Bifurcation Analysis and Electronic Circuit for Sprott Jerk System

In this paper, the Sprott jerk system based quadratic function is presented. The dynamics of this system is revealed through equilibrium analysis, phase portrait, bifurcation diagram and Lyapunov exponents. The Sprott system can exhibit a chaotic attractor, which has complex dynamic behavior. Finally, the circuit implementation is carried out to verify the Sprott Jerk system. The comparison between the MATLAB and MultiSIM simulation results demonstrate the effectiveness of the Sprott system.

In this study, the Sprott model is introduced and its dynamical properties are investigated. In Sec. 2, the existence of chaotic behavior in this system is demonstrated. In Section 3, we determine complex behavior, and introduce the MultiSim platform, together with the electronic circuit schema of the system.

Mathematical Model and Dynamical Analysis
A famous dissipative quadratic jerk chaotic system (Sprott, 1997) is given by the differential equation Sprott showed that the differential equation (1) displays chaotic behaviour when a = 2.017.
In system form, Sprott's quadratic jerk system (1) can be expressed as The system (2) is a five-term Sprott Jerk system with one-quadratic nonlinearity. For numerical simulations, we take the initial values of the Sprott Jerk system (2) as x (0) = 0, y (0) = 0 and z (0) = 0.1. By using Wolf (1985) algorithm, the Lyapunov exponents are (0.44999, 0.45448, -2.0125), and the Kaplan-Yorke dimension is D KY = 2.449. So, this system is chaotic behavior.
For the numerical simulations of the novel jerk chaotic system (2), we have taken the parameter values as in the chaotic case (2) and the initial conditions as (2). Figures 1 (a) -(c) show the projections of the orbital space of the x-y plane, the x-z plane and the y-z plane. For the selected parameter set and initial conditions, the Sprott system in equation (2) present a dense strange attractor.
We derived the bifurcation plots with parameter a as it governs the equilibrium points of the proposed system. The parameter a is varied between [2 2.25], and the local maximum of the state variable is plotted as shown in Figures 2(a) and 2(b) which show the corresponding LEs of the system. It can be seen from Figures 2(a) and 2(b), under the change of parameter a, the chaos state and periodic state of the system appear alternately.
The equilibria of the Sprott Jerk system (2) are found by setting The Sprott Jerk system has one equilibrium point E 0 (-2.017, 0, 0). For the equilibrium point E 0 (-2.017, 0, 0), the Jacobi matrix becomes: To obtain its eigenvalues, let det |J 1 -λ| = 0. Then, the characteristic equation has the following form: Solving the above characteristic equation, the eigenvalues are found as 1 2,3 11 1, 3 22 i   

 
Here λ 1 is a negative real number, λ 2 and λ 1 are conjugate pair of complex eigenvalues having positive real parts. That means the equilibrium E 0 (-2.017, 0, 0) is a saddle point. So, this equilibrium point is unstable

Electronic Circuit
The circuit in Fig. 3 has been designed by using methods in Sambas et al., 2021a), which are based on five operational amplifiers. As can be seen in Fig. 3, X, Y and Z are the voltages at the operational amplifiers U 1 , U 2 and U 3 .
For circuit implementation, we rescale the state variables of the Sprott Jerk system (2) as follows: X = 1/5 x, Y = 1/5 y, Z = 1/5 z. The rescaled chaotic system is given as follows: Applying the Kirchhoff laws, the circuit presented in Figure 3 is described by the following equations: We get the value of electronic components R 1 = R 2 = R 5 = 400 kΩ, R 3 = 198.3 kΩ, R 4 = 80 kΩ, R 7 = R 8 = R 9 = R 10 = 100 kΩ, and C 1 = C 2 = C 3 = 1 nF. Figure 4 illustrates phase portraits which are obtained from the designed circuit. As it can be seen from the MultiSim outputs in Figure 4 and MATLAB simulation in Figure 1, the results are similar.

Conclusion
In this study, we have studied a Sprott Jerk model with quadratic function which is obtained obtained by Sprott (1997). Dynamical properties of this model were analyzed by the help of Lyapunov exponents' spectrum and bifurcation diagram. Finally, the MATLAB simulations and MultiSim simulation are performed to verify the theoretical model. For hardware electronic circuit can be investigated in our future works.