Dynamical Properties and Finite-Time Hybrid Projective Synchronization Using Fractional Nonsingular Sliding Mode Surface in Fractional-Order Two-Stage Colpitts Oscillators

1 Laboratory of Electronics and of Signal Processing, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon 2 Research Group on Experimental and Applied Physics for Sustainable Development (EAPhySuD), P.O. Box 412, Dschang, Cameroon 3 Laboratoire de Mécanique et de Modélisation des Systèmes, Département de Physique, Faculté des Sciences, Université de Dschang, B.P. 67, Dschang, Cameroon


Introduction
Fractional calculus has an about 300-year-old history, but its applications to physics and engineering are rather recent [1].Many systems are known to display fractional-order dynamics, such as viscoelastic systems, dielectric polarization, and electromagnetic waves [2][3][4], just to name some.
For some decades, there is a growing interest in investigating the chaotic behavior and dynamics of fractional-order dynamic systems; this can be understood as it has been found that fractional-order systems possess memory and display more sophisticated dynamics compared to their integralorder counterparts, something that is of great significance in secure communication [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].It has been shown that several chaotic systems can remain chaotic when their models become fractional [5].A three-dimensional fractional-order modified hybrid optical system is presented in [10] where it was shown that Hopf bifurcation occurs on the proposed system when the fractional order varies and passes a sequence of critical values.Despite these many examples the bifurcation of fractional-order nonlinear system has been studied using solely the Caputo derivative definition and limited to threedimensional systems.
On the other hand, in the past two decades, a new direction of chaos research has emerged to address the more challenging problem of chaos synchronization due to its potential applications in laser physics, chemical reactions, secure communication, biomedicine, and so on [21][22][23].The thrust of research within this area aims at achieving a master-slave synchronization between two chaotic systems by choosing various kinds of methods following the pioneering work of Pecora and Carroll [24].In [11], based on the stability theory, a novel fractional-order controller is presented for the control and synchronization of the fractional-order Lorenz chaotic system via the fractional derivative.In [12], based on tracking control and the stability theory of nonlinear fractional-order systems, Zhou et al. propose the multidriveone response synchronization technique which is simple and theoretically rigorous.In [13], the hybrid projective synchronization of different dimensional chaotic fractionalorder systems is investigated based on the stability theory of linear fractional-order systems.Based on the fractionalorder stability theory and control tracking, Zhou and Zhu investigate the function projective synchronization for the fractional-order chaotic systems [7].In [25], based on the stability theory of the fractional-order system, the authors studies on projective synchronization of the new fractionalorder chaotic system through designing the suitable nonlinear controller are investigated.All these examples perfectly clarify the importance of the fractional-order stability theory, but these works have a common drawback from a practical point of view: the lack of knowledge of analytical time of synchronization.
In recent years some researchers have applied finite-time control and synchronization of fractional-order systems by means of the fractional-order Lyapunov stability theory.In [26], the authors design a novel fractional-order nonsingular sliding mode controller for robust synchronization problem of a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances, the stability of a novel fractional-order integral type sliding surface is proven, based on fractional-order Lyapunov stability theory, a robust sliding control law is derived to guarantee the occurrence of the sliding motion in finite time.In [27], using a nonsingular sliding mode surface, the authors study the finite-time synchronization problem of fractional-order chaotic/hyperchaotic systems in the presence of both model uncertainties and external disturbances.These last works will serve as our handbook in this paper.
It is important to seek bifurcations such as those of Hopf in systems because they are routes towards chaos in such systems.Concerning the system with fractional-order, very little work emphasizes the conditions leading to Hopf bifurcation.Moreover modified projective synchronization is a general case of the simple synchronization.But it had been shown that the finite-time synchronization has very great applications on a practical point of view.Finally, to the best of our knowledge there are few works interested in fractional-order hybrid projective synchronization in finite time and using a fractional nonsingular terminal sliding mode surface; the importance of this surface as that used in [27] is that it supports the robustness of the synchronization in the presence of the disturbances and uncertainties, and that it is easily stabilisable with zero and in finite time.
Motivated by the above discussion, at first in the present work, we propose to tackle the problem of bifurcation of a four-dimensional fractional-order nonlinear system.The two-stage well-studied Colpitts oscillator presented in [28] is a good candidate for the study, due to its broad band in frequency domain.Secondly we study the finite-time hybrid projective synchronization problem of fractional-order twostage Colpitts oscillator in the presence of both model uncertainties and external disturbances.The modified nonsingular terminal sliding mode surface is introduced; its finite-time stability to zero is proved via the Lyapunov stability.So on basis of fractional-order Lyapunov stability theory, a robust control law is designed to force the trajectories of the synchronization error system onto the sliding surface within a finite time and remain on it forever.Numerical simulations demonstrate the applicability and efficiency of the nonlinear control law and verify the theoretical results of the paper.
The rest of this paper is organized as follows.In Section 2, the fractional-order system developed around a two-stage Colpitts oscillator is proposed and its dynamics studied.The numerical results of the dynamics are presented and discussed in Section 3, while the next section is devoted to the synchronization of two two-stage Colpitts oscillators.Finally, Section 5 concludes this work.

The Fractional-Order of a Two-Stage Colpitts Oscillator
2.1.Dynamics of the System.The proposed four-dimensional fractional-order system under study described by the set of ( 1) is obtained by modifying the integer-order two-stage Colpitts oscillator proposed in [28]: Here, the parameters  1 ,  2 , , and  are positive reals, () = exp(−) − 1, and  = ( 1 ,  2 ,  3 ,  4 ) is the fractional order.
Let  be the lowest common multiple of the denominators   's.The zero solution of system (3) is globally asymptotically stable in the Lyapunov sense if all roots 's of the equation

Hopf Bifurcation.
One of the basic differences between the dynamical behavior of fractional-order systems and that of integer-order systems is that the limit set of a trajectory of integer-order system such as a limit cycle is solution for the system under consideration, while in the case of fractionalorder systems, such a limit set of a trajectory may not be solution for this system [18].In [19], the authors claimed that there are no periodic orbits in fractional-order systems, and in [20], an example is given where the solutions of the system are also not periodic but do converge to periodic signals, confirming in both cases what has been stipulated in [18].
In the present paper, we consider the final state of trajectory that appears at the Hopf bifurcation (after suppression of the transitory state).It is also not a periodic solution of the fractional-order system (1) but attracts nearby solutions.
Let us consider the following four-dimensional fractional-order commensurate system: where  ∈]0, 2[,  ∈ IR 4 , and suppose that  is an equilibrium point of this system.In the integer case ( = 1), the stability of  is related to the sign of Re(  ),  = 1, 2, 3, 4, where   are the eigenvalues of the Jacobian matrix /|  .If Re(  ) < 0 for all  = 1, 2, 3, 4, then  is locally asymptotically stable.If there exists an  for which Re(  ) > 0, then  is unstable.

Numerical Results
For numerical calculation of fractional-order derivatives, methods defined with (8) to (10) are usually used.The Grünwald-Letnikov (GL) method [14] is given in the following equation: where [⋅] indicates the integer part.The Riemann-Liouville (RL) definition follows as where Γ(⋅) is the gamma function.The Caputo definition of fractional derivatives can also be recalled as Based on the fact that for a wide class of functions the three definitions-GL (8), RL (9), and Caputo's (10)are equivalent if () = 0, we can then use relation (11) derived from the GL definition (8).The new relation for the explicit numerical approximation of th derivative at the points ℎ ( = 1, 2, . ..) has the following form: where   is the "memory length, "   = ℎ, with ℎ the time step of calculation and  () ( = 0, 1, . . ., ) the binomial coefficients.For their calculation we can use for instance the following expression: The binomial coefficients  () ( = 0, 1, . . ., ) can also be expressed using a factorial.The gamma function Γ() = ( − 1)! can allow the generalization of the binomial coefficient to noninteger argument.Thus, relation (12) can be rewritten as follows: When  < 1.150, the equilibrium point  is a locally asymptotically stable focus; the neighbors trajectories converge to .This is supported by the negative sign of the largest Lyapunov exponents.For 1.150 <  < 1.635, system (1) undergoes a Hopf bifurcation as mentioned above.The fixed point  becomes unstable, and a period-one limit cycle appears.A period-two limit cycle follows for  ≈ 1.635, leading to a new bifurcation at  ≈ 1.763, as the system undergoes a period-four bifurcation.This bifurcations scenario continues through a period-height limit cycle for  ≈ 1.791 up to a critical value of  ≈ 1.820 corresponding to the appearance of a chaotic attractor.This chaotic behavior is confirmed by the existence of positive largest Lyapunov exponents.Figure 3 depicts the phase portraits presenting routes to chaos according to the abovementioned parameter values.

Bifurcation and Chaos versus the Fractional Order 𝑞.
The fractional order  is taken as control parameter, while  is fixed at  = 1.9.The critical Hopf bifurcation value is localized at  * ≈ 0.8557, using the previously proposed conditions.The resulting bifurcation diagram (Figure 4(a)) for the second variable of the set of ( 1) is plotted as a function of the fractional order  and corresponding largest Lyapunov exponents in Figure 4(b).When  < 0.8557, the equilibrium point  is a locally asymptotically stable focus confirmed by the negative sign of the largest Lyapunov exponents; the neighbors trajectories converge to this origin.For  = 0.8557, system (1) undergoes a Hopf bifurcation as mentioned above.The fixed point  becomes unstable, and a period-1 limit cycle appears for 0.8557 <  < 0.9307.As the fractional order parameter nears the value  ≈ 0.9307, a new bifurcation occurs for period-2 limit cycle.This is followed by a period-4 limit cycle at  ≈ 0.9454.This bifurcation scenario continues up to a critical value  ≈ 0.953 where a chaotic attractor appears, sustained by the existence of positive largest Lyapunov exponents.For a periodic steady state, all spikes in the power spectrum are harmonically related to the fundamental, whereas a broadband noise like power spectrum is associated with a chaotic steady state.The periodicity of the attractor (i.e., total number of frequencies in a wave) is deduced by counting the number of spikes located at the left-hand side of the highest spike (the latter is included).Indeed, we have obtained the complete scenarios to chaos presented in Figure 5. Specifically, the following scenario was observed when monitoring the control parameter: fixed point behavior → period-1 → period-2 → period-4 → chaos.

Finite-Time Hybrid Projective Synchronization of Two Fractional-Order Two-Stage Colpitts Oscillators
4.1.Analytic Results.This section is devoted to the finite-time hybrid projective synchronization of the drive and response commensurate fractional order of a two-stage Colpitts system using a robust fractional nonsingular terminal sliding mode controller, for  = 0.96.The drive system is defined as follows: The drive system ( 14) can be written as well as Accordingly, the response system takes the following form: =   (,   ) + Δ  (,   ) +   +   ,  = 1, . . ., 4, (16) where   ∈ IR 4 represents the nonlinear controllers and Δ  (,   ) and   ∈ IR 4 represent unknown model uncertainty and external disturbances of the system.By subtracting (15) from ( 16) and setting the following set of equations defining the errors is obtained: Theorem 3 (see [27]).Let  = 0 be an equilibrium point for the nonautonomous fractional-order system    = (, ), where (, ) satisfies the Lipschitz condition with Lipschitz constant  > 0 and  ∈ (0 1).Assume that there exists a Lyapunov function (, ) satisfying where  1 ,  2 ,  3 , and  are positive constants.Then the equilibrium point of the system    = (, ) is Mittag-Leffler stable.
where   are the sliding surface parameters to be introduced later and   are the scaling factor to content in synchronization error.The nonsingular terminal sliding surface as defined in (20) present a major advantage on that proposed in [27], namely its dependence on scaling factor   no matter the initial conditions which, allows to improve the synchronization time between the driven system and response system.For the existence of the sliding mode it is necessary and sufficient that   () = 0 and Ṡ  () = 0 [27].Therefore, the dynamics of the proposed nonsingular terminal sliding mode can be obtained as Theorem 6.The system (20) is finite time stable and its trajectories converge to the equilibrium () = 0 in a finitetime,  1 , determined by where  = min(  ) and  = min(  ).
Proof.Consider the candidate following positive definite Lyapunov function: The time derivative of the Lyapunov function along the trajectories of ( 21) is Using Lemma 5, one can obtain Hence, according to Theorem 3, the error system (21) will converge to zero asymptotically.In order to show that the sliding motion occurs in finite time, we can obtain the convergence time as follows.
From inequality (25) we have What leads us to Taking integral of both sides of ( 27) from 0 to  1 and letting (  1 ) = 0, we have Therefore, the state trajectories of the error system A control law which forces the error trajectories to go onto the sliding surface within a finite time and remain on it forever is designed as follows: = 1, . . ., 4, where   are the sliding surface parameters to be introduced later,   are the scaling factor to content in synchronization error, and   and   are positive constants.
Theorem 7. If the error system (18) is controlled with control law (29), then the states of the system will move toward the sliding surface and will approach the sliding surface   () = 0 in a finite time,  2 , given by   () = ( ) , and the parameters of (29) are selected as follows:

Conclusion
In this paper the dynamics and synchronization of a proposed four-dimensional fractional-order two-stage Colpitts oscillator have been investigated using analytical and numerical methods.The analytic method proved the existence of the Hopf bifurcation as well as the beach of the control parameter for which the system is stable.On the basis of fractional Lyapunov stability theory we determined with success the conditions under which the synchronization of two systems is achieved.For numerical simulation we used the Grünwald-Letnikov method, the largest Lyapunov exponents, and the bifurcation diagrams to show the period-doubling bifurcation routes to chaos as well as the Hopf bifurcation.The numerical analysis validates the conditions of Hopf bifurcation.For the finite-time hybrid projective synchronization the numerical investigation validates also the analytic conditions which achieve synchronization.Numerical simulations have been used to show the effectiveness of the proposed synchronization techniques.
Parameter  and the Fractional Order .In this subsection, we consider the parameter values ( 1 ,  2 ,  3 ) = (1.25, 1, 1.75) for the search for the Hopf bifurcation around the equilibrium point .

Figure 1 :
Figure 1: Critical values  * versus the fractional order  * .This curve depicts the couples of values for which the Hopf bifurcation occurs in the system.

) 3 . 1 .
Bifurcation and Chaos versus the Parameter .In this subsection, the dynamical behavior of system (1) is numerically investigated by means of bifurcation diagram and largest Lyapunov exponents, which measure the exponential rates of divergence or convergence of nearby trajectories in phase space.For  taken as control parameter and the following other parameter values, fractional order  = 0.96,  1 = 1.25,  2 = 1.00 and  = 1.175, the critical Hopf bifurcation value is localized at  * = 1.150 (see Figure2(a)) and confirmed by the largest Lyapunov exponents presented in Figure 2(b).

Figure 2 :Figure 3 :Figure 4 :
Figure 2: Bifurcation diagram expressing the (a) dynamics of the system variable  2 and (b) largest Lyapunov exponent, both as a function of , with  = 0.96.
Figure 7  shows us that when   is far from one, the synchronization time turns fast to zero; this decreasing numerical time of synchronization justifies