are presented to show the multistable nature of the coupled systems. [ABSTRACT FROM AUTHOR]

This work constructs an autonomous chaotic system with cubic nonlinearity. Some fundamental dynamical properties, such as equilibrium point, phase portrait, Lyapunov exponent, Kaplan–Yorke dimension, and bifurcation of the new system are investigated numerically and theoretically. Analyses show that this autonomous system holds rich dynamics and exhibits periodic, multi-periodic and chaos behavior with single-scroll, double-scroll and four-scroll. [ABSTRACT FROM AUTHOR]

The dynamics of network traffic are complex and nonlinear, and chaotic behaviors and their prediction, which play an important role in local area networks (LANs), are studied in detail, using the largest Lyapunov exponent. With the introduction of phase space reconstruction based on the time sequence, the high-dimensional traffic is projected onto the low dimension reconstructed phase space, and a reduced dynamic system is obtained from the dynamic system viewpoint. Then, a numerical method for computing the largest Lyapunov exponent of the low-dimensional dynamic system is presented. Further, the longest predictable time, which is related to chaotic behaviors in the system, is studied using the largest Lyapunov exponent, and theWolf method is used to predict the evolution of the traffic in a local area network by both Dot and Interval predictions, and a reliable result is obtained by the presented method. As the conclusion, the results show that the largest Lyapunov exponent can be used to describe the sensitivity of the trajectory in the reconstructed phase space to the initial values. Moreover, Dot Prediction can effectively predict the flow burst. The numerical simulation also shows that the presented method is feasible and efficient for predicting the complex dynamic behaviors in LAN traffic, especially for congestion and attack in networks, which are the main two complex phenomena behaving as chaos in networks. [ABSTRACT FROM AUTHOR]

A scheme is proposed to classify the basins for attractors of dynamical systems in arbitrary dimensions. There are four basic classes depending on their size and extent, and each class can be further quantified to facilitate comparisons. The calculation uses a Monte Carlo method and is applied to numerous common dissipative chaotic maps and flows in various dimensions. [ABSTRACT FROM AUTHOR]

In this paper, a new fourth-order memristive chaotic system is constructed on the basis of Chua's circuit. Chaotic behaviors are verified through a series of dynamical analyses, including Lyapunov exponent analysis, bifurcation analysis, and phase diagram analysis. In addition, chaos attractors in the newly-proposed system are implemented by hardware circuits. [ABSTRACT FROM AUTHOR]

We define auto- and cross-correlation functions capable of capturing dynamical characteristics induced by local phase-space structures in a general dynamical system. These correlation functions are calculated in the standard map for a range of values of the nonlinearity parameter k. Using a model of noninteracting particles, each evolving according to the same standard map dynamics and located initially at specific phase-space regions, we show that for 0.6

A new three-dimensional autonomous chaotic system is proposed in this paper. It analyzes dynamic behaviors of this new system, including the stability of equilibria, the system dissipativity, poincare maps, Lyapunov exponent, bifurcation, wave forms, and spectrum. Numeral simulation is performed in Matlab. A chaotic circuit is designed, and the corresponding circuit simulation is conducted. The results show that the chaotic attractor exists in the new system. The synchronization problem of the chaotic system is solved with a synchronization method, and its synchronization circuit is designed. Numeral simulation and circuit simulation show that the synchronization problem of this chaotic system proposed is solved. [ABSTRACT FROM AUTHOR]

This paper reports the finding of a new six-dimensional (6D) autonomous hyperchaotic system, which is obtained by coupling a 1D linear system and a 5D hyperchaotic system that is constructed by adding a linear feedback controller and a nonlinear feedback controller to the Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest is that it has a hyperchaotic attractor with four positive Lyapunov exponents and a unique equilibrium in a large range of parameters. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, power spectrum and Poincaré projections verifies the existence of the hyperchaotic and chaotic attractors. In addition, stability of the hyperbolic equilibrium is analyzed and two complete mathematical characterizations for 6D Hopf bifurcation are given. [ABSTRACT FROM AUTHOR]