Autonomous Jerk Oscillator with Cosine Hyperbolic Nonlinearity : Analysis , FPGA Implementation , and Synchronization

1Center for Nonlinear Dynamics, Defense University, Ethiopia 2Department of Mechanical, Petroleum and Gas Engineering, Faculty of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon 3Department of Physics, Higher Teacher Training College, The University of Bamenda, P.O. Box 39 Bamenda, Cameroon 4Laboratory of Electronics and Signal Processing (LETS), Department of Physics, Faculty of Science, University of Dschang, P. O. Box 67, Dschang, Cameroon 5School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi, Vietnam


Introduction
There has been a significant increase in studies of chaotic oscillators over the past decades because of their complex behaviors and their promising applications [ -].There are increasing works on chaos in jerk oscillators [ -].It is worth noting that if the scalar () is denoted as a physical variable at time t, the third derivative Motived by published results related to jerk oscillators, some questions arose as to know, e.g., if a jerk oscillator with cosine hyperbolic nonlinearity can exhibit multistability or how such oscillator can synchronize.The aim of our work is to explore aspects of the unanswered questions.Our paper is structured as follows.Section 2 is devoted to the theoretical analysis of proposed autonomous jerk oscillator with a cosine hyperbolic term.FPGA implementation of proposed autonomous jerk oscillator is presented in Section 3. In Section 4, synchronization of unidirectional coupled jerk oscillators is studied by applying adaptive sliding mode control method.Finally, Section 5 concludes our work.
Figure 2 indicated that system (1a), (1b), and (1c) can display unbounded orbits and periodic and chaotic behaviors.In The bifurcation diagram of the output () in Figure 2(a) shows that the trajectories of system (1a), (1b), and (1c) converge to the equilibrium point  1 up to  ≈ 2.269 where a Hopf bifurcation occurs followed by a limit cycle motion (i.e., period-1 oscillations) and period-doubling to chaos interspersed with periodic windows.The largest Lyapunov exponent shown in Figure 3(b) confirms the chaotic behavior found in Figure 3(a).The chaotic behavior is illustrated in Figure 4 for a specific value of parameter .
The trajectories of chaotic attractor are swirling around one of the two equilibrium points in Figure 4.This is a signature of one-scroll chaotic attractor.
Secondly, for  = 3.3, sample results showing the bifurcation diagram versus  and the related plot of Largest Lyapunov exponent are provided in Figure 5.
By varying parameter  from 2.87 to 3.8, the bifurcation diagram of the output () in Figure 5(a) displays chaotic behavior interspersed with periodic windows.For  > 3.18, period-12-oscillations, period-6-oscillations, and period-3oscillations are found, respectively, up to  < 3.2012 followed by reverse period-doubling to chaos interspersed with periodic windows.By varying parameter  from 3.8 to 2.87 [see red dots of Figure 5(a)], the output () displays the same dynamical behaviors as shown by black dots of Figure 5(a) in the ranges 2.87 ≤  ≤ 3.182 and 3.2012 ≤  ≤ 3.8, while, in the range 3.182 <  < 3.2012, the output () presents chaotic behavior.By comparing black dots of Figure 5(a) and red dots of Figure 5(a), one can notice that system (1a), (1b), and (1c) displays coexistence of period-6-oscillations and chaotic attractors in the range 3.182 <  < 3.188 and coexistence of period-3-oscillations and chaotic attractors in the range 3.188 ≤  < 3.2012.The largest Lyapunov exponent shown in Figure 5(b) confirms the chaotic behavior found in Figure 5(a).Figure 6 depicts the phase portraits of coexisting attractors found in Figure 5(a) for specific values of parameter .
In Figure 7, red, green, and black regions contain initial conditions that lead to unbounded orbits, periodic and chaotic attractors, respectively.One can see from the basin of attraction that the possibility of occurrence of unbounded orbits is greater than the ones of periodic and chaotic attractors.

FPGA Implementation of Proposed Autonomous Jerk Oscillator
There have been many literatures about implementation of chaotic oscillators using FPGA like FPGA based multiscroll attractor discussed in [ -], digital chaotic oscillators and  In this section we propose to implement the proposed autonomous jerk oscillator using FPGA.The major task in this implementation is to decide the type of numerical method to solve the dynamical system.We choose the forward Euler method [ ] to solve the proposed autonomous jerk oscillator and the set of discretized system equations is given as follows: where ,  are the system parameters and ℎ is the step size for the discrete numerical solution.To implement the hyperbolic cosine function, we use the Taylor series expansion with the first three polynomials as To achieve the accuracy we select the step size ℎ = 0.001 and as per the IEEE754 standards, we use the calculation of 32bit finite integer instead of floating-point operation.By using the arithmetic operations multiplying, adding or subtracting the system (8a), (8b), and (8c) is implemented and the initial conditions for the next iteration can be obtained.By repeating this process of iteration we can derive the discrete chaotic system.The absolute value operation only needs to set the first bit of the 32-bit integer to 0. The discrete state equations are implemented using hardware and software cosimulations and the needed basic arithmetic operators are implemented using the Xilinx system generator toolkit.We used the Kintex 7 chipset (xc7k160tfbg)) for the cosimulations and Matlab Simulink is used to plot the phase plots.The latency of the arithmetic blocks is kept at 3 and the maximum clock frequency of the FPGA used is 437 MHz.The register-transfer level (RTL) schematic of hyperbolic cosine function given by ( 9) is not shown while the RTL schematic of proposed autonomous jerk oscillator is shown in Figure 8.
Table 1 shows the resources utilized by the proposed autonomous jerk oscillator for implementing in Kintex-7.
Figure 9 shows the 2D phase portraits of the proposed autonomous jerk oscillator.
Figure 10 shows the coexisting attractors exhibited by the FPGA implemented proposed autonomous jerk oscillator for two different initial conditions.
One-scroll chaotic attractor and coexistence of attractors are clearly seen from Figures 9 and 10

Synchronization of Unidirectional Coupled Proposed Autonomous Jerk Oscillators Using Adaptive Sliding Mode Control Method
Here we present synchronization results of coupled chaotic proposed jerk oscillators in order to promote chaos-based synchronization designs of this type of proposed jerk oscillator.Synchronization of chaotic systems has applications in secure communication and cryptography.The highly sensitive nature of chaotic oscillators to initial conditions makes it difficult to synchronize the oscillators with uncertainties and disturbance.Some well-known ways to synchronize chaotic oscillators are using active control method [ , ] adaptive control method [ , ], extended back stepping control method [ , ], sliding mode control method [ , ], adaptive sliding mode control method [ -], etc. Sliding mode control method is applied to provide robustness in the face of internal and external disturbances.In section, we use adaptive sliding mode control method for synchronization of unidirectional coupled identical proposed autonomous jerk oscillators using adaptive sliding mode control method.
where (), () are  × 1 row vector and (), () are  ×  matrix elements of the master and slave systems, respectively, ,  are the unknown parameters of the systems, and () is the controller to synchronize the systems.The control objective is to synchronize the slave system with initial condition (0) with master system of initial condition (0) such that the synchronization errors (12) approaches zero. lim The sliding mode controller design for synchronizing the two systems involves selection of sliding surface for the desired dynamics and designing the reaching law such that any point on the phase space is brought to the sliding surface in the presence of uncertainties.
Let us define the proportional integral sliding surface [ ] as where  is the proportional constant vector The first derivative of the sliding surface is derived as For the existence of the sliding mode, it is necessary and sufficient that the sliding surface and its first derivative should be equal to zero.The error dynamics can be derived as In order to avoid the chattering phenomenon caused by discontinuous control signals, the adaptive controller used to synchronize the master and slave system is chosen as where , ,  are positive gain values, ĉ, d are parameter estimates of master and slave systems, and  is the sliding surface.
Using (16) in (15), the error dynamics simplifies to The stability of the proposed controller can be analyzed using the Lyapunov candidate function: The dynamics of the Lyapunov candidate function can be derived as follows: Using ( 17), (16), and ( 14) in (19), Let us define the parameter estimate laws as where   ,   are positive constants.Using (21a) and (21b) in (20), we can solve the Lyapunov function dynamics as follows: where  and  are all positive, and V is negative definite.

Numerical Verifications.
For numerical validation of the proposed synchronization method, we use the proposed autonomous jerk oscillators as master and slave and apply the adaptive sliding mode control to achieve the synchronization.Let us define the master system as where ,  are the system parameters.The slave system with the adaptive sliding mode controllers (  ) are defined as The parameters of the slave system are assumed to be unknown with parameter estimates â, b.Using the master system (23a), (23b), and (23c) and slave system (24a), (24b), and (24c) in the error dynamics can be derived as follows: Let the adaptive sliding mode controllers be chosen as follows: where   > 0,   > 0 are the sliding surface gains and   is the controller gain for  = , ,  Using ( 17) and ( 18) with ( 13), the parameter update laws can be defined as follows: where ȧ , ḃ are the dynamics of the parameter estimates â, b.
For numerical simulations, we take the initial conditions of the master system (25a), (25b), and (25c) as Figure 11 shows the dynamics of the synchronization errors and synchronized states of the master and slave systems and estimated parameters.When t>0.3, the synchronization error variables converge to zero with exponentially asymptotical speed [see Figure 11(a)] and thereby guaranteeing the synchronization between the master system (23a), (23b), and (23c) and slave system (24a), (24b), and (24c) [see Figure 11

Conclusion
In this paper, an autonomous jerk oscillator with a cosine hyperbolic nonlinearity and two parameters was proposed and studied.It was demonstrated that the Hopf bifurcation occurs near the equilibrium point as one of jerk oscillator parameter crosses the critical value.The proposed autonomous jerk oscillator exhibits periodic attractors, onescroll chaotic attractors,, and coexistence between chaotic and periodic attractors.Period-doubling route to chaos and reverse period-doubling route to chaos was found in proposed autonomous jerk oscillator with the variation of each of its two parameters.Further, the proposed autonomous jerk oscillator was implemented using Field Programmable Gate Array in order to show that the proposed autonomous jerk oscillator is hardware realizable.Finally, synchronization of unidirectional coupled identical proposed autonomous jerk oscillators was achieved using adaptive sliding mode control method.

2 AdvancesFigure 1 :
Figure 1: (Color online) Stability diagram of equilibrium points  1,2 versus the parameter  for  = 3. Solid black lines indicate the stable branches and the dashed red lines the unstable branches.

Figure 3 ,
Figure 3, we firstly fix  = 3 and plot the bifurcation diagram with respect to  and the related largest Lyapunov exponent.The bifurcation diagram of the output () in Figure2(a) shows that the trajectories of system (1a), (1b), and (1c) converge to the equilibrium point  1 up to  ≈ 2.269 where a Hopf bifurcation occurs followed by a limit cycle motion (i.e., period-1 oscillations) and period-doubling to chaos interspersed with periodic windows.The largest Lyapunov exponent shown in Figure3(b) confirms the chaotic behavior found in Figure3(a).The chaotic behavior is illustrated in Figure4for a specific value of parameter .The trajectories of chaotic attractor are swirling around one of the two equilibrium points in Figure4.This is a signature of one-scroll chaotic attractor.Secondly, for  = 3.3, sample results showing the bifurcation diagram versus  and the related plot of Largest Lyapunov exponent are provided in Figure5.By varying parameter  from 2.87 to 3.8, the bifurcation diagram of the output () in Figure5(a) displays chaotic behavior interspersed with periodic windows.For  > 3.18, period-12-oscillations, period-6-oscillations, and period-3oscillations are found, respectively, up to  < 3.2012 followed by reverse period-doubling to chaos interspersed with periodic windows.By varying parameter  from 3.8 to 2.87 [see red dots of Figure5(a)], the output () displays the same dynamical behaviors as shown by black dots of Figure5(a) in the ranges 2.87 ≤  ≤ 3.182 and 3.2012 ≤  ≤ 3.8, while, in the range 3.182 <  < 3.2012, the output () presents chaotic behavior.By comparing black dots of Figure5(a) and red dots of Figure5(a), one can notice that system (1a), (1b), and (1c) displays coexistence of period-6-oscillations and chaotic attractors in the range 3.182 <  < 3.188 and coexistence of period-3-oscillations and chaotic attractors in the range 3.188 ≤  < 3.2012.The largest Lyapunov exponent shown in Figure5(b) confirms the chaotic behavior found in Figure5(a).Figure6depicts the phase portraits of coexisting attractors found in Figure5(a) for specific values of parameter .For  = 3.185, system (1a), (1b), and (1c) can exhibit either one-scroll chaotic attractor or period-6-oscillations depending on initial conditions as shown in Figure6(a), while, in Figure6(b) for  = 3.189, system (1a), (1b), and (1c) can exhibit either one-scroll chaotic attractor or period-3oscillations depending on initial conditions.Figure7presents the basin of attraction of system (1a), (1b), and (1c) in the plane  = 0 for  = 3.3 and  = 3.189.In Figure7, red, green, and black regions contain initial conditions that lead to unbounded orbits, periodic and chaotic attractors, respectively.One can see from the basin of attraction that the possibility of occurrence of unbounded orbits is greater than the ones of periodic and chaotic attractors.

Figure 3 :Figure 4 :
Figure 3: The bifurcation diagrams depicting the local maxima (black dots) and local minima (gray dots) of () (a) and the largest Lyapunov exponents (b) versus the parameter  for  = 3.

Figure 5 :Figure 6 :
Figure 5: (Color online) The bifurcation diagrams depicting the local maxima of () (a) and the largest Lyapunov exponents (b) versus the parameter  for  = 3.3.In the graph (a), two sets of data corresponding, respectively, to increasing (black dot) and decreasing (red dot) values of  are superimposed.Largest Lyapunov exponents are obtained by scanning parameter  upwards (black line) and downwards (red line).

Figure 7 :
Figure 7: (Color online) Cross section of the basin of attraction of system (1a), (1b), and (1c) in the x-z-plane at  = 0 for  = 3.3 and  = 3.189.Initial conditions in the black region lead to unbounded orbits, those in the green region lead to the periodic attractor and those in the red region lead to the strange attractor.

Figure 11 :
Figure 11: Time history of synchronization error variables (a), synchronized states (b), and estimated parameters (c).

Table 1 :
Resources consumed by the FPGA implemented of proposed autonomous jerk oscillator.