Hysteresis Modeling and Synchronization of a Class of RC-OTA Hysteretic-Jounce-Chaotic Oscillators

A class of RC-OTA hysteretic-chaotic oscillators has been previously reported using electronics; therefore, hysteresis is realized by an electronic circuit. To obtain a mathematical model of this RC-OTA chaotic-electronic device, hysteresis modeling turns an important issue. Here, we develop a new mathematical hysteretic model proposing a new jounce-chaotic oscillator. Chaosity test is proved using Poincaré theory. After that, a synchronization scheme is granted to synchronize our new jounce-chaotic oscillator (the transmitter) to a dynamics second-order system (the receiver).


Introduction
During recent years, many chaotic oscillators have been proposed. Some of them are based on hysteresis feedback ( [11], [9]). However, these oscillators have been designed from electronic point of view. Nevertheless, hysteresis modeling is an important issue in mechanical and structural systems ( [7]). Some hysteresis models are developed invoking physical laws. Meanwhile, others are heuristic ones. Moreover, [8] reported chaotic behavior in structures with hysteresis, in which hysteresis is governed by the well-known Bouc-Wen model. However, this Bouc-Wen model is not appropriate for the class of RC-OTA chaotic oscillators because it has more parameters than needed. Following this line of engineering modeling, we propose a new dynamic-hysteretic model which is able to capture hysteresis behavior for a class of RC-OTA hysteretic-chaotic oscillators designed, for instance, in [9] (where a pioneer study is given in [2]).
In mechanics, a jerk function is the time derivative of acceleration which, actually, is a third-order dynamic system. According to [10], some forms of jerk functions present chaos. And the time derivative of a jerk function might be called jounce ( [10]), which is a fourth-order dynamic system. Using our dynamic-hysteretic model, and inspired by the RC-OTA architecture, we propose a fourth-order chaotic system named jounce-chaotic oscillator, and whose chaosity test is realized thought Poincaré theory.
By the other hand, chaos synchronization has gained an important attention among scientist (see, for instance, [1], [4], and [13], among others). This because some fields of engineering issues, e.g. secure communications, use chaos synchronization ( [13], [1], and [3]). So far, many of the synchronization systems developed are granted to achieve chaos synchronization between two identical chaotic systems. The case of two different systems seems not to be complete developed ( [13]). We also present a synchronization scheme where our jounce-chaotic system is synchronized with a second order system. The structure of the paper is as follows. Section two presents a review of the simple mathematical model of a class of RC-OTA hysteretic-chaotic oscillators. Our new dynamic-hysteretic model is presented in Section three together with numerical experiments. Our new jouncechaotic system is commented too. In Section four, our synchronization design is granted. Finally, Section five presents the conclusions.

RC-OTA hysteretic chaotic systems
According to [9], a dimensionless dynamic model of a class of RC-OTA hysteretic-chaotic oscillator is given by: where δ and p are the system parameters. The hysteresis function h(x) is shown in Fig. 1. This system presents chaos with δ = 0.05, and p = 1 ([9]).

Hysteresis modeling and a jounce-chaotic system
Hysteresis behavior is recognized as a system with memory. One way to capture hysteresis is by using a dynamic system. For instance, the new hysteretic system:ż can reproduce the hysteretic behavior shown in Fig. 2, where a and b are the hysteresis curve parameters. The speed transition between b and −b is governed by the positive parameter α; sgn(·) is the signum function. For instance, if a = b = 1 and α = 10, the system (2) is: z = 10(−z + sgn(x + sgn(z))).
With a small change on the initial condition of x 1 from 0 to 0.001 (the remaining initial conditions were the same than the previous numerical experiment), the numerical results are displayed in Fig. 9. We thus have a dynamic system that presents bounded trajectory solutions that is highly sensitive to initial conditions and whose signals are random-like. These are the properties of a chaotic system.

Synchronization
Let us introduce the following system: Hysteresis Modeling and Synchronization of a Class of RC-OTA Hysteretic-Jounce-Chaotic Oscillators   where z 1 := z 1 (t), z 2 := z 2 (t), and x 2 := x 2 (t) arrive from the jounce-chaotic system (4)- (7). At this point, the jounce-chaotic system (4)-(7) represents the transmitter and the system (8)-(9) the receiver. Fig 10 gives a schematic representation of our the synchronization design. It is said that the receiver is synchronized with the transmitter if y 1 (t) converges to x 1 (t) and y 2 (t) converges to x 2 (t) as time goes on (and for any initial conditions   y 1 (0), x 1 (0), y 2 (0), and x 2 (0)) 1 . This fact can be proved as follows. Consider the signal errors given by Then, after some basic manipulations, we obtain: The above system represents an exponential stable dynamics. Simulation results are shown in Fig. 11. Remark 1 On synchronization of chaotic systems, it is used to test the synchronization performance by adding a noisy signal to the lines of the channel communication. This noisy signal is a kind of common noise because it is induced simultaneously on each communication lines ( [5]). But, from the technological point of view, common noisy signals are easily to remove via Instrumentation Amplifiers. For instance, according to ( [5], page 85), the common noise induced

Conclusions
A new hysteretic-jounce-chaotic system has been designed along with a synchronization scheme. According to numerical experiments, chaos synchronization between two different systems can be achieved. This fact was theoretically proved too.