A COMMON PHENOMENON IN CHAOTIC SYSTEMS LINKED BY TIME DELAY

We present an algorithm for extracting basis functions from the chaotic Lorenz system along with timing and bit-sequence statistics. Previous work focused on modifying Lorenz waveforms and extracting the basis function of a single state variable. Importantly, these efforts initiated the development of solvable chaotic systems with simple matched filters, which are suitable for many spread spectrum applications. However, few solvable chaotic systems are known, and they are highly dependent upon an engineered basis function. Non-solvable, Lorenz signals are often used to test time-series prediction schemes and are also central to efforts to maximize spectral efficiency by joining radar and communication waveforms. Here, we provide extracted basis functions for all three Lorenz state variables, their timing statistics, and their bit-sequence statistics. Further, we outline a detailed algorithm suitable for the extraction of basis functions from many chaotic systems such as the Lorenz system. These results promote the search for engineered basis functions in solvable chaotic systems, provide tools for joining radar and communication waveforms, and give an algorithmic process for modifying chaotic Lorenz waveforms to quantify the performance of chaotic time-series forecasting methods. The results presented here provide engineered test signals compatible with quantitative analysis of predicted amplitudes and regular timing.

— The problem of chaos synchronization is to design a coupling between two chaotic systems (master-slave/drive-response systems configuration) such that the chaotic time evaluation becomes ideal and the output of the slave (response) system asymptotically follows the output of the master (drive) system. This paper has addressed the chaos synchronization problem of two chaotic systems using the Nonlinear Control Techniques, based on Lyapunov stability theory. It has been shown that the proposed schemes have outstanding transient performances and that analytically as well as graphically, synchronization is asymptotically globally stable. Suitable feedback controllers are designed to stabilize the closed-loop system at the origin. All simulation results are carried out to corroborate the effectiveness of the proposed methodologies by using Mathematica 9. Keywords-Synchronization; Lyapunov Stability Theory; Nonlinear Control; Routh-Hurwitz Criterion I. I NTRODUCTION Synchronization of chaotic systems is a process where two (or many) chaotic systems eventually progress identically for different initial conditions in all future states. This means that the dynamical state of one of the system is completely dictated by the dynamical state of the other system [1]. Chaos Synchronization between two chaotic systems is one of the most primary procedures in complex systems’ control and has wide potential applications in different fields [2-6]. After a pioneering work on chaos synchronization [1], synchronization of chaotic dynamical systems has received a great interest among researchers in nonlinear sciences for more than two decades [7]. Until now, diverse techniques have been proposed and applied successfully to synchronize two identical (or nearly identical) as well as nonidentical chaotic systems [8-13]. Notable among those, the Nonlinear control algorithm [7, 9] is one of the effectual techniques for synchronizing two chaotic systems [7]. Nonlinear control techniques take the advantage of the given nonlinear system dynamics to produce high-performance designs. No Lyapunov exponents or gain matrix are required for its execution. These qualities allow the designer to focus on the synchronization problem, leaving troublesome model manipulations [9]. Edward Lorenz, a meteorologist and mathematician, is known to be the pioneer of chaos theory. In the 1960s, Lorenz made his historical discovery by observing weather phenomena particularly in convections of fluids [14]. Lorenz took different mathematical models of fluid convection and simplified them into a system of ordinary differential equations and came up with a 3-D chaotic attractor for the first time, what is now known as the popular Lorenz equations [14]. After the exceptional discovery of E. Lorenz on chaotic attractor, chaos has become an interesting topic for many researchers. During the last three decades, remarkable research has been done on chaos which explored its different applications, features and fundamental properties [15]. The significance of the 3-D differential equations is that relatively simple systems could exhibit rather complex or specifically chaotic behavior. The 3-dimensional chaotic systems have many potential applications in different scientific fields such as chemical reactions, secure communications, biological systems and nonlinear circuits [15]. Due to a wide range of applications of 3-D chaotic systems, various systems such as the Chen system, Rossler system, Liu system, Qi system, Tigan system and Lu system [16-19] have been proposed and applied successfully to many practical systems and have shown some effective outcomes. Recently, a new 3-D autonomous chaotic system based on a quadratic exponential nonlinear term and a quadratic cross product term has been proposed and studied [20]. A quadratic exponential nonlinear term was added to the third equation while eliminating the second term from the second equation and a nonlinear term from the third equation of the Lorenz System [20]. The new 3-D chaotic system is topologically different from the Lorenz System. The two-scroll attractor from the new system exhibits multiplex chaotic dynamics. The nonlinear dynamical properties of the new

Blowout bifurcation in chaotic dynamical systems occurs when a chaotic attractor, lying in some invariant subspace, becomes transversely unstable. We establish quantitative characterization of the blowout bifurcation by unstable periodic orbits embedded in the chaotic attractor. We argue that the bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits . There are two distinct groups of periodic orbits: one transversely stable and another transversely unstable. The bifurcation occurs when some properly weighted transverse eigenvalues of these two groups are balanced. @S1063-651X~97!50902-0# PACS number~s!: 05.45.1b Recently, a novel type of bifurcation has been discovered in chaotic dynamical systems @1,2#. This is the so-called ‘‘blowout bifurcation’’ that occurs in dynamical systems with a symmetric invariant subspace. Let S be the invariant subspace in which there is a chaotic attractor. Since S is invariant, initial conditions in S result in trajectories that remain in S forever. Whether the chaotic attractor in S is also an attractor in the full phase space depends on the sign of the largest Lyapunov exponent L’ computed for trajectories in S with respect to perturbations in the subspace T which is transverse to S. When L’ is negative, S attracts trajectories transversely in the vicinity of S and, hence, the chaotic attractor in S is an attractor in the full phase space. If L’ is positive, trajectories in the neighborhood of S are repelled away from it and, consequently, the attractor in S is transversely unstable and it is hence not an attractor in the full phase space. Blowout bifurcation occurs when L’ changes from negative to positive values. There are distinct physical phenomena associated with the blowout bifurcation. For example, near the bifurcation point where L ’ is negative, if there are other attractors in the phase space, then typically, the basin of the chaotic attractor in S is riddled @3#. When L ’ is slightly positive, if there are no other attractors in the phase space, the dynamics in the transverse subspace T exhibits an extreme type of temporally intermittent bursting behavior, the on-off intermittency @4,5#. Recent study has also revealed that blowout bifurcation can lead to symmetry breaking in chaotic systems @6#. In the study of chaos theory, it is important to be able to understand a bifurcation in terms of unstable periodic orbits of the system because the knowledge of periodic orbits usually yields a great deal of information about the dynamics @7‐9#. Periodic orbits are known to be responsible for many different types of bifurcations in chaotic systems. For example, the period-doubling bifurcation @10# and the saddlenode bifurcation are bifurcations of periodic orbits. Catastrophic events in chaotic systems such as crises @11# and basin boundary metamorphoses @12# are triggered by collision of periodic orbits, usually of low period, embedded in different dynamical invariant sets. The birth of Wada basin boundaries, meaning common boundaries of more than two basins of attraction, is caused by a saddle-node bifurcation on the basin boundary @13#. More recent study indicates that the riddling bifurcation, bifurcation that gives birth to a riddled basin, is triggered by the loss of transverse stability of some periodic orbit of low period embedded in the chaotic attractor in S @14#. In view of the role of periodic orbits played in these major bifurcations, it is desirable to study the blowout bifurcation by periodic orbits. In this regard, Ashwin, Buescu, and Stewart have noticed that as a system parameter changes towards the blowout bifurcation point, more and more atypical invariant measures become transversely unstable @2#. At the bifurcation, the natural measure of the chaotic attractor in S becomes unstable. In this paper, we establish a quantitative characterization of the blowout bifurcation by unstable periodic orbits embedded in the chaotic attractor in the invariant subspace S .I n particular, we argue that near the bifurcation, there exist two groups of periodic orbits S s and S u , each having an infinite number of members, one transversely stable and another transversely unstable, respectively. The sign of the largest transverse Lyapunov exponent L’ is determined by the relative weights of S s and S u : L’ is negative ~positive! when S s ~S u! weighs over S u ~S s!. ~A precise definition of the ‘‘weights’’ will be described in the sequel. ! At the bifurcation, the weights of S s and S u are balanced. In contrast to most known bifurcations in chaotic systems that usually involve only one or a few periodic orbits @10‐14#, blowout bifurcation is induced by a change in the transverse stability of an infinite number of unstable periodic orbits . The num

Performance evaluation of a self-synchronizing Lorenz chaotic system is formulated as a stochastic differential equation problem. Based on stochastic calculus, we provide a rigorous formulation of the numerical evaluation and analysis of the self-synchronization capability and error probabilities of two chaotic Lorenz communication systems with additive white Gaussian noise disturbance. By using the Ito theorem, we are able to analyze the first two moments behavior of the self-synchronization error of a drive-response Lorenz chaotic system. The moment stability condition of the synchronization error dynamic is explicitly derived. These results provide further understanding on the robust self-synchronization ability of the Lorenz system to noise. Various time-scaling factors affecting the speed of system evolution are also discussed. Moreover, an approximate model of the variance of the sufficient statistic of the chaotic communication is derived, which permits a comparison of the chaotic communication system performance to the conventional binary pulse amplitude modulation communication system. Due to synchronization difficulties of chaotic systems, known synchronization-based chaotic communication system performance is quite poor. Thus, alternative synchronization-free chaotic communication systems are needed in the future, The use of a stochastic calculus approach as considered here, however, is still applicable if the considered chaotic communication system is governed by nonlinear stochastic differential equations.

In this research work, time-delay adaptive synchronization and adaptive anti-synchronization of chaotic fractional order systems are analyzed via the Caputo fractional derivative, and the prob-lem of synchronization and anti-synchronization of chaotic systems of variable fractional order is solved by using the fractional order PID control law, the adaptive laws of variable-order frac-tional calculus, and a control law deduced from Lyapunov’s theory extended to systems of time-delay variable-order fractional calculus. In this research work, two important problems are solved in the control area: The first problem is described in which deals with syn-chro-nization of chaotic systems of adaptive fractional order with time delay, this problem is solved by using the fractional order PID control law and adaptative laws. The second problem is de-scribed in which deals with anti-synchronization of chaotic systems of adaptive frac-tional order with time delay, and this problem is solved by using the fractional order PID con-trol law and adaptative laws.

In the chaotic Lorenz system, Chen system and R\"ossler system, their equilibria are unstable and the number of the equilibria are no more than three. This paper shows how to construct some simple chaotic systems that can have any preassigned number of equilibria. First, a chaotic system with no equilibrium is presented and discussed. Then, a methodology is presented by adding symmetry to a new chaotic system with only one stable equilibrium, to show that chaotic systems with any preassigned number of equilibria can be generated. By adjusting the only parameter in these systems, one can further control the stability of their equilibria. This result reveals an intrinsic relationship of the global dynamical behaviors with the number and stability of the equilibria of a chaotic system.

Adaptive synchronization of two time-delayed fractional-order chaotic systems with different structure and different order

An adaptive fuzzy observer-based approach for chaotic synchronization

The article aims to study the reduced-order anti-synchronization between projections of fractional order hyperchaotic and chaotic systems using active control method. The technique is successfully applied for the pair of systems viz., fractional order hyperchaotic Lorenz system and fractional order chaotic Genesio-Tesi system. The sufficient conditions for achieving anti-synchronization between these two systems are derived via the Laplace transformation theory. The fractional derivative is described in Caputo sense. Applying the fractional calculus theory and computer simulation technique, it is found that hyperchaos and chaos exists in the fractional order Lorenz system and fractional order Genesio-Tesi system with order less than 4 and 3 respectively. The lowest fractional orders of hyperchaotic Lorenz system and chaotic Genesio-Tesi system are 3.92 and 2.79 respectively. Numerical simulation results which are carried out using Adams-Bashforth-Moulton method, shows that the method is reliable and effective for reduced order anti-synchronization.

Accurate parameter identification in chaotic dynamical systems constitutes a challenging inverse problem due to extreme sensitivity to initial conditions, pronounced nonlinearity, and highly multimodal error landscapes. To address these challenges, this study proposes a global-best-guided electric eel foraging optimization algorithm (g-EEFO), which enhances the original EEFO framework by embedding a behavior-aware and phase-dependent global learning mechanism. Unlike existing EEFO variants that rely solely on stochastic foraging dynamics, g-EEFO integrates global-best information as a soft cooperative signal that modulates the interacting, resting, hunting, and migrating behaviors without overriding them. In this way, global guidance acts as a directional bias rather than a dominant attractor, preserving ecological diversity while strengthening convergence coherence. For the first time, EEFO and its improved variant are applied to chaotic system parameter estimation. The proposed method is evaluated on two representative models: the classical Lorenz system and a structurally richer memristive chaotic system. Extensive numerical experiments, including statistical analysis, convergence profiling, boxplot distributions, and parameter-evolution trajectories, demonstrate the clear superiority of g-EEFO over several state-of-the-art metaheuristics. For the Lorenz system, g-EEFO achieves a best mean squared error of [Formula: see text], which is six to twenty orders of magnitude lower than competing methods, while maintaining an exceptionally small standard deviation ([Formula: see text]). For the memristive system, g-EEFO attains a best error of [Formula: see text], again outperforming all benchmarks by several orders of magnitude and exhibiting the highest run-to-run stability. In both cases, the estimated parameters match the true system values with near-perfect precision. These results confirm that the proposed behavior-aware global guidance fundamentally reshapes the search dynamics of EEFO, yielding substantial gains in convergence stability, numerical accuracy, and robustness. The g-EEFO therefore provides a powerful and reliable alternative for chaotic parameter identification and nonlinear system reconstruction across diverse dynamical regimes.

In this paper, the authors investigate the anti-synchronization of chaotic Pan, Lorenz, and Lu systems and anti-synchronization of Liu and Cai systems. Global exponential synchronization of Pan, Lorenz, Lu, Liu, and Cai chaotic systems are established by using Lyapunov stability theory. Numerical simulations are performed to illustrate the effectiveness of the proposed synchronization schemes for Pan, Lorenz, Lu, Liu, and Cai chaotic systems.

This extensive study introduces the Rosenbrock method (RosM) for numerically integrating the chaotic Lorenz system, with a focus on its ability to preserve the system’s intrinsic dynamical and structural symmetries. The Lorenz system exhibits significant symmetry, most notably an inversion symmetry (x,y,z)⟶(−x,−y,z), which is a fundamental feature of its chaotic attractor. We lay forth the algorithm and, after systematic comparisons to explicit Runge–Kutta higher-order schemes and semi-analytically obtained solutions, show that the second-order Rosenbrock method performs with excellent accuracy and stability. Crucially, we demonstrate that RosM reliably preserves the system’s symmetry over long-term integration, a property where some explicit methods can exhibit subtle drift. We give a formal error characterization, assess the computational efficiency, and verify the method via bifurcation analysis to support that RosM is a robust and symmetry-aware tool for simulating chaotic systems.

In this paper, a simple control method is proposed for stabilizing unstable equilibria of two typical classes of chaotic systems. For piecewise-linear chaotic systems, such as Chua's circuit, the control parameters can be selected via the pole placement technique from the linear control theory. For general nonlinear chaotic systems with continuously differentiable nonlinearities, particularly polynomial chaotic systems such as the Ro/spl uml/ssler system, Lorenz system, Chen's system, and the modified Chua's circuit with cubic nonlinearity, the control parameters can be chosen according to the pole placement technique and some additional theories of nonlinear ordinary differential equations. The criteria for the design of the control parameters are also investigated. This method is demonstrated to be highly robust against system parametric variations. To verify the effectiveness of the method, it is applied to both the original and the modified chaotic Chua's circuits, where satisfactory control performance is observed in simulations.

The feedback control of a delayed dynamical system, which also includes various chaotic systems with time delays, is investigated. On the basis of stability analysis of a nonautonomous system with delays, some simple yet less conservative criteria are obtained for feedback control in a delayed dynamical system. Finally, the theoretical result is applied to a typical class of chaotic Lorenz system and Chua circuit with delays. Numerical simulations are also given to verify the theoretical results.

Construction of chaotic quantum magnets and matrix Lorenz systems S-boxes and their applications