A new 4-D hyperchaotic Lü system with a curve equilibrium, its bifurcation analysis, multistability, circuit simulation and synchronization via integral sliding mode control

Chaos in nonlinear dynamics occurs widely in physics, chemistry, biology, ecology, secure communications, cryptosystems and many scientific branches. Chaotic systems have important applications in science and engineering. In this work, we discuss the dynamics and qualitative properties of the 3-D chemical chaotic reactor obtained by Huang and Yang (2005). The phase portraits of the chemical reactor system are depicted and the qualitative properties of the chemical system are discussed. The Lyapunov exponents of the chemical chaotic reactor system are obtained as \(L_1 = 0.2001\), \(L_2 = 0\) and \(L_3 = -10.8295\). Also, the Kaplan-Yorke dimension of the chemical chaotic reactor system is obtained as \(D_{KY} = 2.0185\), which shows the complexity of the system. Since the sum of the Lyapunov exponents is negative, the chemical chaotic reactor system is dissipative. We show that the chemical chaotic reactor system has three unstable equilibrium points and a stable equilibrium point. Control and synchronization of chaotic systems are important research problems in chaos theory. Sliding mode control is an important method used to solve various problems in control systems engineering. In robust control systems, the sliding mode control is often adopted due to its inherent advantages of easy realization, fast response and good transient performance as well as insensitivity to parameter uncertainties and disturbance. Next, using integral sliding mode control, we design adaptive control and synchronization schemes for the chemical chaotic reactor system. The main adaptive control and synchronization results are established using Lyapunov stability theory. MATLAB simulations are shown to illustrate all the main results of this work.

A novel financial system with one stable and two unstable equilibrium points: Dynamics, coexisting attractors, complexity analysis and synchronization using integral sliding mode control

Anti-synchronization is an important type of synchronization of a pair of chaotic systems called the master and slave systems. The anti-synchronization characterizes the asymptotic vanishing of the sum of the states of the master and slave systems. In other words, anti-synchronization of master and slave system is said to occur when the states of the synchronized systems have the same absolute values but opposite signs. Anti-synchronization has applications in science and engineering. This work derives a general result for the anti-synchronization of identical chaotic systems using sliding mode control. The main result has been proved using Lyapunov stability theory. Sliding mode control (SMC) is well-known as a robust approach and useful for controller design in systems with parameter uncertainties. Next, as an application of the main result, anti-synchronizing controller has been designed for Vaidyanathan–Madhavan chaotic systems (2013). The Lyapunov exponents of the Vaidyanathan–Madhavan chaotic system are found as \(L_1 = 3.2226, L_2 = 0\) and \(L_3 = -30.3406\) and the Lyapunov dimension of the novel chaotic system is found as \(D_L = 2.1095\). The maximal Lyapunov exponent of the Vaidyanathan–Madhavan chaotic system is \(L_1 = 3.2226\). As an application of the general result derived in this work, a sliding mode controller is derived for the anti-synchronization of the identical Vaidyanathan–Madhavan chaotic systems. MATLAB simulations have been provided to illustrate the qualitative properties of the novel 3-D chaotic system and the anti-synchronizer results for the identical novel 3-D chaotic systems.

In this work, we report a new chaotic population biology system with one prey and two predators. Our new chaotic population model is derived by introducing two nonlinear interaction terms between the prey and predator-2 to the Samardzija-Greller population biology system (1988).We show that the new chaotic population biology system has a greater value of Maximal Lyapunov Exponent (MLE) than the Maximal Lyapunov Exponent (MLE) of the Samardzija- Greller population biology system (1988).We carry out a detailed bifurcation analysis of the new chaotic population biology system with one prey and two predators. We also show that the new chaotic population biology model exhibits multistability with coexisting chaotic attractors. Next, we use the integral sliding mode control (ISMC) for the complete synchronization of the new chaotic population biology system with itself, taken as the master and slave chaotic population biology systems. Finally, for practical use of the new chaotic population biology system, we design an electronic circuit design using Multisim (Version 14.0).

In this researchwork,we propose a new5-D highly hyperchaotic system with three quadratic nonlinear terms. A novel feature of the new hyperchaotic system is that it has the maximal Lyapunov exponent (MLE) given by ��1 = 10.5374 which implies that the proposed hyperchaotic system has high complexity. Another novel feature of the new hyperchaotic system is that the system exhibits a line of equilibrium points, which shows that the new hyperchaotic system has hidden attractors. We carry out a detailed bifurcation analysis of the new hyperchaotic system with a line equilibrium. Using Multisim, we build an electronic circuit of the new 5-D hyperchaotic system with a line equilibrium. As a control application, we use integral sliding mode control (ISMC) to achieve global asymptotic stabilization of the new 5-D highly hyperchaotic system with a line equilibrium. Lyapunov control theory has been used to establish the stabilization results for the new 5-D hyperchaotic system. MATLAB simulation results are shown to illustrate the main results of this research work.

— 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

Memristor-based systems and their potential applications, in which memristor is both a nonlinear element and a memory element, have been received significant attention in the control literature. In this work, we study a memristor-based hyperchaotic system with hidden attractors. First, we study the dynamic properties of the memristor-based hyperchaotic system such as equilibria, Lyapunov exponents, Kaplan-Yorke dimension, etc. We obtain the Lyapunov exponents of the memristor-based system as \(L_1 = 0.2205\), \(L_2 = 0.0305\), \(L_3 = 0\) and \(L_4 = -10.7862\). Since there are two positive Lyapunov exponents, the memristor-based system is hyperchaotic. Also, the Kaplan-Yorke fractional dimension of the memristor-based hyperchaotic system is obtained as \(D_{KY} = 3.0233\), which shows the high complexity of the system. We show that the memristor-based hyperchaotic system has no equilibrium point, which shows that the system has a hidden attractor. Control and synchronization of chaotic and hyperchaotic systems are important research problems in chaos theory. Sliding mode control is an important method used to solve various problems in control systems engineering. In robust control systems, the sliding mode control is often adopted due to its inherent advantages of easy realization, fast response and good transient performance as well as insensitivity to parameter uncertainties and disturbance. Next, using integral sliding mode control, we design adaptive control and synchronization schemes for the memristor-based hyperchaotic system. The main adaptive control and synchronization results are established using Lyapunov stability theory. MATLAB simulations are shown to illustrate all the main results of this work.

In this paper, an adaptive integral sliding mode control scheme is developed to study the control and synchronization of the hyper chaotic Zhou system with unknown parameters. First, an integral sliding mode control is designed which can stabilize the hyper chaotic Zhou system with known parameters to its unstable equilibrium at the origin. In the next step, the aim is synchronization of two identical hyper chaotic master and slave Zhou systems with unknown parameters. The master and slave system can be considered as a simple dynamical network with two nodes and one link. We propose an adaptive scheme which is led to a new type adaptive controller for this dynamical network with hyper chaotic Zhou system as oscillators which can identify unknown parameters of the master system and synchronize master and slave systems. Simulation results have shown that the proposed method has an excellent convergence from both speed and accuracy points of view and it outperforms Sundarapandian Vaidyanathan's scheme, a well-recognized scheme in the area.

Dual and dual-cross synchronizations in chaotic systems

Hybrid phase synchronization is a new type of synchronization of a pair of chaotic systems called the master and slave systems. In hybrid phase synchronization, the odd numbered states of the master and slave systems are completely synchronized (CS), while their even numbered states are anti-synchronized (AS). The hybrid phase synchronization has applications in secure communications and cryptosystems. This work derives a new result for the hybrid phase synchronization of identical chaotic systems using sliding mode control. The main result has been proved using Lyapunov stability theory. Sliding mode control (SMC) is well-known as a robust approach and useful for controller design in systems with parameter uncertainties. As an application of this general result, a sliding mode controller is derived for the hybrid phase synchronization of the identical 3-D Vaidyanathan chaotic systems (2014). MATLAB simulations have been provided to illustrate the Vaidyanathan system and the hybrid synchronizer results for the identical Vaidyanathan systems.

This paper announces a novel three-dimensional chaotic system with line equilibrium and discusses its dynamic properties such as Lyapunov exponents, phase portraits, equilibrium points, bifurcation diagram, multistability and coexisting attractors. New synchronization results based on integral sliding mode control (ISMC) are also derived for the new chaotic system with line equilibrium. In addition, an electronic circuit implementation of the new chaotic system with line equilibrium is reported and a good qualitative agreement is exhibited between the MATLAB simulations of the theoretical model and the MultiSim results. We also display the implementation of the Field-Programmable Gate Array (FPGA) based Pseudo-Random Number Generator (PRNG) by using the new chaotic system. The throughput of the proposed FPGA based new chaotic PRNG is 462.731 Mbps. Randomness analysis of the generated numbers has been performed with respect to the NIST-800-22 tests and they have successfully passed all of the tests. Finally, an image encryption algorithm based on the pixel-level scrambling, bit-level scrambling, and pixel value diffusion is proposed. The experimental results show that the encryption algorithm not only shuffles the pixel positions of the image, but also replaces the pixel values with different values, which can effectively resist various attacks such as brute force attack and differential attack.

This paper introduces an enhanced five-dimensional Chaotic Supply Chain Model (5DCSCM) by incorporating a transport lag variable into a previously established four-dimensional model. The newly added differential equation in the transit dynamics of the supply chain model captures the inherent lag between customer demand and the physical response in transportation, modeled as a first-order transport lag system. Through comprehensive numerical simulations, the influence of various system parameters—including customer demand rate, delivery efficiency, information distortion, contingency reserve, safety stock, and transportation lag—are examined. The study utilizes bifurcation diagrams and a Lyapunov Exponent (LE) to investigate tran-sitions between periodic and chaotic behavior. Additionally, the model is extended with offset boosting control, allowing for controlled amplitude adjustment without altering the underlying chaotic dynamics. Offset boosting control (OBC) is useful in chaotic supply chain systems because it stabilizes inventory and order fluctuations by counter-acting the amplification of small disturbances, reducing the bullwhip effect, and im-proving overall system reliability and responsiveness. As an application, integral sliding mode control (ISMC) technique has been applied to achieve complete synchronization between a pair of the 5DCSCM. Synchronization based on ISMC is useful in chaotic supply chain systems because it ensures robust coordination between different tiers, suppresses chaos-induced fluctuations, and maintains stable inventory and order patterns even under disturbances and uncertainties.

This paper reports a new five-dimensional four-wing hyperchaotic system with hidden attractor. First, this paper discusses the dynamic properties of the new four-wing system with a detailed bifurcation analysis, coexistence of attractors and multistability, offset boosting, Lyapunov exponents, etc. It is shown that the new four-wing system has no rest point and thus it exhibits hidden attractor. The new four-wing system exhibits two positive Lyapunov characteristic exponents and a large value of Kaplan-Yorke dimension indicating high complexity of the system. We realise the dynamic equations of the new four-wing system with an electronic circuit and simulations via MultiSIM. As a control application, we derive new results for the complete synchronisation of the new four-wing system via integral sliding mode control. MATLAB simulations are adequately provided to illustrate modelling and applications of the new four-wing system with hyperchaotic four-wing attractor.

This paper reports a new five-dimensional four-wing hyperchaotic system with hidden attractor. First, this paper discusses the dynamic properties of the new four-wing system with a detailed bifurcation analysis, coexistence of attractors and multistability, offset boosting, Lyapunov exponents, etc. It is shown that the new four-wing system has no rest point and thus it exhibits hidden attractor. The new four-wing system exhibits two positive Lyapunov characteristic exponents and a large value of Kaplan-Yorke dimension indicating high complexity of the system. We realise the dynamic equations of the new four-wing system with an electronic circuit and simulations via MultiSIM. As a control application, we derive new results for the complete synchronisation of the new four-wing system via integral sliding mode control. MATLAB simulations are adequately provided to illustrate modelling and applications of the new four-wing system with hyperchaotic four-wing attractor.

Chapter 8 - A new thermally excited chaotic jerk system, its dynamical analysis, adaptive backstepping control, and circuit simulation