Fractional calculus can describe physical phenomena and mathematical models more accurately than the classical integer calculus^{[1– 3]} and has many applications in interdisciplinary fields.^{[4– 9]} The applications of fractional calculus to physics, ^[5] engineering, ^[6] signal processing, ^[7] and control processing^[8] have attracted a great deal of attention recently. In comparison with traditional integer-order control, the fractional-order control is more general and becoming more popular because of its flexibility and integrity.^[9] Intensive studies confirmed theoretically and physically that the chaotic and hyperchaotic behaviors exist the nonlinear fractional-order systems.^{[3, 8– 15]} Meanwhile, synchronization of fractional-order chaotic systems also has attracted increasing attention due to its potential applications in secure communication^{[16, 17]} and digital cryptography.^{[18– 20]} A variety of control schemes have been proposed to control and synchronize fractional-order chaos in infinite or finite time, ^{[21, 22]} such as active control, ^{[23, 24]} sliding mode control, ^{[25– 27]} adaptive control, ^[28] passive control, ^[29] and impulsive control.^{[30– 34]} Both fractional-order models and controllers have more degrees of freedom besides memory and hereditary effects.^[8] A consensus is arising that chaotic synchronization of fractional-order systems has more advantages and real objects than that of integer order systems.

Impulsive control is a control paradigm based on impulsive differential equations and has been applied in some areas with inherited impulsive properties, such as neurobiology and genetics, economics, and ecology. In realizations, the control actions are imposed on a control plant at certain discrete instants. It has been found that it is important on synchronizing complex systems via impulsive control, ^{[31– 43]} and is more suitable for digital implementation of secure communication in comparison with other schemes.^[41] Since the impulsive control allows the stabilization of the controlled plant using only small impulsive actions, the overall control cost would be reduced. Therefore, it is more suitable for dealing with the systems that cannot endure continuous strong disturbance.^[35] Heretofore, most impulsive controllers are designed in the framework of Lyapunov asymptotic stability based on the comparisons method developed in Ref. [41]. Different restriction relations are necessary due to different designs and repetitive calculations. For example, one has to compute the maximum eigenvalues of some matrices and then examine whether they meet some assigned inequalities. In this case, previous works would rather check these restriction relations many times and estimate the error vectors of states than utilize the comparison method directly. Furthermore, in order to seek the control gains, the nonlinear functions in the drive system and response system should be immersed in some rigorous mathematical constraints. The estimations of Lipschitz conditions of nonlinear functions and their bounded norms in Euclidean space are two common conditions. However, it is very hard to work out the local Lipschitz constant for every nonlinear equation in practice. In addition, the initial difference between two systems is huge but the final controlled errors cannot be constant although small. The bounded norms of them are always evaluated with the boundaries of orbits in the phase space. As a result, these loose approaches always trigger improper outcomes. For instance, the coupling strengths may be very big or unequal, which in experience may bring out unexpected dynamical behavior, such as desynchronization bursts. Such a desynchronization burst is usually overlooked or difficult to be found because it always starts after a long synchronization moment. The combination of adaptive and impulsive controls partly makes up for the aforementioned imperfections and is easy to achieve synchronization. Alternatively, the variable feedback strength can be automatically adapted to completely synchronize two chaotic systems. In an augment system, the adaption terms are linearly relying on the errors in variables but vanish once the synchronization is achieved. Although this combination scheme is very analytical and has been extended extensively, ^{[33, 44– 49]} the feedback strength is never considered to be updated with an adaptive law. This is because both the transient and the final feedback strengths at every impulsive constant are not sure of falling into the admissible region. Furthermore as far as we know, the adaptive– impulsive synchronization on fractional-order chaotic systems is rarely considered to this day.

In this paper, a novel adaptive– impulse synchronization scheme is proposed to achieve synchronization of fractional-order chaotic systems. The nonlinear equations in the chaotic systems considered here suffer unknown local Lipschitz constants. In order to guarantee the convergence is no less than an expected exponential rate, a combined feedback strength design is created. At the impulsive instants, rational constant feedback strengths neutralize the updated transient and the ultimate feedback strengths such that the symmetric axes can shift freely. Whereby, the combined feedback strength at every impulsive instant is surely falling in the admissible region. Both impulse intervals and the feedback strength are derived from one criterion that is also associated with unknown local Lipschitz constants, and the rate of convergence we expected. The criterion is completely adaptive to implement in locating the admissible region of control strengths to ensure the occurrence of synchronization. All of the unknown local Lipschitz constants will be identified dynamically with adaptive laws in the process of achieving synchronization, rather than estimated from the attractors bounds in the initial state. Finally, an autonomous hyperchaotic system and a non-autonomous fractional-order chaotic system are employed to demonstrate the effectiveness and the feasibility of the proposed scheme.

Consider a fractional-order chaotic system with initial conditions,

where, k = 0, 1, … , m − 1. x(t) ∈ ℝ ⁿ, n ∈ ℕ is the state vector of the system. Assume that the nonlinear function f(t, x(t)) is bounded in the phase space Ω , Ω ⊆ ℝ ⁿ × ℝ ⁺ . f(t, x(t)): ℝ ⁿ × ℝ ⁺ → ℝ ⁿ is supposed to be -Lipschitz, that is, there is a Lipschitz constant λ _i > 0, such that | f_i(t, x₁) − f_i(t, x₂)| ≤ λ _i| x₁ − x₂| , i = 1, 2, … , n, for every pair of points (t, x₁) and (t, x₂) in Ω , where

Remark 1 All the chaotic systems (both in integer order and their fractional-order corresponding parts) can be represented in the form of system (3), such as Chua circuits, Lorenz system, Chen system, Lü system, the unified system and Duffing oscillators.^[26] It is hard to work out the accurate Lipschitz constant for every function in a nonlinear system although its attractors are bounded in Ω . Previous researches evaluated the local Lipschitz constants in Ω , which is also an imprecise evaluation from the boundaries of attractors. As stated before, such an evaluation in the initial state is so loose that it is unfit for a dynamical process. To avoid such an imperfection, we assume that all the local Lipschitz constants are unknown a priori. The unknown Lipschitz constants λ ̃ _i(t), i = 1, 2, … , n, would be identified dynamically in the course of synchronization, lim_{t→ ∞} λ ̃ _i(t) = λ _i, i = 1, 2, … , n, and

In order to achieve synchronization, system (3) is taken as the drive system, while the impulsive-controlled response system is represented as follows,

where y(t) ∈ ℝ ⁿ, n ∈ ℕ is the state vector of the response system. The time sequence {t_ı } of impulsive instants is set as 0 ≤ t₀ < t₁ < … < t_ı < … , and t_ı → ∞ as ı → ∞ . The impulse interval here is not equidistant, is the right limit of y(t) at t = t_ı , while is the left limit and is assumed to be left continuous, i.e.,

Define the error state as e(t) = y(t) − x(t). Subtracting system (3) from system (4), we obtain the error dynamical system between them as follows:

The task is to design an adaptive– impulsive controller to achieve the asymptotic synchronization of the drive system (3) and the response system (4), that is, lim_{t→ ∞} ‖ e(t)‖ = 0. As a result, U(ı ) = Δ e(t_ı ) in (4) is the sole controller at , ı = 1, 2, … . The i-th control action is described as follows,

where the feedback strength k(t) + k̃ is presented in a combined form, and k̃ is a positive constant concomitant of k(t), which is affirmed that ❘ k(t)❘ > k̃ . More details on k̃ will be given below. The feedback strength k(t) is negative and follows an adaptive law, which is described as

where γ ∈ {0}∪ ℝ ⁺ and γ is only configured with an appropriate positive constant at , ı = 1, 2, … . Hereby both k(t) and its concomitant k̃ are only actuated at , ı = 1, 2, … . In this case, k(t) and k̃ can be rewritten as k_ı (t) and k̃ _ı respectively at the impulse instants.

However, to the best of our knowledge, there is no efficient scheme to deal with the stability of the fractional-order impulsive differential equation (5a).^{[32, 33]} In order to settle such a problem, equation (4a) will be converted to a corresponding integer-order one by introducing a new function, μ (x(t)) = ẋ (t) − D^α x(t) ^[33]. Both x(t) and D^α x(t) can be taken from the simulation of system (3). As a result, systems (4) and (5) become

and,

Theorem 1 Suppose that the adaptive feedback strength k_ı (t) is bounded with the lower bound − e^{− ρ /2} − k̃ _ı − 1 and the upper bound e^{− ρ /2} − k̃ _ı − 1, where − ρ (ρ > 0) is the least convergence exponential rate we expected. Meanwhile, the adaptive law for estimating the Lipschitz constant λ ̃ _i is given as

where η _i ∈ ℝ ⁺ . The error state e(t) in Eq. (9a) will be exponentially stable at a rate no less than e^{− ρ} if the adaptive controller (6) is actuated at , ı = 1, 2, … .

Proof The candidate Lyapunov function is

Clearly, λ ̃ _i(t) is an increasing function such that one has λ ̃ _i(t_{ı − 1}) ≤ λ ̃ _i(t) ≤ λ ̃ _i(t_ı ). In the impulsive interval t ∈ (t_{ı − 1}, t_ı ], the combined feedback strength k_ı (t) + k̃ _ı only is adapted at , ı = 1, 2, … . Thus, the Dini derivative of V(t, e(t)) along the trajectories of the error system (9a) is,

Therefore in the interval of t ∈ (t_{ı − 1}, t_ı ], ı = 1, 2, … , it is represented as

Due to the instant control action (6) at , ı = 1, 2, … , one has

From expressions (10) and (12), we can deduce the following formulae. When ı = 1,

Next, when ı = 2,

By repeating the same method, the generalized form is,

where ı = 1, 2, … .

To guarantee the convergence is exponential, we let

As a consequence,

It can be concluded here that , e(t)) converges to 0 as ı → ∞ . In the meantime, the adaptive law (10) is also settled exponentially to a modest Lipschitz constant λ _i.

Recalling expression (13), we have

as t → ∞ .

As shown in expression (19), the convergence rate of error dynamical system (9a) is no less than e^{− ρ ı} . For any given in the interval t ∈ (t_{ı − 1}, t_ı ], ı = 1, 2, … , it can be derived as

The impulsive interval in inequality (21) can be as large as possible, but k_ı (t) approaches to − (k̃ _ı + 1) in the meantime. However, such an ideal strength, k_ı (t) = − (k̃ _ı + 1) in the axis of symmetry, should be avoided because the actual control only activates at the impulsive constants t_ı , ı = 1, 2, … . In this case, the most rational interval for the combined feedback k_ı (t) + k̃ _ı is

with assuming that t_ı − t_{ı − 1} ≪ ∞ .

On the other hand, the impulsive interval is determined by the adapted control strength and the identified Lyapunov constant cooperatively at the current impulsive instant. The relationship between them is described as

The impulsive interval τ _ı is positive, which implies that

Hereby, the infimum and the supremum of k_ı (t) + k̃ _ı are − 2 and 0 respectively. In the process of synchronization, the adaptive law for feedback strength k_ı (t) will be updated to a new integration

at an impulsive action. Whereby the concomitant constant k̃ _ı can be configured with a positive number in a band depicted in inequality (24), whilst it satisfies k_ı (t) + k̃ _ı ≠ − 1, and so on. Unifying it with the integration of Eq. (10) and the convergence rate we expected, the next impulsive interval τ _ı is determined. That is to say the next impulsive instant is fixed. Obviously, the impulsive actions terminate if τ _ı < h, where h is the time-step we used for the simulations in the impulsive intervals.

Remark 2 It should be noted here that the axis of symmetry is not controlled by any unpredictable coefficients. It can shift freely to any positive constants according to the transient and ultimate adaptive feedback strengths. One important thing is that regardless of transient time, the costs on the synchronization of two identical chaotic systems are very low. This is an important reason why we confined the combined feedback strength k_ı (t) + k̃ _ı in the admissible region (− 2, − 1) ∪ (− 1, 0) randomly although the endpoints cannot present at any moment. With an expected convergence at exponential rate ρ , the choice on the coefficient in the adaptive law (7) plays an important role because it determines the estimation of k_ı (t), while guaranteeing the impulsive interval 0 < τ _ı ≪ ∞ . Such an almost insolvable problem has to be handled by a posteriori means.^[55]

In this paper, a novel adaptive– impulsive scheme is proposed for synchronizing fractional-order chaotic systems. The nonlinear functions considered here are supposed to satisfy the local Lipschitz conditions without needing to be known a priori. The combination of adaptive control and impulsive control offers a control strategy gathering the advantages of both. The combined feedback strength design is created to shift the symmetric axis freely according to the updated transient feedback strength. The synchronization converges at an expected exponential rate. All of the unknown Lipschitz constants are also updated to their rational values once the synchronization occurs.