**2. The Birkhoff-Shaw chaotic system**

As it is known, chaos theory studies systems that present three very important features [46, 47]:

**•** its periodic orbits must be dense,

the world because it is an interesting phenomenon with a broad range of applications, such as in various complex physical, chemical and biological systems [1–9], in secure and

In synchronization two or more systems with chaotic behaviour can adjust a given of their motion property to a common behaviour (equal trajectories or phase locking), due to forcing or coupling [14]. However, having two chaotic systems being synchronized, it is a major surprise, due to the exponential divergence of the nearby trajectories of the systems. Nevertheless, nowadays the phenomenon of synchronization of coupled chaotic oscillators is well-

Synchronization theory has begun studying in the 1980s and early 1990s by Fujisaka and Yamada [15], Pikovsky [16], Pecora and Carroll [17]. Onwards, a great number of research works based on synchronization of nonlinear systems has risen and many synchronization schemes depending on the nature of the coupling schemes and of the interacting systems have been presented. Complete or full chaotic synchronization [18–23], phase synchronization [24, 25], lag synchronization [26, 27], generalized synchronization [28], antisynchronization [29, 30], anti-phase synchronization [31–36], projective synchronization [37], anticipating [38] and inverse lag synchronization [39] are the most interesting types of synchronization, which have

This chapter deals with two of the aforementioned cases: the complete synchronization and the antisynchronization. In the case of complete synchronization, two identically coupled chaotic systems have a perfect coincidence of their chaotic trajectories, i.e., *x*1(*t*) = *x*2(*t*) as *t* → ∞. In the case of antisynchronization, for initial conditions chosen from large regions in the phase space two coupled systems *x*1 and *x*2, can be synchronized in amplitude, but with

From our knowledge, chaotic systems exhibit high sensitivity on initial conditions or system's parameters and if they are identical and start from almost the same initial conditions, they follow trajectories which rapidly become uncorrelated. That is why many techniques exist to obtain chaotic synchronization. So, many of these techniques for coupling two or more nonlinear chaotic systems can be mainly divided into two classes: unidirectional coupling and bidirectional or mutual coupling [40]. In the first case, only the first system, the master system, drives the second one, the slave system, while in the second case, each system's dynamic

Furthermore, the subject of synchronization between coupled chaotic systems, especially in the last decade, plays a crucial role in the field of neuronal dynamics [6, 41]. Neural signals in the brain are observed to be chaotic and it is worth considering further their possible synchronization [42–46]. These signals are produced by nerve membranes exhibiting their own nonlinear dynamics, which generate and propagate action potentials. Such nonlinear dynam-

So, motivated by the aforementioned fact, the Birkhoff-Shaw system [45], which exhibits the structure of beaks and wings, typically observed in chaotic neuronal models, is chosen for use in this chapter. It is a second order non-autonomous dynamical system with rich dynamical

ics in nerve membranes can produce chaos in neurons and related bifurcations.

broadband communication system [10, 11] and in cryptography [12, 13].

been investigated numerically and experimentally by many research groups.

studied theoretically and proven experimentally.

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opposite sign, that is *x*1(*t*) = –*x*2(*t*) as *t* → ∞.

behaviour influences the dynamics of the other.


In more details, the periodic orbits of a chaotic system have to be dense and that means that the trajectory of a dynamical system is dense, if it comes arbitrarily close to any point in the domain. The second feature of chaotic systems, the topological mixing, means that the chaotic trajectory at the phase space will move over time so that each designated area of this trajectory will eventually cover part of any particular region. Additionally, the third feature, which is the most important feature of chaotic systems, is the sensitivity on initial conditions. When a small variation on a system's initial conditions exists, a totally different chaotic trajectory will be produced.

Here, as it is mentioned above, the well-known non-autonomous chaotic system of Birkhoff-Shaw, which has been proposed by Shaw in 1981 [45], is used. The Birkhoff-Shaw system is described by the 2-D system of differential equations:

$$\begin{cases} \dot{\mathbf{x}} = a\mathbf{y} + \mathbf{x} - c\mathbf{x}\mathbf{y}^2\\ \dot{\mathbf{y}} = -\mathbf{x} - B\cos(dt) \end{cases} \tag{1}$$

where *x* and *y* are the states variables and *a*, *B*, *c* and *d* are positive parameters.

**Figure 1.** Bifurcation diagram of *x* versus *B*, for *a* = 1, *c* = 0.1 and *d* = 1.

**Figure 2.** Bifurcation diagram of *x* versus *B*, for *a* = 1, *c* = 0.2 and *d* = 1.

In this section, the system's dynamic behaviour is investigated numerically by employing a fourth order Runge-Kutta algorithm. As a first step in this approach, the bifurcation diagram and the Lyapunov exponents, which are very useful tools from nonlinear theory, are used. In **Figures 1**–**8**, two sets of bifurcation diagrams of the variable *x* versus the parameter *B*, for *c* = 0.1 and *c* = 0.2 and for various values of the parameter *d*, are displayed. The above bifurcation diagrams show the richness of system's dynamical behaviour. Apart from limit cycles, system (1) has quasiperiodicity and chaos, which makes the system's control a difficult target in practical applications where a particular dynamic is desired.

**Figure 3.** Bifurcation diagram of *x* versus *B*, for *a* = 1, *c* = 0.1 and *d* = 1.5.

where *x* and *y* are the states variables and *a*, *B*, *c* and *d* are positive parameters.

**Figure 1.** Bifurcation diagram of *x* versus *B*, for *a* = 1, *c* = 0.1 and *d* = 1.

296 Nonlinear Systems - Design, Analysis, Estimation and Control

**Figure 2.** Bifurcation diagram of *x* versus *B*, for *a* = 1, *c* = 0.2 and *d* = 1.

In this section, the system's dynamic behaviour is investigated numerically by employing a fourth order Runge-Kutta algorithm. As a first step in this approach, the bifurcation diagram and the Lyapunov exponents, which are very useful tools from nonlinear theory, are used. In

**Figure 4.** Bifurcation diagram of *x* versus *B*, for *a* = 1, *c* = 0.2 and *d* = 1.5.

**Figure 5.** Bifurcation diagram of *x* versus *B*, for *a* = 1, *c* = 0.1 and *d* = 2.

**Figure 6.** Bifurcation diagram of *x* versus *B*, for *a* = 1, *c* = 0.2 and *d* = 2.

**Figure 7.** Bifurcation diagram of *x* versus *B*, for *a* = 1, *c* = 0.1 and *d* = 3.

**Figure 5.** Bifurcation diagram of *x* versus *B*, for *a* = 1, *c* = 0.1 and *d* = 2.

298 Nonlinear Systems - Design, Analysis, Estimation and Control

**Figure 6.** Bifurcation diagram of *x* versus *B*, for *a* = 1, *c* = 0.2 and *d* = 2.

**Figure 8.** Bifurcation diagram of *x* versus *B*, for *a* = 1, *c* = 0.2 and *d* = 3.

In greater detail, having small values of the parameter *d* (i.e. *d* = 1) the system begins from a quasiperiodic state and as the amplitude *B* of the external force increases, the system passes to a stable periodic behaviour of period-1 (**Figures 1** and **2**). For example, in the case of *a* = 1, *c* = 0.1 and *d* = 1, the Lyapunov exponents (LEs) for two respective values of *B* in the regions of quasiperiodic and periodic regions are:

**•** for *B* = 0.1 (quasiperiodic state): LE1 = 0.000, LE2 = 0.000, LE3 = -1.516

**•** for *B* = 2 (periodic state): LE1 = 0.000, LE2 = -0.996, LE3 = -80.998

According to the nonlinear theory, if the number of zeros of LEs is one or two then the system is in periodic or quasiperiodic behaviour, respectively. So, the calculation of Lyapunov exponents plays a crucial role to the estimation of the dynamic behaviour of the proposed system.

However, as the value of the parameter *d* increases the system's complexity is also increased. For *d* = 1.5 (**Figures 3** and **4**) in both cases of *c* = 0.1 and *c* = 0.2, the range of quasiperiodic region has been significantly enlarged, as compared to the previous case (*d* = 1). Nevertheless, with the end of this region, system's behaviour alternates between periodic and chaotic ones. The chaotic regions are detected by finding one positive Lypaunov exponent (i.e. for *a* = 1, *B* = 2.8, *c* = 0.1 and *d* = 1.5, the Lypaunov exponents are: LE1 = 0.157, LE2 = 0.000, LE3 = -1.626). Finally, the system passes from a quasiperiodic state to a stable periodic (period-1) one again.

System's behaviour remains almost the same as the value of parameter *d* (i.e. *d* = 2) increases (**Figures 5** and **6**). However, two important conclusions could be drawn. The first is that the chaotic regions have been enlarged, while the second is that the quasiperiodic region, before the final system's periodic state, has been significantly decreased.

Finally, if the value of parameter *d* has been further increased(i.e. *d* = 3) then the chaotic regions have also been increased while the respective periodic regions have been significantly decreased. Also, the system suddenly passes from chaotic to the final periodic behaviour, as it is shown in the bifurcation diagram of **Figures 7** and **8**.

In these diagrams, the region of period-3 dominates, which is characteristic of system's chaotic behaviour. Also, this region reveals two more important phenomena from nonlinear theory. Firstly, this window of period-3 begins with a sudden transition from a chaotic to periodic behaviour, which in this case is known as *Intermittency* [48] and ends with an *Interior Crisis* [49, 50] that causes intermittency induced from crisis.

In **Figures 9**–**12**, the phase portraits for various values of the parameter *B*, in the case of *a* = 1, *c* = 0.2 and *d* = 3, are presented. In more details, **Figure 9** shows the quasiperiodic attractor, that the system is in for low values of the amplitude *B* (*B* = 0.5) of the external sinusoidal source, while **Figures 10** and **12** display the system's periodic attractors of period-3 (*B* = 3) and period-1 (*B* = 9), respectively. Finally, in **Figure 11** the system's chaotic attractor for *B* = 7 is presented.

Synchronization Phenomena in Coupled Birkhoff-Shaw Chaotic Systems Using Nonlinear Controllers http://dx.doi.org/10.5772/64811 301

**Figure 9.** Phase portrait of *y* versus *x*, for *a* = 1, *c* = 0.2, *d* = 3 and *B* = 0.5 (quasiperiodic behaviour).

In greater detail, having small values of the parameter *d* (i.e. *d* = 1) the system begins from a quasiperiodic state and as the amplitude *B* of the external force increases, the system passes to a stable periodic behaviour of period-1 (**Figures 1** and **2**). For example, in the case of *a* = 1, *c* = 0.1 and *d* = 1, the Lyapunov exponents (LEs) for two respective values of *B* in the regions of

According to the nonlinear theory, if the number of zeros of LEs is one or two then the system is in periodic or quasiperiodic behaviour, respectively. So, the calculation of Lyapunov exponents plays a crucial role to the estimation of the dynamic behaviour of the proposed

However, as the value of the parameter *d* increases the system's complexity is also increased. For *d* = 1.5 (**Figures 3** and **4**) in both cases of *c* = 0.1 and *c* = 0.2, the range of quasiperiodic region has been significantly enlarged, as compared to the previous case (*d* = 1). Nevertheless, with the end of this region, system's behaviour alternates between periodic and chaotic ones. The chaotic regions are detected by finding one positive Lypaunov exponent (i.e. for *a* = 1, *B* = 2.8, *c* = 0.1 and *d* = 1.5, the Lypaunov exponents are: LE1 = 0.157, LE2 = 0.000, LE3 = -1.626). Finally, the system passes from a quasiperiodic state to a stable peri-

System's behaviour remains almost the same as the value of parameter *d* (i.e. *d* = 2) increases (**Figures 5** and **6**). However, two important conclusions could be drawn. The first is that the chaotic regions have been enlarged, while the second is that the quasiperiodic region, before

Finally, if the value of parameter *d* has been further increased(i.e. *d* = 3) then the chaotic regions have also been increased while the respective periodic regions have been significantly decreased. Also, the system suddenly passes from chaotic to the final periodic behaviour, as

In these diagrams, the region of period-3 dominates, which is characteristic of system's chaotic behaviour. Also, this region reveals two more important phenomena from nonlinear theory. Firstly, this window of period-3 begins with a sudden transition from a chaotic to periodic behaviour, which in this case is known as *Intermittency* [48] and ends with an *Interior Crisis* [49,

In **Figures 9**–**12**, the phase portraits for various values of the parameter *B*, in the case of *a* = 1, *c* = 0.2 and *d* = 3, are presented. In more details, **Figure 9** shows the quasiperiodic attractor, that the system is in for low values of the amplitude *B* (*B* = 0.5) of the external sinusoidal source, while **Figures 10** and **12** display the system's periodic attractors of period-3 (*B* = 3) and period-1 (*B* = 9), respectively. Finally, in **Figure 11** the system's chaotic attractor for *B*

**•** for *B* = 0.1 (quasiperiodic state): LE1 = 0.000, LE2 = 0.000, LE3 = -1.516

**•** for *B* = 2 (periodic state): LE1 = 0.000, LE2 = -0.996, LE3 = -80.998

the final system's periodic state, has been significantly decreased.

it is shown in the bifurcation diagram of **Figures 7** and **8**.

50] that causes intermittency induced from crisis.

quasiperiodic and periodic regions are:

300 Nonlinear Systems - Design, Analysis, Estimation and Control

system.

odic (period-1) one again.

= 7 is presented.

**Figure 10.** Phase portrait of *y* versus *x*, for *a* = 1, *c* = 0.2, *d* = 3 and *B* = 3 (periodic behaviour).

**Figure 11.** Phase portrait of *y* versus *x*, for *a* = 1, *c* = 0.2, *d* = 3 and *B* = 7 (chaotic behaviour).

**Figure 12.** Phase portrait of *y* versus *x*, for *a* = 1, *c* = 0.2, *d* = 3 and *B* = 9 (periodic behaviour).
