Advances in Mathematical Physics
Volume 2018 (2018), Article ID 7273531, 12 pages
https://doi.org/10.1155/2018/7273531

Research Article

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

Karthikeyan Rajagopal,¹ Sifeu Takougang Kingni,² Gaetan Fautso Kuiate,³ Victor Kamdoum Tamba,⁴ and Viet-Thanh Pham⁵

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

Correspondence should be addressed to Gaetan Fautso Kuiate; fautso_kuiate@yahoo.com

Received 22 May 2018; Revised 17 July 2018; Accepted 30 July 2018; Published 17 September 2018

Academic Editor: Dimitrios J. Frantzeskakis

Copyright © 2018 Karthikeyan Rajagopal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A two-parameter autonomous jerk oscillator with a cosine hyperbolic nonlinearity is proposed in this paper. Firstly, the stability of equilibrium points of proposed autonomous jerk oscillator is investigated by analyzing the characteristic equation and the existence of Hopf bifurcation is verified using one of the two parameters as a bifurcation parameter. By tuning its two parameters, various dynamical behaviors are found in the proposed autonomous jerk oscillator including periodic attractor, one-scroll chaotic attractor, and coexistence between chaotic and periodic attractors. The proposed autonomous jerk oscillator has period-doubling route to chaos with the variation of one of its parameters and reverse period-doubling route to chaos with the variation of its other parameter. The proposed autonomous jerk oscillator is modelled on Field Programmable Gate Array (FPGA) and the FPGA chip statistics and phase portraits are derived. The chaotic and coexistence of attractors generated in the proposed autonomous jerk oscillator are confirmed by FPGA implementation of the proposed autonomous jerk oscillator. A good qualitative agreement is illustrated between the numerical and FPGA results. Finally synchronization of unidirectional coupled identical proposed autonomous jerk oscillators is achieved using adaptive sliding mode control method.

1. 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 [1–9]. There are increasing works on chaos in jerk oscillators [10–12]. It is worth noting that if the scalar is denoted as a physical variable at time t, the third derivative presents the jerk [13]. Different jerk oscillators with chaos were summarized by Sprott [14]. By using Josephson junctions, Yalcin constructed a general jerk circuit with multiscroll and hypercube attractors [15]. Multiscroll attractors in jerk circuits were presented by Ma et al. [16]. An elegant chaotic oscillators based on jerk equation was implemented in a circuit where the nonlinearity was provided by a single diode [17]. A simple chaotic jerk circuit was used in a sound encryption scheme [18]. Especially, a three-dimensional novel jerk chaotic oscillator with two hyperbolic sinusoidal nonlinearities was reported in [19].

Researchers have shown an increased interest in multistability [20–26]. Multistability is associated with the presence of coexisting attractors with different initial conditions [20]. Jerk oscillators can exhibit multistability despite their simplicity [27–31]. Recently, Njitacke et al. discovered the coexistence of multiple attractors and crisis route to chaos in a chaotic jerk circuit [27] and a simple autonomous jerk oscillator with multiple attractors was introduced in [28]. Kengne et al. performed an analysis of a jerk circuit including a pair of semiconductor diodes connected in antiparallel and investigated its multiple attractors [29]. It is interesting that, by applying the diode bridge memristor into the original jerk circuit, authors proposed a novel hybrid diode based jerk circuit [30]. This work revealed that the hybrid diode based jerk circuit exhibits rich dynamical behaviors including multiple coexisting self-excited attractors. Derived from the autonomous jerk circuit, an autonomous memristor-based jerk circuit was constructed [31]. Interestingly, for the same values of system parameters, the coexistence of four different attractors was obtained in such a memristor-based jerk circuit.

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.

2. Theoretical Analysis of Proposed Autonomous Jerk Oscillator

In chapter three of “Elegant Chaos: Algebraically Simple flow” book published in 2010, Sprott proposed a list of sixteen autonomous chaotic jerk oscillators with different nonlinearities called memory oscillators [14]. The nonlinearities of memory oscillators include quadratic, cubic, quintic, absolute, maximum, sign, exponential, sine, and tangent hyperbolic functions. Inspired by [14], in this work we introduce an autonomous jerk oscillator with a cosine hyperbolic nonlinearity described by

where are state variables of the oscillator and two positive parameters. System (1a), (1b), and (1c) can be converted to a jerk oscillator, as follows:

System (1a), (1b), and (1c) is dissipative because . The equilibrium points of system (1a), (1b), and (1c) are obtained by solving , , , which gives

Equation (3b) cannot be solved analytically. We thus use the Newton-Raphson method to find the value of . Depending on the value of parameter , (3b) presents no roots for and one root () for . Therefore, for , system (1a), (1b), and (1c) has no equilibrium points while, for , it has two equilibrium points and with (see Figure 1). The characteristic equation evaluated at the equilibrium point is

According to the Routh-Hurwitz criterion, the real parts of all the roots of (4) are negative if and only if

The stability analysis of equilibrium point as function of the parameter of system (1a), (1b), and (1c) is depicted in Figure 1.

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

From Figure 1, one can notice that equilibrium point is always unstable for , whereas the equilibrium point is stable for and unstable for . Since the equilibrium points is stable for but it changes stability properties at , system (1a), (1b), and (1c) can exhibit a Hopf bifurcation from equilibrium point when the parameter varies.

Theorem 1. System (1a), (1b), and (1c) undergoes a Hopf bifurcation at the equilibrium point if and the parameter passes through the critical value .

Proof. Substituting into (4) and separating real and imaginary parts, we obtain

Differentiating both sides of (4) with respect to , we can obtain

and

then

All the conditions for Hopf bifurcation to occur are met; therefore system (1a), (1b), and (1c) undergoes a Hopf bifurcation at when and periodic solutions will exist in a neighbourhood of the point (provided that holds). If , the critical value is . For , the trajectories of jerk system (1a), (1b), and (1c) converge to the equilibrium point whereas, for , system (1a), (1b), and (1c) exhibits a limit cycle (not shown).
We plot in Figure 2 a two-parameter () bifurcation diagram depicting the dynamical behaviors of system (1a), (1b), and (1c).
Figure 2 indicated that system (1a), (1b), and (1c) can display unbounded orbits and periodic and chaotic behaviors. In Figure 3, we firstly fix and plot the bifurcation diagram with respect to and the related largest Lyapunov exponent.
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 up to 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 , 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 , period-12-oscillations, period-6-oscillations, and period-3-oscillations are found, respectively, up to 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 and , while, in the range , 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 and coexistence of period-3-oscillations and chaotic attractors in the range . 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 .
For , system (1a), (1b), and (1c) can exhibit either one-scroll chaotic attractor or period-6-oscillations depending on initial conditions as shown in Figure 6(a), while, in Figure 6(b) for , system (1a), (1b), and (1c) can exhibit either one-scroll chaotic attractor or period-3-oscillations depending on initial conditions. Figure 7 presents the basin of attraction of system (1a), (1b), and (1c) in the plane for and .
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.

Figure 2: (Color online) Two-parameter bifurcation diagram depicting the dynamical behaviors of system (1a), (1b), and (1c). Unbounded orbits are in red; periodic oscillations are in green and chaos is in black.

(a)

(b)

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 .

Figure 4: 2D phase portrait of system (1a), (1b), and (1c) for and . The initial conditions are (x(0), y(0), z(0)) = (1, 0, 0.5).

(a)

(b)

Figure 5: (Color online) The bifurcation diagrams depicting the local maxima of (a) and the largest Lyapunov exponents (b) versus the parameter for . 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).

(a)

(b)

Figure 6: Phase portraits in the plane for specific values of parameter : (a) and (b) . The initial conditions are (x(0), y(0), z(0)) = (1.0, 0, 0.5) in left column and (x(0), y(0), z(0)) = (3.0, 0, 0.5) in right column. The other parameter value is .

Figure 7: (Color online) Cross section of the basin of attraction of system (1a), (1b), and (1c) in the x-z-plane at for and . 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.

3. 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 [32–39], digital chaotic oscillators and its FPGA model [36, 40], FPGA based four-wing chaotic attractor [37], and image cryptography with chaotic ciphers and its FPGA implementations and FPGA implemented memristor-based chaotic attractors [41]. Hyperchaotic oscillators with complex mathematical models are also implemented using FPGA and show that the FPGA performs better in power efficiency [42, 43]. More complex synchronization algorithms are also implemented in FPGA such as synchronization of Ikeda oscillators [44], genetically optimized PID and sliding mode control based synchronizations [45, 46], and adaptive sliding mode synchronization [47]. These FPGA realizations are mainly based on hardware-software cosimulations [45–47] and with experimental analysis [44]. FPGA are also used in implementing various chaos-based encryption schemes like chaos-based pseudorandom number generator [48]. 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 [40] to solve the proposed autonomous jerk oscillator and the set of discretized system equations is given as follows:

where are the system parameters and 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 and as per the IEEE754 standards, we use the calculation of 32-bit 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.

(a)

(b)

Figure 8: (Color online) RTL schematic of proposed autonomous jerk oscillator implemented in Kintex 7.

Table 1 shows the resources utilized by the proposed autonomous jerk oscillator for implementing in Kintex-7.

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

Resource
Utilization
Available
Utilization

LUT
880
101400
0.87

FF
522
202800
0.26

DSP
20
600
3.33

IO
97
285
34.04

BUFG
1
32
3.13

Figure 9 shows the 2D phase portraits of the proposed autonomous jerk oscillator.

(a)

(b)

(c)

Figure 9: (Color online) Phase portraits of the FPGA implemented of proposed autonomous jerk oscillator with hardware-software co-simulation for parameter values , and initial conditions with .

Figure 10 shows the coexisting attractors exhibited by the FPGA implemented proposed autonomous jerk oscillator for two different initial conditions.

(a)

(b)

Figure 10: (Color online) Phase portraits in the plane of the FPGA implemented proposed autonomous jerk oscillator for specific values of parameter : (a) and (b) for the initial conditions (x(0), y(0), z(0)) = (1.0, 0, 0.5) in left column and (x(0), y(0), z(0)) = (3.0, 0, 0.5) in right column with the value of the parameter . The step size is taken as .

One-scroll chaotic attractor and coexistence of attractors are clearly seen from Figures 9 and 10, respectively. By comparing Figures 9 and 10 with Figures 9 and 10, it can be noticed that there is a good qualitative agreement between the numerical simulations and FPGA results.

4. 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 [49, 50] adaptive control method [51, 52], extended back stepping control method [53, 54], sliding mode control method [55, 56], adaptive sliding mode control method [56–59], 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.

4.1. Problem Statement

Let the generalized master system be defined as

and the slave system as

where are 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 with master system of initial condition such that the synchronization errors (12) approaches zero.

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 [60] 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, 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 is negative definite.

4.2. 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 . 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 , 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 are the dynamics of the parameter estimates . For numerical simulations, we take the initial conditions of the master system (25a), (25b), and (25c) as , slave system as , parameter estimates as , and the proportional constants . Figure 11 shows the dynamics of the synchronization errors and synchronized states of the master and slave systems and estimated parameters.

(a)

(b)

(c)

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

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(b)]. The unknown parameters of slave system (24a), (24b), and (24c) are simultaneously successfully estimated to their values [see Figure 11(c)].

5. 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, one-scroll 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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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