Active Control and Electronic Simulation of a Novel Fractional Order Chaotic Jerk System

The chaotic jerk system represents a class of nonlinear systems with unique, varied, and interesting characteristics and behaviour. In this study, we proposed a modiﬁed fractional-order chaotic Jerk system by the introduction of two novel nonlinear terms and two constant terms. The non-linear terms introduced are the exponential and tangential functions while the two parameters are 𝑎 and 𝑏 . Stability analysis of the fractional-order Jerk system was used to establish the parameter ranges for chaotic behaviour in the system. Chaos in the system was conﬁrmed via the Lyapunov exponents. The bifurcation diagram with respect to the parameters, 𝑎 and 𝑏 , as well as the fractional order, 𝛼 were examined. The synchronization of the system was implemented using the active control method. Finally, the studied novel circuit was realized using analog components for experimental validation.


Introduction
Chaos is defined as the "aperiodic long-termed behavior in a deterministic system that exhibits sensitive dependence on initial conditions" [69].The sensitive dependence on initial conditions ensures that future predictions of the system trajectory are limited.The first chaotic system, the Lorenz system, was discovered in a system of differential equations representing weather patterns [36].The Lorenz system consists of three differential equations with two nonlinearities, seven terms, and three parameters.Rössler [59] proposed a three dimensional chaotic system with seven terms and one nonlinearity.A simple three dimensional chaotic system with one parameter and nonlinearity has also been reported [67].Several systems have also been discovered with unique properties.These include Chua [14], Chen [35], Duffing oscillator, and others.Researchers have come up with several simulated or real life systems, represented by a set of differential equations, that have been found to be chaotic.This include neuron models [29,26], population [5], plant and animal diseases [64,39], 3 ring laser [27], substance abuse [8], and chemical reactions [50].The study of chaotic systems began with systems represented by a set of first order differential equations.However, real life systems are best described by other mathematical expressions other than first order differential equations.The methods of chaos analysis were soon extended to the study of discrete chaotic equations where the present state is dependent on previous states of the systems.These models include discrete population models [15].Realizing that natural systems come as signals rather than a set of equations, the study of chaotic time series has gained traction in recent times.Delay differential equations have one or more of the dynamic variable at one or more times in the past.Common chaotic delay differential equations include Mackey and Glass [37] and Ikeda system [28].Partial differential equations represent a set of systems that exists in space and time.Chaotic systems represented by partial differential equations include the Kuramoto-Sivashinsky system [49].Fractional order systems represent a set of differential equations with memory.This memory is succinctly captured by the order of the differential equation which is fractional.Some of the new models are mixtures of different mathematical representations such as discrete fractional order systems [24], fractional time delayed chaotic systems [57].The study of chaos has also been extended to discrete signals like climate and weather [25].
Fractional calculus has been in existence for over 300 years but just gaining interest in Physics and Engineering [12].This trend can be attribtued to new methods of solutions and developments in computers.Recent developments in chaotic fractional calculus include design of robust and non-robust controllers [12], linear relationship between fractal dimensions and the order of fractional calculus [79], and numerical solutions [75].Various approaches for the numerical solutions of fractional calculus have been discussed in details [10,17].Fractional chaos has been found relevant for describing systems in energy harvesting [34], nanotechnology [11], population models [7], neuronal dynamics [41], [2].Various forms of fractional chaotic systems have also been introduced.These includes: fractional discrete systems [24], fractional time delay systems [77], fractional order partial differential equations [63].
Several fractional operators have been introduced for fractional order systems.The most common operators are the Caputo and Riemann-Liouville operators [20].New operators proposed include: the -fractional order derivative [80], -Hilfer fractional derivative [65], and Atangana-Baleanu fractional operator [6].Some of the advances in fractional calculus include the development of novel methods of solving fractional equations.Saadatmandi and Dehghan [60] proposed a numerical algorith for the solution of fractional equations based on the truncated Legendre series.Their approach includes reducing the fractional equation to a set of algebraic equations by extending the Legendre operational matrix to fractional calculus.Jafari et al. [30], rather than a truncated Legendre series, used Legendre wavelets to reduce the fractional order systems to algebraic equations.Doha et al. [18] used Jacobi matrix as opposed to Legendre matrix to reduce the fractional order equations to algebraic equations.Other approaches introduced for solving fractional order systems of equations include Haar wavelet approach [33], Adomian decomposition method [22], fractional order Bernoulli functions [53], second kind Chebyshev wavelet method [76], and shifted Jacobi polynomials [19].
Detecting or characterizing chaos is not easy or straightforward.This became imperative as arguments began to rise as to whether chaos arose from round-off errors [52].Common indicators of chaos in a system include phase space, bifurcation diagram, power spectra, and Lyapunov exponents [1].The power spectra of a chaotic system displays wide spectrum instead of a finite number of peaks.Phase space trajectories that show fractality, non-periodic, are presumed to be chaotic [72].Bifurcation diagrams show changes in a system as parameters of the system changes.It gives information about the route to chaos in a system.A generally acceptable test for chaos is the Lyapunov exponent.It represents the rate of separation of close trajectories.Chaotic systems are characterized by at least one positive Lyapunov exponent and hyperchaotic system with more than one positive Lyapunov exponent.Different algorithms for the computation of Lyapunov exponents include Wolf method [78].
A direct implication of chaos is that the trajectories of two identical systems with different initial conditions will be different.Considering chaos in natural and physical systems, this can either be good or bad.For instance, abnormal synchronization of neurons has been found to be responsible for epilepsy [32].In such situations, it is desirable for two chaotic systems, either similar or dissimilar, to follow the same trajectory.Synchronization between two dynamical systems "refers to the tendency to have the same dynamical behaviour" [23], [44], [45].Carroll et al. [13] implemented the first chaos synchronization scheme in 1987.Since then, the field of synchronization has evolved very fast.The different types of synchronization include complete synchronization, lag synchronization, phase synchronization, projective synchronization, and generalized synchronization [46].The different synchronization techniques include active control, backstepping control, Open plus close loop [43].Synchronization can involve two [46], and three or more systems [3].Furthermore, synchronization can be between systems of different dimensions either as reduced [47] or increased order [48].One of the leading applications of chaos synchronization is cryptography and secure communication.
According to Schot [62], a jerk system is defined as "the time rate of change of acceleration".Jerk systems are of the form ⃛ = ( , ̇ , ̈ ).The first Jerk system was proposed by Malasoma [38] while the simplest Jerk system is existence was reported by Sprott [66].The fractional form of an exponential jerk system has been analysed and applied to random number generation [55].Rajagopal et al. [54] introduced a jerk system with fractional order and time delay.Introducing a piecewise exponential nonlinearity, a novel Jerk system was discovered [71].Several researchers have implemented electronic circuits for the jerk system [68,58].Autonomous dynamical equations can be rewritten as jerk equations under certain conditions.Vaidyanathan et al. [74] proposed a Jerk system with three nonlinearities and one equilibrium point.A Jerk system with two quadratic nonlinearities has also been identified and implemented electronically [73].Kengne et al. [31] introduced a novel Jerk system with two nonlinearities, six terms, and two parameters that exhibit multiple co-existing attractors.For practical relevance in engineering, there is the need to identify and investigate Jerk systems with minimal nonlinearities and system parameters with rich and robust dynamics.Sambas et al. [61] developed a Jerk system with one quadratic term, six terms, and two system parameters.Prakash et al. [51] introduces a fractional-order memristor-based chaotic jerk system with no equilibrium point.Recently, a fractional order chaotic Jerk system with exponential nonlinearity was applied to image encryption [56].
Jerk systems are gaining significant attention in the scientific community due to their physical and practical importance.There is the challenge of obtaining the simplest Jerk system, albeit, with interesting dynamics for application in secure communication and other areas.There is the search for chaotic model, including Jerk systems, which offers a higher level of complexity and a wider chaotic parameter range that can be useful in secure communication.In this study, we propose a novel fractional order chaotic system.The phase space, bifurcation diagrams and largest Lyapunov exponent of the system will also be explored to investigate its chaotic behaviour.Furthermore, the electronic circuit realization of the system will be presented.Results from the electronic implementation will be compared with the numerical results.Finally, controllers will be designed to synchronize two different chaotic Jerk systems using active control method.The structure of this paper is as follows: In Section 3, we introduced the proposed system.The dynamical characteristics are shown in Section 4, while the electronic implementation is shown in Section 7.

Fractional order formalism
Fractional order systems are systems defined by non-integer order [42].Due to lack of adequate solution, the study of fractional order system has largely been ignored.However, the recent advances in numerical and computational methods have drawn a lot of attention to the study of fractional order systems.Fractional order systems have found applications in fields such as biology, engineering, economics [21,70]

Definition
There are two common definitions for the fractional order differential equations -Caputo and Riemann-Liouville definitions.In this study, we adopt the Caputo definition.The Caputo fractional derivative is defined as where 0 < < 1 is the order of the derivative and is the smallest integer greater than [40].

Solution of fractional order systems
Equation 1 can be written in the general form with initial conditions (0) = 0 ( ) [75].The solution can be written as where +1 is the predictor-corrector given as , are functions of given by Equations 5 and 6 [75,16]. (

System
In this study, we introduce a jerk system with both exponential and hyperbolic nonlinearity as where and are the system parameters and is the fractional order of the system.

Theorem 1 ([4]). The equilibrium point
are the eigenvalues of the Jacobian matrix at the equilibrium point.
Considering the divergence of the vector field for a 3-D system [69] The system is considered dissipative if ∇ ⋅ < 0. For System 7, ∇ ⋅ = tanh 3 2 − 1 < 0, hence it is a dissipative system.

Bifurcation analysis and Lyapunov exponents
The analysis of system (7) was carried out using phase space, bifurcation and two dimensional Lyapunov exponents.The phase space realization of the proposed Jerk system both in 2D and 3D are shown in Figs. 1 and 2. The phase space was found to have different dynamics in the 2 − 3 and 1 − 2 planes.The bifurcation of the system with respect to the fractional order, is presented in Fig. 3.The system was found to be stable in the region 0 < < 0.88 while chaotic behaviour occurs in the region 0.89 ≤ < 1.In Fig. 4, we presented the bifurcation diagram of the jerk system with respect to parameter .The bifurcation diagram suggests that the system will be chaotic when = 0.92, = 0.03, and > 0.1.The bifurcation of the Jerk system with respect to parameter shows a reverse period doubling route to chaos (Fig. 5).The largest Lyapunov exponents for the system was estimated in the -, -b, and -planes as shown in Fig. 6 -8 respectively.According to Fig. 6, the system was found to be non-chaotic in very few regions of the − Lyapunov plane.In Fig. 7, the system was found to lose chaoticity at ≈ 0.3.A small region 0.1 < < 0.15 was also found to have negative 1 values.The Lyapunov exponents in the − plane showed clear distinctions of chaotic and non-chaotic regions when the fractional order was chosen as 0.92.

Active control
In this study, the active control method proposed by Bai and Lonngren [9], will be employed.This method has been found to be efficient for the synchronization in ordinary differential equations [46] and fractional order systems [43].The different synchronization schemes include lag synchronization, projective synchronization, phase synchronization, and complete synchronization.In this study, we apply projective synchronization.In projective synchronization, one of the system follows the trajectory of the other system with a scaling factor, .If = 1, we obtain complete synchronization, however, if = −1, we get anti-synchronization. Function projective synchronization is obtained when is a real function rather than a real number.

Design of controllers
We defined the master system as System ( 7) and the slave system as where the terms 1 ( ), 2 ( ), 3 ( ) are active control functions to be determined.The error terms are then defined as where is the scaling factor.Using this notation, we obtain We define the active control functions as The error dynamics can now be written as The control terms 1 , 2 , 3 are chosen such that the solution of System ( 14) converges to zero as time goes to infinity.Therefore, a constant matrix A is chosen which will control the error dynamics such that the feedback matrix is The elements in matrix A are chosen such that the feedback system must have all of the eigenvalues with negative real parts, to make the system stable.We chose the matrix A as The eigenvalues 1 ( = 1, 2, 3) are chosen to be 1 to achieve stable synchronization between the master and slave system.

Numerical simulations
The initial conditions were chosen as [ 1 , 2 , 3 , 1 , 2 , 3 ] = [1, 0.1, 0.2, 0.5, 0.4, 0.3].The system parameters are also chosen as [ , ] = [0.3,0.03] to ensure chaotic behaviour.All simulations were carried out with fractional order 0.92.The solution of the coupled system was obtained using the Adams-Bashforth-Moulton method described in Section 2.2.The complete synchronization of the systems ( = 1) is shown in Fig. 9.In this case, the controllers were applied at time ≥ 50.It could be observed that before the application of the control function (0 < ≤ 50), the system was not synchronized.However, after the activation of the control functions, the system achieved complete synchronization.The solution of the coupled system with = −1 was also obtained and the results shown in Figure 10.At ≥ 50, the solutions were found to have anti-synchronization behaviour.Furthermore, the efficiency of the controller in simulating projective synchronization was tested by setting = 2.The result, shown in Fig 11, depicts the trajectory of the slave system synchronized to that of the master system by a factor of 2. Thus, we have been able to confirm the efficiency of the proposed controllers for complete synchronization, anti-synchronization, and projective synchronization.

Tracking control
Control is a form of synchronization.In chaos control, rather than force the chaotic system to follow the trajectory of another chaotic system, it is made to track another function.In this study, we aim to control each component of the new system using different functions.This approach is referred to as mixed tracking control.

Design of controllers
We introduce control parameters into the Jerk system (Equation 7).This yields The error function is defined as 0.8 0.85 0.9 0.95 1 Fractional order, where ( = 1, 2, 3) are arbitrary functions.Differentiating Equation 17, we get Substituting Equations 17 and 18 into Equation 16, we get Eliminating terms which cannot be expressed as linear terms in 1 , 2 , 3 and solving for ( ), we get where the parameter ( ) will be obtained later.Substituting Equation 20 into Equation 19, the differential of the error becomes Using the active control method, a constant matrix A is chosen which will control the error dynamics (Equation ( 21)) such that the feedback matrix is The matrix A is chosen to be of the form The eigenvalues ( = 1, 2, 3) are chosen to be negative values to achieve control of each component of the Jerk system to the desired function.

Numerical simulations
The coupled differential equations representing the Jerk equation and controllers was solved using the algorithm described in Section 2.2.The initial conditions for the system was taken as [ 1 , 2 , 3 ] = [10.10.20.5].The system was solved with parameters = 3 and −0.03 with fractional order 0.92.These parameters ensure chaotic behaviour of the system.Three functions 1 = , 2 = 2 , and sin were considered for the 1 , 2 , and 3 components respectively.The values of of = 0.03, − 0.02, and = 2 were chosen.Fig. 12 shows the trajectory of the system when the controllers were activated at = 10.The trajectory of 1 were observed to track to a constant value of 0.03 while the trajectory of 2 tracked a curve.After the activation of the control function at = 10, the trajectory of 3 was observed to follow the preset sinusoidal track.This shows the effectiveness of the control functions.To further demonstrate the performance of the controllers, another set of predefined functions were tested.These includes 1 = , 2 = * , and 3 = tanh , where = 3 and = 0.2.The trajectory of the system with the new functions is shown in Fig. 13.

Electronic Realization
As discussed earlier, the proposed fractional-Order Jerk system with two nonlinear terms and two controlled variables can exhibit extremely rich dynamic behavior.Therefore, this section will be focused on the electronic design and analog implementation so as to validate the numerical and theoretical studies.An electronic circuit realization of system (1) consists of exponential and tangential function, which can be expressed as Choosing suitable time rescale and variables: = , = × 1 (1, 2, 3), we can archive an equivalent electronic version of system(1).The system parameters is defined as: 0.3 = , 0.03 = .The analog implementation consists of diode (IN4001) for realization of exponential nonlinearity, while the second nonlinear term is the hyperbolic tangential function.The tangential module marked − ℎ(.) consists of the input voltage ( ), output voltage ( ), resistors, biploar NPN dual-transistor pair (2N2222), pair of operational amplifiers (TL082CD), . A dc current source as shown in dashed box in Fig. 9, can also be realized with another dualtransistor pair (2N2222), while the input-output of the tangential unit can therefore be expressed as: where is the thermal voltage of the transistor and its value is about 26 mV at room temperature, R=10kΩ, 28 = 0.52 Ω

Conclusion
Fractional calculus represents a class of differential equations with fractional order.The study of fractional calculus is over 300 years old and have evolved rapidly over the last few years.In this chapter, we proposed a novel chaotic fractional order Jerk system with two nonlinearities and two constant parameters.The dynamical analysis of the proposed system was investigated.Using stability analysis, the behaviour of the system under different parameter regimes was studied.Our results showed that parameter , was chaotic for values greater than 0.3.This gives a wide spectrum of parameter space when applied to secure communication.Numerical analysis suggests a dissipative system with one equilibrium point.The stability results obtained numerically were confirmed using bifurcation diagrams.Lyapunov exponent, the signature of chaos, was used to confirm the existence of chaos in the system in different parameter space.We further employed the active control method for projective synchronization.By activating the controller at ≥ 50, the anti-synchronization occur when = −1 and complete synchronization at = 1, respectively.The electronic realization of the new proposed system is designed and implemented.The electronic realization brings the system a step further to practical implementation of the system for secure communication and cryptographic applications.This work can be extended to combination, combination-combination, and neural network synchronization.We have used simple electronic implementation, however, this can be extended to FPGA and integrated circuits for deployment in the field.

Figure 3 :
Figure 3: Bifurcation diagram of the proposed Jerk system with respect to the fractional order with parameters = 0.3 and = 0.03.

Figure 4 :
Figure 4: Bifurcation diagram for the proposed Jerk system with respect to parameter a with parameters = 0.03 and = 0.92.

Figure 5 :
Figure 5: Bifurcation diagram for the proposed Jerk system with respect to parameter with parameters = 3 and = 0.92.

Figure 9 :
Figure 9: Solution of the coupled Jerk systems with active control activated at ≥ 50 when = 1

Figure 10 :
Figure 10: Solution of the coupled Jerk systems with active control activated at ≥ 50 when = −1

Figure 16 :
Figure 16: Phase portraits for the modified Jerk system