A Class of Different Fractional-Order Chaotic (Hyperchaotic) Complex Duffing-Van Der Pol Models and Their Circuits Implementations

Abstract

In this paper, we introduce three versions of fractional-order chaotic (or hyperchaotic) complex Duffing-van der Pol models. The dynamics of these models including their fixed points and their stability is investigated. Using the predictor-corrector method and Lyapunov exponents we calculate numerically the intervals of their parameters at which chaotic, hyperchaotic solutions and solutions that approach fixed points exist. These models appear in several applications in physics and engineering, e.g., viscoelastic beam and electronic circuits. The electronic circuits of these models with different fractional-order are proposed. We determine the approximate transfer functions for novel values of fractional-order and find the equivalent tree shape model (TSM). This TSM is used to build circuits simulations of our models}. A good agreement is found between both numerical and simulations results. Other circuits diagrams can be similarly designed for other fractional-order models.

A novel adaptive synchronization algorithm for a general class of fractional-order complex-valued systems with unknown parameters, and applications to circuit realization and color image encryption

“…The implementation of a circuit differs somewhat from numerical simulation due to constraints posed by hardware limitations. The implementation of circuits for chaotic systems is crucial in the field of nonlinear science due to its practical applications based on chaos [35]. Chaotic systems are highly sensitive to changes in initial values and system parameters, making it challenging to control resistor and capacitor values with high precision during circuit implementation [45].…”

“…Fractional-order complex-valued systems have gained significant attention due to their ability to capture the complex dynamics observed in various real-world phenomena and their potential applications in diverse fields such as physics, engineering, image processing, neural networks, and communications [26,35,36]. In recent years, several techniques based on the theory of complex functions have been developed to investigate various synchronization phenomena in fractional-order complex-valued systems; e.g., finite-time synchronization [36], pinning synchronization [37], adaptive synchronization [38], and complete synchronization [15] are among the synchronization schemes that have been studied extensively.…”

“…The precise modeling and simulation of fractional-order systems using circuit-based approaches play a pivotal role in the advancement of practical applications, serving as a link between theoretical concepts and realworld implementations [45]. One commonly utilized method for simulating fractional-order systems in circuits is through the approximation of their transfer functions using conventional linear circuit components such as resistors and capacitors [35]. This article presents the approximation of transfer functions for various fractionalorder, along with the corresponding resistor and capacitor values in the circuit simulation methodology (CSM).…”

Synchronization and desynchronization of chaotic models with integer, fractional and distributed-orders and a color image encryption application

“…For the first time, chaos synchronization was investigated in 1990 [15]. It was thoroughly introduced in a variety of fields, like physics, engineering, image encryption and neural networks [16][17][18][19]. Numerous synchronization kinds were studied such as combination and combination-combination synchronization [20], while different types of modulus-modulus synchronization among chaotic complex models were stated by Mahmoud et al [18].…”

On the fractional‐order simplified Lorenz models: Dynamics, synchronization, and medical image encryption

“…In the 17th century, the theory of fractional-order derivatives was investigated by Podlubny [1]. Fractional-order differential equations have several applications in various sciences, like physics, biology, and engineering [2][3][4][5]. Many models exist with chaotic and hyperchaotic behaviors in fractional-order models, such as Lorenz model [6], Lü model [7], van der Pol-Duffing model [8], Chen model [9], Burke-Shaw model [10], and the modified Lü, Chen, and Lorenz models [11].…”