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Error in the synchronization scheme between drive (Duffing) and response (Lorenz) in Increased-order synchronization.
Source publication
This paper presents increased and reduced order generalized synchronization (GS) of two different chaotic systems with different order based on active control technique. Through this technique, suitable
control functions were designed to achieve generalized synchronization between: (i) 2D Duffing oscillator
as the drive and 3D Lorenz system as the...
Citations
... Moreover, in control scenarios involving coupled systems with unknown parameters [30,33], additional difficulties may arise in synchronizing these systems if they are not of the same order, i.e., they have different phase space dimensions) [34,35,36]. In same-order (same phase space dimension) synchronization problems, the drive and response systems share similar geometric and topological characteristics [21,30,37]. ...
... As a result, the literature has predominantly focused on chaos synchronization in systems of the same dimension, and relatively fewer research works have focused on different-order synchronization problems, i.e., reduced-order synchronization [39,40,41] or increased-order synchronization [36,42,43]. There are practical scenarios where different-order systems need to be synchronized, such as the synchronization between the heart and lungs, the coordination between thalamic and hippocampal neurons, and the alignment of neuron systems with certain biomechanical systems (e.g., biological implants). ...
In this paper, we address the reduced-order synchronization problem between two chaotic memristive Hindmarsh-Rose (HR) neurons of different orders using two distinct methods. The first method employs the Lyapunov active control technique. Through this technique, we develop appropriate control functions to synchronize a 4D chaotic HR neuron (response system) with the canonical projection of a 5D chaotic HR neuron (drive system). Numerical simulations are provided to demonstrate the effectiveness of this approach. The second method is data-driven and leverages a machine learning-based control technique. Our technique utilizes an ad hoc combination of reservoir computing (RC) algorithms, incorporating reservoir observer (RO), online control (OC), and online predictive control (OPC) algorithms. We anticipate our effective heuristic RC adaptive control algorithm to guide the development of more formally structured and systematic, data-driven RC control approaches to chaotic synchronization problems, and to inspire more data-driven neuromorphic methods for controlling and achieving synchronization in chaotic neural networks in vivo.
... Different synchronization methods and techniques have been used to study synchronization between two similar integer order systems (Ojo and Ogunjo 2012), two dissimilar integer order systems of same dimension (Motallebzadeh et al. 2012;Femat and Solís-Perales 2002), two similar or dissimilar systems with different dimensions Ogunjo 2013), three or more integer order system (compound, combination-combination synchronization Ojo et al. 2014b, a;Mahmoud et al. 2016, discrete systems (Liu 2008Kloeden 2004;Ma et al. 2007), fractional order system of similar dimension (Lu 2005), fractional order synchronization of different dimension (Bhalekar 2014;Khan and Bhat 2017), circuit implementation of synchronization (Adelakun et al. 2017), impulsive anti-synchronization of fractional order (Meng et al. 2018), discrete fractional order systems (Liu 2016), multiswitching (Ogunjo et al. 2018), and synchronization between integer order and fractional order systems (Chen et al. 2012). ...
The paper investigates a new hybrid synchronization called modified hybrid synchronization (MHS) via the active control technique. Using the active control technique, stable controllers which enable the realization of the coexistence of complete synchronization and anti-synchronization in four identical fractional order chaotic systems were derived. Numerical simulations were presented to confirm the effectiveness of the analytical technique.
... But different order systems are less considered for synchronization in literature. Reduced order and increased order synchronization are two noteworthy problems in this domain, as for reference we can study the work described in Ho et al. (2006); Ogunjo (2013); Zheng and Li (2021). In present scenario, the technological advancement has demanded lot of sophisticated control techniques in different engineering fields, such as robotic engineering, aerospace-engineering, biomedical engineering and nano-science etc. Non-linear control techniques, including robust control methods are most popularly exploited all across these fields. ...
In this paper, we tried to achieve synchronization between two chaotic systems of different order in master-slave configuration. We mainly focused on reduced order synchronization case. Achieving synchronization of two non-identical non-linear systems is often a very challenging problem. This paper proposes an adaptive law while utilizing to attain linear matrix inequality (LMI) framework so as to identify the conditions to attain synchronization goal. As all natural systems involve uncertainty and are exposed to external disturbances in real practice. Here a detailed analysis to tackle such dynamical systems which contain such effects, is performed. Using Lyapunov stability theory, the asymptotic convergent behavior of the error dynamics between master and slave systems of proposed class under the control effect is proved. The efficacy of the proposed control scheme is tested on different chaotic systems, provided as example. Finally, detailed simulation results are provided in the end of the article to illustrate the effectiveness of the proposed work.
... Recently, the topic of synchronization between different dimensional chaotic and hyperchaotic systems attract more and more attentions. Until now, many effective control schemes have been introduced to achieve chaos synchronization between dynamical systems with different dimensions such as full state hybrid projective synchronization [7], inverse matrix projective synchronization [8], generalized synchronization [9], inverse generalized synchronization [10], hybrid synchronization [11], − synchronization [12], Q-S synchronization [13], reduced order synchronization [14] and increased order generalized synchronization [15]. Recently, a new type of synchronization, called − φ generalized synchronization, has been proposed to synchronize chaotic and hyperchaotic systems with different dimensions. ...
This paper investigates the − φ generalized synchronization between non-identical fractionalorder systems characterized by different dimensions and different orders. The − φ generalized
synchronization combines the inverse matrix projective synchronization with the generalized
synchronization. In particular, the proposed approach enables − φ generalized synchronization to be achieved between n-dimensional master system and m-dimensional slave system in
different dimensions. The technique, which exploits nonlinear controllers, stability property of
integer-order linear systems and Lyapunov stability theory, proves to be effective in achieving
the − φ generalized synchronization. Finally, the approach is applied between 4-D and 5-D
fractional hyperchaotic systems with the aim to illustrate the capabilities of the novel scheme
proposed herein.
... Q-S synchronization [9]. Reduced order synchronization [10]. Increased order generalized synchronization [11]. ...
... (a, b) = (100, 10), and p = 0.95. As for the slave system, let us also consider the 4-component hyperchaotic fractional order system proposed in [28] with the addition of a control term yielding ...
This paper investigates the F – M synchronization between non-identical fractional-order systems characterized by different dimensions and different orders. F – M synchronization combines the inverse generalized synchronization with the matrix projective synchronization. In particular, the proposed approach enables the F – M synchronization to be achieved between an n-dimensional master system and an m-dimensional slave system. The developed approach is applied to chaotic and hyperchaotic fractional systems with the aim of illustrating its applicability and suitability. A multiple-input multiple-output (MIMO) secure communication system is also developed by using the F – M synchronization and verified through computer simulations.
... Different synchronization methods and techniques have been used to study synchronization between two similar integer order systems [26], two dissimilar integer order systems of same dimension [20,8], two similar or dissimilar systems with different dimensions [27,28,22], three or more integer order system (compound, combination-combination synchronization) [25,24,19], discrete systems [14,12,18], fractional order system of similar dimension [16], fractional order synchronization of different dimension [2,11], circuit implementation of synchronization [1] and synchronization between integer order and fractional order systems [3]. ...
The paper investigates a new hybrid synchronization called modified hybrid synchronization (MHS) via the active control technique. Using the active control technique, stable controllers which enable the realization of the coexistence of complete synchronization, anti-synchronization and project synchronization in four identical fractional order chaotic systems were derived. Numerical simulations were presented to confirm the effectiveness of the analytical technique.
... As a consequence of increasing interest in discussing the synchronization phenomena of chaotic nonlinear dynamical systems numerous synchronization types and schemes have been put forward for consideration for instance generalized synchronization [34], projective synchronization [35], modified projective synchronization [36], function projective synchronization [37], modified-function projective synchronization [38], hybrid synchronization [39] and hybrid function projective synchronization [40]. Few years back another kind of synchronization was developed namely increased-order synchronization and reduced-order synchronization in hyper-chaotic systems [41,42]. In this manuscript, we have considered the reduced order scheme for synchronization of the novel chaotic system. ...
This manuscript investigates a novel 3D autonomous chaotic system which generates two strange attractors. The Lyapunov exponent, bifurcation diagram, Poincaré section, Kaplan–Yorke dimension, equilibria and phase portraits are given to justify the chaotic nature of the system. The novel system displays fixed orbit, periodic orbit, chaotic orbit as the parameter value varies. The reduced order combination synchronization is also performed by considering three identical 3D novel chaotic systems in two parts (a) choosing two third order master systems and one second order slave system which is the projection in the 2D plane. (b) choosing one third order master system and two second order slave systems which are the projection in the 2D plane. Numerical simulations justify the validity of the theoretical results discussed.
... The synchronization of chaotic systems with different dimensions has been studied for integer order systems. Some of the studies include increased order synchronization of two systems (Ogunjo, 2013;Ojo et al., 2014c), hybrid function projective combination synchronization (Ojo et al., 2014b), reduced order function projective combination synchronization (Ojo et al., 2014a), backstepping fuzzy adaptive control (Wang and Fan, 2015), and experimental designs (Adelakun et al., 2017) (Fig. 15.1). ...
... The sensitivity of chaotic system to initial conditions implies that two more systems with different initial conditions will exhibit different dynamics. However, with the addition of appropriate functions, trajectories of similar or different chaotic systems can be made to coincide [24]. This is referred to as synchronization. ...
... Over the years, real life applications of synchronization requires the synchronization of different systems and a given number of systems higher than the traditional two systems. This has given rise to reduced and increased order synchronization [24,35], combination synchronization [26,33,34], combination-combination synchronization [27,29] and compound combination synchronization [28]. ...
... The system has been found to be chaotic when a = 10, b = 8∕3, c = 28 with Lyapunov exponents 1.49, 0, −22.46 indicating a strange attractor. Chaotic synchronization of the Lorenz system has been done using different techniques such as increased and reduced order using Active control [24], complete synchronization using OPCL [14]. Gao et al. [12] introduced the 3D fractional order chaotic Lorenz system with order 0.98. ...
The importance of synchronization schemes in natural and physical systems including communication modes has made chaotic synchronization an important tool for scientist. Synchronization of chaotic systems are usually conducted without considering the efficiency and robustness of the scheme used. In this work, performance evaluation of three different synchronization schemes: Direct Method, Open Plus Closed Loop (OPCL) and Active control is investigated. The active control technique was found to have the best stability and error convergence. Numerical simulations have been conducted to assert the effectiveness of the proposed analytical results.
... Increased or reduced order synchronization is the synchronization of two or more systems with different order. Different order synchronization has been achieved between two systems (Ogunjo 2013) and multiple systems (Ojo et al. 2014a, b). ...
... The pioneering work of Pecora and Carroll (1990) in synchronizing two chaotic systems has led to new innovations such as increased order synchronization (Ogunjo 2013), compound synchronization (Ojo et al. 2016), compound-compound synchronization (Ojo et al. 2015a), fractional order synchronization (Ogunjo et al. 2017) and others. Chua (1971) predicted the memristor as a circuit element. ...
The use of memristor in the realization of chaotic circuits has gained popularity in recent times. This can be attributed to its simplicity over the traditional Chua’s diode. The memristor as a nanometer-scale passive circuit element which can be described as a resistor with memory and possesses nonlinear characteristics. In this chapter, the numerical and experimental dynamics of non-autonomous time delay memristive oscillator which consists of negative conductance and smooth-cubic memristor are reported. Diffusive and negative feed back coupling of combination-combination arrays of the electronic circuits are also presented. The viability of both numerical and electronic simulation are also presented.
