Chaos Synchronization in Discrete-Time Dynamical Systems with Application in Population Dynamics

Study of chaotic synchronization as a fundamental phenomenon in the nonlinear dynamical systems theory has been recently raised many interests in science, engineering, and technology. In this paper, we develop a new mathematical framework in study of chaotic synchronization of discrete-time dynamical systems. In the novel drive-response discrete-time dynamical system which has been coupled using convex link function, we introduce a synchronization threshold which passes that makes the drive-response system lose complete coupling and synchronized behaviors. We provide the application of this type of coupling in synchronized cycles of well-known Ricker model. This model displays a rich cascade of complex dynamics from stable fixed point and cascade of period-doubling bifurcation to chaos. We also numerically verify the effectiveness of the proposed scheme and demonstrate how this type of coupling makes this chaotic system and its corresponding coupled system starting from different initial conditions, quickly get synchronized.


Introduction
Population dynamics can be modeled through the continuous-time system and the discrete-time system. However, when population size is small or that population does not overlap, discrete-time system is more appropriate to use. Discrete-time population models are widely used to describe the dynamics of hosts and parasitoids interactions [1] [2] [3]. There are many simple nonlinear discrete-time biological models which create rich and complex spectrum of dynamics from coexistence of species through periodic cycles to irregular and chaotic behaviors [4] [5] [6] [7] [8]. Chaos and synchronization are two widespread phenomena with application in many disciplines which have been considered as a central topic in nonlinear dynamics [9] [10] [11]. The presence of chaos in population models has been extensively reported by different researchers [4] [5] [6] [7] [8]. The main property of chaotic dynamics is its critical sensitivity to initial conditions, which is responsible for initially neighboring trajectories separating from each other exponentially in the course of time. Synchronization implies that there is a strong correlation between coupled systems and intuitively, it refers to a phenomenon which makes the systems have the same dynamical behavior. Traditionally, synchronization was based upon periodic signals. However, after coming to the chaotic signals, more possibilities and flexibilities have been entered in this area.
Chaos synchronization, has been started by the work of Fujisaka and Yamada [12] in 1983. After 1990, when the possibility of chaos synchronization was understood by researchers, this idea has received many attractions by people in different areas [13] [14] [15] [16] [17]. Synchronized chaos means that for any two chaotic systems for which any two nearby initial points in phase space quickly diverge and become unpredictable, it is possible that these two converge toward each other and evolve with each other in time. Complete synchronization takes place if there is a perfect linking of the chaotic solutions such that they remain in step with each other in time. In 1990, L.M. Pecora and L. Carroll, described a coupling method which constructs a real set of chaotic synchronization circuits [13]. They have applied this common signal to several well-known continuous-time dynamical systems such as Lorenz and Rossler and they claimed that it is possible to use this method with a slight variation for discrete-time dynamical systems. Chaos synchronization has great interest and application in different disciplines like physics [9] and biology [10], and it has been observed in a huge variety of phenomena in nature [11]. Synchronization has an important role in self-organization of organisms' groups in various biological systems [18]. There are several types of synchronization, such as complete synchronization [14], generalized synchronization [19] [20] [21], phase synchronization [22], lag synchronization [23], antisynchronization and projective synchronization [24] and [25].
This study proposes a new way to couple of discrete-time dynamical systems. We study three different types of synchronization for this new coupled system and we present the results related to the local stability of drive response system and we find the attractive set of this novel coupled system. This paper also studies the interesting dynamics of a drive-response Ricker model which has been coupled by convex link function. Our goal is to build an appropriate response system which traces the drive system and finally they evolve in time even in chaotic regime. We explain that how this coupling method can be applied on a general discrete-time dynamical system to get a complete synchronization. Finally, the long term analysis through bifurcation diagrams and also time-series analysis exhibit that this drive-response system which reveals complex dynamics including cascade of period doubling to chaotic solutions, for smaller synchronization threshold, get completely synchronized.

Description of the Coupling Method
In this section, we study the complete synchronization in a general discrete-time drive-response system. Here, we use a convex function to build the proposed drive-response system. To begin with, consider the following discrete-time dynamical system: where n X ∈  is the state vector of drive system at time n, f is a mapping from n  to itself and is continuously differentiable. Next step is to find a perfect linking such that the system (2.1) and new coupled one remain in step with each other in time. To model the response system or coupled system, we use a convex link function as the form and we demonstrate the error between the solutions of the drive system (2.1) and the response system (2.2) by ( ) n n e n Y X = − .

Complete Synchronization Using Contraction Mapping Theorem
To explain the complete synchronization between two systems (2.1) and (2.2), where α is called a contraction constant of F on E [26].
The error between the drive and response system (2.1) and (2.2) has the following form: where, β is a contraction constant.
As we defined before, to get complete synchronization, we need to have Since, we assumed that F is a contraction mapping, it satisfies the following inequality: where, α is a contraction constant. By triangular inequality we can write: where λ is the root of characteristic polynomial or eigenvalue for A. Since, 0 1 α Here, similar to the proof of theorem (2.3), we decompose the nonlinear dynamical system (2.1) and (2.2) into linear and non-linear part. In this case, using triangular inequality we have:

Local Dynamics, Attractors and Attracting Set of Drive-Response System
We continue this section by seeking appropriate closed subset n Ω ∈  in which for drive and response system (2.1) and (2.2); where , the following conditions hold: 1) for all , 2) f is a contraction on Ω .
The Jacobian matrix for drive-response system (2.1)-(2.2) has the following form: We can immediately obtain the following result: Proposition 2.4 Given Jacobian matrix (2.7), for which the following inequality holds: where β is the contraction constant, J ρ is the spectral radius of J and i λ for are the eigenvalues of Jacobian matrix J. Then, the mapping f is a contraction and the drive-response system (2.1)-(2.2) satisfies the complete synchronization properties, i.e.
where, β is a contraction constant. Thus, for contraction constant 0 which means that the drive-response system (2.1)-(2.2) satisfies the complete synchronization properties. The schematic representation for this type of coupling to obtain complete synchronization has been demonstrated in Figure 1.
Recall from the theorem (2.3) that the linearization of a given autonomous drive-response problem discrete dynamical system can be written as the form:  Remark 2. For drive-response system (2.9), the following hold: 2) h is a contraction on Ω × Ω .
Therefore, we have the following statements for system (2.9).
Proposition 2.5 Consider the drive-response system In another word, if (2.8) holds, then the drive-response system (2.9) has a unique fixed point.
Remark 3 According to the well known contraction mapping theorem, the converse of proposition (2.5) does not necessarily hold.
For drive-response system (2.9), we can establish notions of sequential convergence and hence of stability for drive-response system (2.9). We now provide a general theorem which is the result of proposition (2.4).
2) Equilibrium * Z of drive-response system (9) is stable; i.e. for any 0 ε > , there exists Proof. The proof is straightforward. Remark 4 In theorem (2.6), hyperbolicity is a robust property and it is one of the most important assumptions.
It has been experimentally observed that there may be some situations in which the response system is stable but the response system has complex dynamics and the reason is using non differentiable link function or any non-differentiable transport system. In this study, we take the advantages of using a continuous convex link function which can completely control over the behavior of response system and we will numerically show that the response system inherits the same qualitative dynamics as its drive system and even for smaller synchronization threshold, the response system and drive system are almost completely equivalent.

Application of Chaos Synchronization in Population Dynamics
The chaotic behavior may be observed experimentally in natural systems in many scientific areas. Chaos can be defined as irregular and unpredictable time evolution of non linear systems. Main characteristic of chaos is sensitivity on initial conditions and that system does not repeat its past behavior. Despite the fact that chaotic systems are sensitive to initial conditions, it has been experimentally shown that the chaotic oscillators could be coupled. Chaos synchronization occurs when a chaotic oscillator drives another chaotic oscillator and is a very important phenomenon which has been occurred widely in ecological systems [27] [28] [29] [30]. Because of importance of synchronization and its consequences on population dynamics, we study Ricker model and its synchronized system and we present their qualitative dynamics using different dynamical system tools. The Ricker model is one of the most widely-used ecological models which displays regular and irregular complex nonlinear dynamics [31] [32] and its coupled system as the following form:    : Then, at the origin we have The local stability analysis results for the fixed points of (3.1)-(3.2) are summarized as below: 1) The equilibrium point ( ) 0,0 for ln r s < − is a saddle point, and for ln r s > − is an unstable fixed point.
2) The equilibrium point ( ) 3) The equilibrium point ( ) is an unstable point, and for ( ) ( ) To study the global stability of the equilibrium points of both systems, at first we prove that all solutions in the first quadrant 2 +  are eventually bounded. Proof. By induction.
Since ( ) x n x n f x Hence, x n x f M r r  2) is persistent. If system is not persistent, it is called non-persistent [33]. 1) Then for drive-response system (3.1)-(3.2) for ( ) x n x n r Proof. Using theorem (2.6), since Ω × Ω is closed and the conditions of proposition (2.4) are satisfied, therefore the solutions of drive-response system (3.1)-(3.2) are completely synchronized and the error between the solutions converges toward zero.

Phase and Amplitude Synchronization in Population Dynamics
We begin with two important concepts in theory of synchronization of chaotic systems: a) mean phase difference, b) mean amplitude difference. We analyze these two types of synchronizations for drive-response population model (3.1)-(3.2) which have been coupled using the proposed link function. We consider the oscillations of this discrete-time population system as being synchronized if their phases coincide repeatedly and they have identical mean amplitude. These two types of synchronization have been studied widely in science, nature, engineering, or social life [34]- [39] and before using them, we briefly define them.
Definition 3. 5 We call two systems are in phase synchronization if they have equivalent mean phase or they have a constant difference in phase. We define mean phase for two oscillators as where, N τ is the number of cycles within a time τ .
In fact, for two non-identical oscillators, phase synchronization happens when their phases evolve in synchrony but their amplitude remain unsynchronized. Definition 3. 6 We call two systems are in amplitude synchronization if they have identical mean amplitude. We define mean amplitude for two oscillators as  Figure 2 demonstrates the mean phase difference, i.e.  (3.2). As we can see, using these two tools, we can numerically catch the threshold at which the systems (3.1) and (3.2) satisfy the phase synchronization and amplitude synchronization properties.

Complete Synchronization in Population Dynamics
In this section, we study the complete synchronization for drive-response system (3.1)-(3.2) using some qualitative methods which have been used frequently to detect chaos. In order to understand some dynamical behaviors of systems including period doubling bifurcations and chaotic oscillations, we picked a single parameter bifurcation, which can demonstrate how dependence is the dynamics of the systems on a certain parameter. Figure 3 demonstrates the solutions of drive system (3.1) (red color) and response system (3.2) (black color) with different initial conditions and some interesting r values while 0.5 s = . For 1.3 r = and 0.5 s = both drive and response system (3.1)-(3.2) evidently, exhibit the sigmoidal approach to carrying capacity reminiscent of the logistic model. For 1.9 r = , we have damped oscillations toward steady state which is because of two biological phenomena at first, the population which started below carrying capacity does not smoothly approach steady state through a phenomenon called as overcompensation and then this follows by the second phenomenon called undershooting, which is due to further overshooting. If we increase r further, for the value 2.1 r = we see that these damped oscillations follows by a 2-cycle pattern and they are diverging from the steady states. Similarly, for 2.2 r = and 2.4 r = , the oscillations follow a 2-cycle pattern. Finally, for 3.8 r = we can see the occurrence of unpredictable, irregular and chaotic oscillations.  x and its coupled 2 x in time with two different initial conditions for drive-response system (3.1)-(3.2) when 0.5 s = and 10 k = , drive system (red color) and response system (black color).
In Figure 4 and for the case 0.95 s = , we have almost the same dynamics as we had for 0.5 s = . As we can see, for larger values of threshold s, we do not get completely synchronized cycles. Indeed, displayed dynamics in Figure 3 and Figure 4 are not special to the Ricker model, but are common features of discrete time population models.
For Figure 5, we explain some general properties which have been shared between the drive-response system (3.1)-(3.2) in common. As we can see, with increasing r the value of carrying capacity k is increasing. However, for 2 r ≈ the branch corresponding to stable steady states bifurcates through period-doubling bifurcation into a 2-cycle. As we increase r further, the interval over which we have a new period reduces, and as we know this smaller and smaller windows are called Feigenbaum cascade which after them the dynamics become aperiodic. As we expected, the bifurcation diagram of drive-response system (3.1)-(3.2) for greater values of s, shows the same types of dynamics for drive and response system, but not completely synchronized.
The Poincare section and power spectrum of drive-response system (3.1)-(3.2) have been displayed in Figure 6 for the case 0.95 s = , 3 r = and 10 k = . Basically, the Poincare section can be constructed by sampling the phase portrait which helps to simplify the complicated dynamical systems. It is known that periodic behavior corresponds to a fixed point in Poincare section and any chaotic dynamics can be detected by set of distinct points in Poincare map. Moreover, the wideband chaotic signals and periodic signals can be easily distinguished from each other using the frequency spectra. Therefore, as we understood from bifurcation diagram, drive-response system (3.1)-(3.2) experiences the chaotic dynamics for 3 r = and as we expected for large values of s we can not establish a complete synchronization.

Conclusion
Synchronization in population dynamics can lead to arising complex dynamics and understanding the synchronization of oscillations is crucially important in   T. Azizi, G. Kerr this area. In this study, we developed a new drive-response system by defining a convex continuous link function which maps the orbits of the drive system keeping the same qualitative properties such as stability and periodicity into the orbits of its coupled system. As has been shown by L. M. Pecora and T. L. Carroll, in 1990, two Lorenz systems with the property of sensitive dependence on the initial conditions could be synchronized starting from different initial states. We extended this result into discrete-time dynamical systems and we have shown that by using the concept of convex function, we can force the orbits of a discrete-time drive-response system starting from different initial conditions to get synchronized and we observed that this coupling method can be successful for drive-response system (3.1)-(.3.2) to get a complete synchronization when the synchronization threshold has smaller values, closer to zero. We also changed the values of synchronization threshold s in its range between (0,1) and we observed that the response system (3.2) for larger values of synchronization threshold s is not completely synchronized with its original drive system (3.1) and when we increased the values of synchronization threshold s, we also noticed that the qualitative behaviors of both systems remain the same, even though, we do not get a complete synchronization between the solutions of drive and response system (3.1)-(3.2). In chaotic regime, for larger values of synchronization threshold s, closer to one, we could not get a complete synchronization. But, for smaller synchronization threshold s, closer to zero, we have shown that two systems are in complete synchronization even though the dynamics is chaotic.