Stability and neimark - sacker bifurcation for a discrete system of one - scroll chaotic attractor with fractional order

This present work investigates the dynamical behaviors of a new form of fractional order three dimensional system with a chaotic attractor of the one-scroll structure and its discretized counterpart. Firstly, existence and parametric conditions for local stability analysis of steady states of the model are addressed. Then the bifurcation theory is applied to investigate the presence of Neimark - Sacker (NS) bifurcations at the coexistence steady state taking the time delay as a bifurcation parameter in the discrete fractional order system. Also the trajectories, phase diagrams, limit cycles, bifurcation diagrams, attractors for period - 1, 2, 3, 4, 5 and a chaotic attractor with one scroll are exhibited for biologically meaningful sets of parameter values in the discretized system. Finally, several numerical examples are presented to assure the validity of the theoretical results and further rich dynamics of the model is explored as well.


Introduction
Over the past few centuries the explosion in scientific knowledge and technology has been a contributing factor and also to some extent a direct consequence in advancement of mathematical modeling. The development of more contemporary form of mathematical modelinghas further enhanced our understanding of the world around us. A periodic motivation for those who develop and study mathematical models is the desire to form prediction [1].The system of equations, that describes physical problems or phenomenonshas been applied in physical and chemical science, economics, engineering and mathematical biology. Also the subject of mathematical modeling involves physical intuition, formulation of equations, solution methods, and analysis leading to accurate prediction. During the past centuries, several types of mathematical models have been developed to investigate the natural and social processes that enlarged over time. These models are referred as dynamical systems. Dynamical systems are divided into two general categories, i.e. deterministic models and stochastic models respectively. Deterministic models are ordinarily employed when the number of quantities involved in the process being modeled is relatively small and all the underlying scientific principles are fairly well understood [4,5].
The rest of this work is organized as follows: In section 2, we obtain the discretization of the model with piecewise constant arguments from the continuous fractional -order dynamical system (5). The existence of steady states of (5) is discussed in section 3. In section 4, we discuss the various 2. New form of fractional order system and its discretization The fractional order systems have risen to prominence due to its distinct features namely they are realistic and practical in their approach, it has non-local property,as it takes into consideration the past and distributed effects of the model which the integer differential equations lack, and these systems are convenient to model biological systems with memory effect. The non-local property implies that the subsequent status of the model depends not only upon its present stage but also upon all its previous stages. Few mathematicians showed that a time delay could have considerable influence over the local stability of coexistence steady state and the cause of periodic doubling bifurcation (PDB) and Neimark -Sacker bifurcation (NSB) [7]. Lorenz and Roseler systems discussed the nonlinear dynamics, all kinds of bifurcations, synchronization and existence of chaotic attractors in a fractional order continuous dynamical system [8]. The motivation for this research work comes from the paper [9]. The model under consideration is a new form of fractional order continuous three dimensional system: , the system (1) is a classical integer order system and Here  ,  ,  and  are the system parameters, q is the fractional order and >0 h is the step size. Now, the system (1) is discretized with piecewise constant arguments process [2], [3] are given as

D x t x t h h y t h h D y t x t h h y t h h z t h h D z t x t h h y t h h z t h h
First, we take 0< th  , so 0 ( / ) <1 th  . Thus, we have The solution of (3) is . Then we obtain which have the following solution  x nh x nh y nh qq t nh y t y nh x nh y nh z nh qq

Steady states of model (5)
First, we find the steady states of system (5) from Obviously, the model (5) has always two non negative steady states, (i) (5) has coexistence steady state

Dynamical nature of system (5)
In this section, we investigate the nonlinear dynamical behavior of the discretized new form of fractional-order system (5). Now, we study the criteria for stability analysis in the neighborhood of In view of the local stability analysis for discrete fractional order system (5), the following theorem is presented. Theorem 4.1. [6] We consider the cubic polynomial equation of the form: where 1  , 2  and 3  are constants. The roots of the cubic equation (8)     are as in (12).

Numerical results
In this section, we provide some numerical examples for the qualitative dynamical nature of a new form of discrete fractional order system (5) to verify the analytical results in Section 4. From the numerical study, it is clearly shown that the approximate solutions We observe that time plot is oscillatory but convergent. The corresponding trajectory spirals moving inwards and it approaches to steady state 1 S . In this case, we get 1 S is sink and the system (5) In this case, we observe that      S is unstable and the system (5) is chaotic attractor, see figure 5. In dynamical systems, existence of chaos implies that the scrolls of a chaotic attractor are developed only around the saddle states of index 2. Furthermore, the saddle states of index 1 are responsible for connecting the scrolls [9]. Also the approximate solutions t x and t y depend on the system parameters q and  are displayed in figure 6. From the above analysis we can see that the coexisting steady state 1 S is a saddle states of index 2. Therefore, the system (5) is chaotic with one -scroll and it is displayed on xy  plane, see figure 6.

Neimark -Sacker bifurcation analysis
In this section, we discuss the parametric conditions for the existence of Neimark-Sacker bifurcation (NSB) at the coexistence steady state and attractors for different values of  to support the analytical analysis and the complex dynamics of a new form of discrete fractional order system (5) with the help of numerical simulations. In order to discuss the NSB for the system (5) at the coexisting steady state 1 S , we choose  as bifurcation parameter. From (11) it is easy to see that ( ) = 0 F  must have a complex conjugate root with modulus one. Clearly equation (11) will have two pure imaginary roots and one real root. Let