Generalized convergence analysis of the fractional order systems

Ahmad Ruzitalab; Mohammad Hadi Farahi; Gholamhossien Erjaee

doi:10.1515/phys-2018-0055

Article Open Access

Generalized convergence analysis of the fractional order systems

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Published/Copyright: July 17, 2018

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From the journal Open Physics Volume 16 Issue 1

Abstract

The aim of the present work is to generalize the contraction theory for the analysis of the convergence of fractional order systems for both continuous-time and discrete-time systems. Contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another. The result of this study is a generalization of the Lyapunov matrix equation and linear eigenvalue analysis. The proposed approach gives a necessary and sufficient condition for exponential and global convergence of nonlinear fractional order systems. The examples elucidate that the theory is very straightforward and exact.

Keywords: Contraction theory; Stability; Convergence; Fractional order systems

PACS: 02.30.Xx; 02.30.Yy; 05.45.Xt; 47.27.ed

1 Introduction

Stability analysis is well-known as a base in control theory and many methods have been proposed to check this property. One of the major tools among these methods is Lyapunov theory. In this regard, it is confirmed that determining a Lyapunov function is fundamental in the stability analysis and control design of nonlinear systems (see for example [1,2]). In the last two decades, Lohmiller and Slotine created a new technique from fluid mechanics and differential geometry known as contraction theory, for evaluation of stability [3]. Revisiting the contraction concept could result in introducing the suitable Riemann metrics [3,4]. The idea behind this theory is that stability can be assessed through checking the nearby trajectories’ convergence, rather than by finding some Lyapunov functions, or by global state transformation using feedback linearization (see [5]). Lohmiller and Slotine used the contraction theory for analyzing the stability of and designing a control system for nonlinear chemical processes [6]. Wang and Slotine used the contraction theory to attain exact and global results for studying the synchronization of two or more coupled systems [7]. Jouffroy and Fossen, using contraction theory, introduced a methodology for analysis of differential nonlinear stability [8]. Pham and colleagues derived a stochastic version of the theory of nonlinear contraction, which provides a bound for the mean square distance between any two trajectories of a stochastically contracting system [9]. Rayguru in his paper designed a novel disturbance observer based dynamic surface controller using contraction framework [10]. The novelty of the proposed approach in the paper of Blocher et al. for the learning of robot point-to-point motions, is that they guarantee the stability of a learned dynamical system via Contraction theory [11].

Fractional derivatives and integrals have been gaining more and more interest of scientists due to their extensive applications in different directions of science, social science, engineering and finance [12, 13, 14, 15, 16, 17, 18, 19, 20]. In this context, Rostamy et al. in order to solve multi-term order fractional differential equations, utilized new matrices based on the Bernstein Polynomials basis, to reduce the equations to a system of algebraic equations [21]. Kumar et al. in [22] studied the fractional model of Lienard’s equation by constituting a numerical algorithm based on the fractional homotopy analysis transform method. They also, discussed the uniqueness and convergence analysis of the solution of Lienard’s equation. Kumar et al. presented a time-fractional modified Kawahara equation through a fractional derivative with exponential kernel [23]. In another study, they presented a new non-integer model for convective straight fins with temperature-dependent thermal conductivity associated with Caputo-Fabrizio fractional derivative [24]. They also, presented a new numerical scheme based on a combination of a q-homotopy analysis approach and a Laplace transform approach to examine the fractional order Fitzhugh-Nagumo equation which describes the transmission of nerve impulses [25]. Inc et al. reduced the time fractional Cahn-Allen and time fractional Klein-Gordon to respective nonlinear ordinary differential equations of fractional order. They solved the reduced fractional ODEs using an explicit power series method and investigated the convergence analysis for the obtained explicit solutions [26]. Kumar et al. presented a new fractional extension of a regularized long-wave equation; this is a very important mathematical model in physical sciences, which unfolds the nature of shallow water waves and ion acoustic plasma waves [27].

As mentioned above, Contraction Theory and Fractional Order Systems (FOSs) are two subjects of much interest in last two decades. In this context, applying the contraction theory, Kamal et al. have aimed to design a universally exponentially stable controller for fractional-order systems [28]. Bandyopadhyay and Kamal reconsidered the theory of contraction by substituting the integer order variation of the system state by the fractional-order variation [29]. The major benefit of that methodology is its applicability in the evaluation of the stability of non-differentiable systems and FOSs and the design of a fractional order controller [29]. A sufficient condition is acquired by revisiting contraction theory in [29] for the system’s exponential convergence.

In this article, a generalization of the FOSs convergence analysis is presented. The contraction theory in [29] is extended through the application of a general definition of differential length. The result is a fractional generalization for the Lyapunov matrix equation and linear eigenvalue analysis, providing a necessary and sufficient condition for the exponential convergence in a system.

This article is presented in the following way: In Sect. 2, the contraction analysis of FOSs by applying a fractional order infinitesimal variation is described. The generalization of convergence analysis of FOSs is presented in Sect. 3. In Sect. 4, the contraction method for discrete-time FOSs is considered. Numerical examples are given to illustrate the theory.

2 Contraction analysis of FOSs by fractional order infinitesimal variation

Stability analysis using differential approximation, is well-known as a base in control theory. The advantage of the contraction method is that, it yields global and exact results of stability analysis for nonlinear systems. This section presents a summary of the basic results of Bandyopadhyay and Kamal [29], to which reference can be made for more details.

Consider an autonomous dynamical system

x˙=fxtx0=x0(1)

where f is a nonlinear vector field and x(t) is an n-dimensional state vector. The dynamic system (1) can be stated in the fractional derivative form

x˙=t0RLDt1x=f(x(t))

where t0RLDt1 indicates the Riemann-Liouville (R-L) fractional derivative, which for the order α ∈ ℝ⁺, m – 1 < α < m, m ∈ ℕ, is defined as [29]:

t0RLDtαft=DmD−m−αft=dmdtm1Γm−α∫0tfτt−τα−m+1dτ.(2)

For simplicity, we omit the left superscript RL and subscript t₀ and assume Dtα to be the Riemann-Liouville α-order fractional derivative operator that exists and is continuous. It is clear that for the system (1), δẋ = δf(x(t)) = ∂f∂xδx, the fractional virtual dynamics is represented by

δαDt1x=δαfxt=DxαfxtDxαx−1δαx,(3)

where δ^αx and δ^αẋ are termed as virtual displacement and virtual velocity respectively [29]. Now, consider an FOS:

Dtαxt=fxt.(4)

Taking Dt1−α in both sides of the dynamical system (4), one can write (see [29] Page 195, [30] Page 29):

Dt1xt=x˙t=Dt1−αfxt,

now by (3) we have

δαDt1x=DxαDt1−αfxtDxαx−1δαx.(5)

Theorem 1

If the matrixDxαDt1−αfxtDxαx−1is uniformly negative definite (UND), all the solution trajectories of the system (4) converge exponentially to a single trajectory, discounting the initial conditions.

Proof

The time derivative of squared distance between the two neighboring trajectories will be

Dt1δαxTδαx=2δαxTDxαDt1−αfxt(Dxαx)−1δαx≤2λm(x,t)δαxTδαx(6)

where λ_m(x,t) is the largest eigenvalue of the symmetric part of

DxαDt1−αfxtDxαx−1.

Therefore, from (6) one can find

δαx≤δαx0e∫0tλmx,tdt.(7)

If λ _m(x,t) is strictly uniformly negative then any infinitesimal length ∥δ^α x∥ converges exponentially to zero as t tends to infinity. Thus all the solution trajectories of the system (4) converge exponentially to a single trajectory, irrespective of the initial conditions. □

Note that by a matrix A being UND, we mean that

∃β>0suchthat12A+AT≤−βI<0,

by convention, the above can be written as A≤ – βI < 0.

Example 1

Consider the following FOS:

Dtαx=u.(8)

In order to design a controller u that is able to stabilize the system (8), the convergence condition of Theorem 1, must be stablished. Therefore,

DxαDt1−α(u)Dxαx−1

must be UND in the whole state space. There are many possible values of u which satisfies the above convergence condition. One can choose

u=Dtα−1Dx−α(−kx1−α),

where k > 0. Substituting the value of u in the convergence condition, one can get [29, 30]:

DxαDt1−α(u)Dxαx−1=DxαDt1−α(Dtα−1Dx−α(−kx1−α))Dxαx−1=−kx1−αx1−αΓ(2−α)=−kΓ(2−α)<0

Therefore, the proposed controller stabilizes the system (8).

This example shows the advantage of the contraction theory for analyzing the stability of fractional order systems and designing the fractional order controller.

Remark 1

Consider the linear time-invariant (LTI) FOS:

Dtαx=Ax.(9)

ApplyingDt1−αto both sides, one can write [29, 30]:

x˙=Dt1−αAx.

Consider two neighboring trajectories of the above equation and the virtual displacement δ x between them. This leads to the following:

δx˙=∂Dt1−αAx∂xδx.

The rate of change of the squared distance(δ x)^Tδ x between these two trajectories is given by

d(δx)Tδxdt=2(δx)Tδx˙=2(δx)T∂Dt1−αAx∂xδx.

Consider the following R-L derivative:

Dt1−αAx=1Γ(1−(1−α))ddt∫0tAx(τ)(t−τ)1−αdτ,

thus we have

∂Dt1−αAx∂x=1Γ(α)ddt∫0tA(t−τ)1−αdτ=Atα−1Γ(α).(10)

So the system(9)is stable ifAtα−1Γ(α)is UND. Since t^α–1and Γ (α) are positive numbers, thus, one concludes that(9)is stable if A is UND.

We can see that if α = 1, thenAtα−1Γ(α) A, which is the Jacobian for an integer order LTI system.

Remark 2

An application of the contraction method is its use in studying the synchronization of a given nonlinear system coupled with a contracting virtual system in order to conclude the convergence and stability of the original system rather than through finding a Lyapunov function.

Example 2

Consider the following nonlinear FOS:

Dtαx1=−x1+x1x2Dtαx2=−x12−x2

or, in the matrix equation form:

Dtαx1Dtαx2=−1x1−x1−1x1x2.

One can consider the following virtual y-system:

Dtαy1Dtαy2=−1x1−x1−1y1y2.

Since, the matrix

−1x1−x1−1

is UND, according to Remark 1, the virtual system is contracting with two particular solutions, namely

y1y2=00andy1y2=x1x2.

Since the virtual system is contracting, all the trajectories and especially two particular solutions converge to each other. Therefore, the arbitrary x1x2 tends to 00 and the original x-system is stable.

One can see that this stability analysis is intuitive and much simpler than finding lyapunov function.

Definition 1

Given the FOS Dtαx(t) = f(x(t)), a region Ω ⊆ ℝⁿ of the state space is entitled as a contraction (semi-contraction) region, if the matrix DxαDt1−αfxtDxαx−1 is UND (uniformly negative semi-definite) in that region.

3 Generalization of the convergence analysis of FOS

An extension of Theorem 1 can be deduced by the use of a broad description of differential length. The result can be considered as a generalization for the fractional type of Lyapunov matrix equation and linear eigenvalue analysis. Furthermore, it gives a necessary and sufficient condition for exponential convergence.

3.1 General definition of length

If the coordinate system of x is transformed to the coordinate system of z, and the vectors δ^α x and δ^αz are respectively virtual displacements between two neighboring trajectories of x and z coordinate systems, then δ^αz, using the coordinate transformation can be expressed as

δαz=Θδαx(t)(11)

where Θ(x, t) is an invertible square matrix. Therefore, a generalization of the squared length is as follows:

(δαz)Tδαz=(δαx(t))TM(x,t)δαx(t)(12)

where M(x,t) = Θ^TΘ should be a symmetric, uniformly positive definite (UPD) and continuously differentiable metric (in other words, a Riemannian metric). If these conditions hold for M, and δ^αz converge exponentially to 0, then δ^αx converges exponentially to 0.

The time derivatives of the left and right hand sides of equation (12) lead to a generalization of the linear eigenvalue analysis (Section 3.1.1), and a generalized fractional form of the Lyapunov equation (Section 3.1.2), respectively.

3.1.1 Generalized eigenvalue analysis

Theorem 2

For the systemDtαxt = f(x(t)), if the matrix

F=Θ˙+ΘDxαDt1−αfxDxαx−1Θ−1

is UND, where Θ is defined in(11), then all system trajectories converge globally to a single trajectory exponentially regardless of the initial conditions, and the rate of global exponential convergence is equal to the largest eigenvalues of the symmetric part of F.

Proof

Using the property of variation that Dt1δαx=δαDt1x, and considering (5), by getting derivatives from both sides of δ^αz = Θ δ^αx, we have

Dt1δαz=Θ˙δαx+ΘDt1δαx=Θ˙δαx+Θδαx˙=Θ˙δαx+ΘDxαDt1−αfxDxαx−1δαx=Θ˙+ΘDxαDt1−αfxDxαx−1δαx=Θ˙+ΘDxαDt1−αfxDxαx−1Θ−1δαz=Fδαz(13)

where

F=Θ˙+ΘDxαDt1−αfxDxαx−1Θ−1.(14)

Hence, by (13) the rate of change in squared length, which quantifies the contraction rate of the volume, is represented as

Dt1(δαz)Tδαz=2δαzTDt1δαz=2δαzTFδαz(15)

So, as in the proof of Theorem 1, δ^αz and thus δ^αx, converge to 0 in regions where F is UND. □

In (14), F is called the generalized Jacobian for FOS (4).

3.1.2 Metric analysis

Theorem 3

The systemDtαxt = f(x(t)) is contracting if ∃ β_M > 0, such that

DxαDt1−αfxDxαx−1TM+M˙+MDxαDt1−αfxDxαx−1≤−βMM.(16)

Proof

Remember that by (12), (δ^αz) ^Tδ^αz = (δ^αx(t))^T M(x,t)δ^αx(t). The rate of change of the right hand side by using (5) is:

ddt(δαxtTM(x,t)δαx(t))=δαx(t)TDxαDt1−αfxDxαx−1TM+M˙+MDxαDt1−αfxDxαx−1δαx(t)=δαxtTΦδαxt,(17)

where

Φ=DxαDt1−αfxDxαx−1TM+M˙+MDxαDt1−αfxDxαx−1.(18)

So that exponential convergence to a single trajectory can be concluded in an area identified by Φ ≤ –β_MM (where β_M is a positive constant). □

From Theorems 2 and 3, one can conclude that the region identified by Φ ≤ –β_M M, is the region for which F in (14) is UND.

3.2 Generalized contraction analysis

The above subsection results in the subsequent generalized definition, superseding Definition 1.

Definition 2

Given the FOS Dtαx(t) = f(x(t)), a region of the state space is called a contraction (semi-contraction) region in accordance with a UPD metric M(x,t) = Θ^TΘ, if equivalently F or Φ are UND (semi-definite) in that region.

The result of generalized convergence can be expressed, as follows:

Corollary 1

Consider the FOSDtαx(t) = f(x(t)) and a ball of constant radius with respect to the metric M(x,t), positioned at a given trajectory and confined at all times in a contraction region. Any trajectory, which starts in that ball remains in that ball and converges exponentially to the given trajectory. In addition, if the whole state space is a contraction region, global exponential convergence to the given trajectory is guaranteed.

Proof

Immediate from Theorems 2 and 3 and Definition 2. □

The converse of Theorem 2 is also valid.

Theorem 4

Any exponentially convergent FOS is contracting in respect of an appropriate metric.

Proof

Suppose that the system (4) is exponentially convergant, which implies that there exist β > 0 and k ≥ 1, such that along any system trajectory x(t) and for any t ≥ 0,

(δαx)Tδαx≤k(δαx0)Tδαx0e−βt.

Suppose the metric M(x(t),t) satisfies the following Lyapunov form ordinary differential equation:

M˙=−βMM−DxαDt1−αfxDxαx−1TM−MDxαDt1−αfxDxαx−1Mt=0=kI,(19)

by substituting (19) in (17), and solving the differential equation (17), it is easy to find that

k(δαx0)Tδαx0e−βt=δαxTMδαx

Now by the assumption that (δ^αx)^Tδ^α x ≤ k(δ^αx₀)^Tδ^αx₀e^{–β t}, we find that

(δαx)Tδαx≤δαxTMδαx(20)

Since (20) holds for any δ^αx, this concludes that M is UPD. Therefore, with respect to an appropriate metric,any exponentially convergent system is contracting. □

Theorems 2 and 4 correspond to necessary and sufficient conditions for the exponential convergence of FOS.

Example 3

Consider the autonomous differential equation

Dtαx=−ksign(x)=f(x)(21)

where k > 0 and sign (x) is defined as

signx=+1ifx>00ifx=0−1ifx<0(22)

Consider the differential coordinate transformation δ^αz = Θ δ^αx (where Θ is constant). To check the stability of system (21), we calculate the value of F in (14) as:

F=Θ˙+ΘDxαDt1−αfxDxαx−1Θ−1=Θ˙+ΘDxαDt1−α−ksign(x)Dxαx−1Θ−1=Θ˙+Θ−ksign(x)tα−1x−α(Γ(2−α))(Γ(1−α))(Γ(α))x1−αΘ−1=Θ−kxxtα−1(Γ(2−α))(Γ(1−α))(Γ(α))xΘ−1=Θ−ktα−1(Γ(2−α))(Γ(1−α))(Γ(α))xΘ−1<0

since −ktα−1(Γ(2−α))(Γ(1−α))(Γ(α))x <0.Therefore, any two trajectories of the FOS (21) converge to each other.

As we can see in this Example, one of the main advantages of using the fractional order variation in the contraction theory is that it also works for analyzing the stability of non-differentiable systems.

4 Contraction method for discrete-time FOS

Consider the following integer order discrete-time system:

xk+1=f(x(k))(23)

where f(⋅) is a smooth nonlinear vector function. The discrete-time fractional-order system (DFOS) can be represented as follows, which for more details the reader is referred to [31]:

xk+1=fxk+α−1xk+∑p=1kCpx(k−p)(24)

where C_p = (–1)^pαp+1. Since all the absolute values of the coefficients decrease as the iteration p increases for each fractional-order α [31], in order to decrease the calculations and the needed space for each computing, we can use a finite truncation to approximate a DFOS as follows:

xk+1=fxk+α−1xk+∑p=1LCpx(k−p)(25)

where L denotes the truncation length and it is selected appropriately according to a practical problem.

Theorem 5

Exponential convergence of system(25)is guaranteed if

∂gk∂xkT∂gk∂xk−I(26)

be UND, where

gk=fxk+α−1xk+∑p=1LCpx(k−p).(27)

Proof

The associated virtual dynamics of (25) is

δxk+1=∂gk∂xkδxk(28)

so that the virtual length dynamics is

δxTk+1δxk+1=δxTk∂gk∂xkT∂gk∂xkδxk,(29)

therefore, the rate of change of the left hand side is

δxTk+1δxk+1−δxTkδxk=δxTk∂gk∂xkT∂gk∂xkδxk−δxTkδxk=δxTk∂gk∂xkT∂gk∂xk−Iδxk

thus, trajectories will exponentially converge to a single trajectory, if

∂gk∂xkT∂gk∂xk−I(30)

be UND. □

Corollary 2

For the linear DFOS

xk+1=Axk+α−1xk+∑p=1LCpx(k−p)(31)

trajectories will exponentially converge to a single trajectory, if

BTB−I

be UND, where

B=A+α−1I.

Proof

We have

gk=Axk+α−1xk+∑p=1LCpx(k−p)=(A+α−1I)xk+∑p=1LCpx(k−p)(32)

therefore,

∂gk∂xk=A+α−1I,

which concludes the proof. □

In the discrete-time version, using the generalized virtual displacement

δzk=Θkxk,kδxk(33)

and by relation (28) we have:

δzTk+1δzk+1=δxTk∂gk∂xkTΘk+1TΘk+1∂gk∂xkδxk=δzTkFkTFkδzk(34)

where

Fk=Θk+1∂gk∂xkΘk−1(35)

is the fractional order discrete-time generalized Jacobian. Now, we can provide the following generalized definition of a contraction region for DFOS.

Definition 3

Given the DFOS: x(k+1) =g _k(x(k)), with g_k given in (27), a region of the state space is recognized as a contraction region in respect of a UPD metric M_k(x(k),k) = ΘkTΘk, if in that region

∃β>0,FkTFk−I≤−βI<0

where Fk=Θk+1∂gk∂xkΘk−1.

Remark 3

Corollary 1 can be immediately changed to the discrete version.

Example 4

Consider the fractional order discrete-time Logistic system [31]:

xk+1=μxk1−xk +α−1xk+∑p=1LCpxk−p.(36)

By putting x(k+1) = x(k), the fixed point (equilibrium point) will be

x∗=2−μ−α±(μ+α−2)2+4μc−2μ

where

c=∑p=1LCpx(k−p)

Let

gk=μx(k)(1−x(k))+α−1xk+∑p=1LCpx(k−p)

therefore,

∂gk∂xk=μ+α−1−2μxk

and the fractional order discrete-time Logistic system will be convergent if

∂gk∂xkT∂gk∂xk−I<0.

Numerically, choosing α = 0.4, μ = 2 and L = 50, the fixed point will be x^* = 0.4228 and

∂gk∂xk2−1=−0.9151<0

therefore, the system will converge to x = 0.4228 for each arbitrary initial point (see Figure 1).

This example, shows the simplicity of contraction theory for analyzing the convergence of DFOSs.

$Figure 1 Convergence of fractional order discrete-time Logistic system for three arbitrary initial points 0.2, 0.5 and 0.7.$

Figure 1

Convergence of fractional order discrete-time Logistic system for three arbitrary initial points 0.2, 0.5 and 0.7.

Conclusion

In this article, applying a broad description of differential length, we have generalized the fractional order contraction theory. The proposed approach is useful for analyzing the stability of non-differentiable systems and also FOSs; furthermore, it leads to necessary and sufficient conditions for exponential convergence of an FOS. The theory was also stated for the case of discrete-time FOS. Numerical examples illustrate the proposed method.

Competing interests: The authors declare that they have no competing interests.
Authors’ contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

References

[1] Slotine J.J.E., Li W., Applied nonlinear control, Vol. 199, No. 1, Englewood Cliffs, NJ: Prentice hall, 1991Search in Google Scholar

[2] Khalil H.K., Noninear systems, Prentice-Hall, New Jersey, 2(5), 5-1, 1996Search in Google Scholar

[3] Lohmiller W., Slotine J.J., On contraction analysis for nonlinear systems, Automatica, 1998, 34(6), 683-696.10.1016/S0005-1098(98)00019-3Search in Google Scholar

[4] Aylward E.M., Parrilo P.A., Slotine J.J., Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming, Automatica, 2008, 44(8), 2163-2170.10.1016/j.automatica.2007.12.012Search in Google Scholar

[5] Angeli D.A., Lyapunov approach to incremental stability properties, IEEE T. Automat. Contr., 2002, 47(3), 410-421.10.1109/9.989067Search in Google Scholar

[6] Lohmiller W., Slotine J.J., Nonlinear process control using contraction theory, AIChE J., 2000, 46(3), 588-596.10.1002/aic.690460317Search in Google Scholar

[7] Wang W., Slotine J.J., On partial contraction analysis for coupled nonlinear oscillators, Biol. Cybern., 2005, 92(1), 38-53.10.1007/s00422-004-0527-xSearch in Google Scholar PubMed

[8] Jouffroy J., Fossen T.I., A tutorial on incremental stability analysis using contraction theory, Modeling, Identification and control, 2010, 31(3), 93-126.10.4173/mic.2010.3.2Search in Google Scholar

[9] Pham Q.C., Tabareau N., Slotine J.J., A contraction theory approach to stochastic incremental stability, IEEE T. Automat. Contr., 2009, 54(4), 816-820.10.1109/TAC.2008.2009619Search in Google Scholar

[10] Rayguru M.M., A Contraction Theory Approach for Analysis of Performance Recovery in Dynamic Surface Control, arXiv preprint arXiv:1511.00120. 2015Search in Google Scholar

[11] Blocher C., Saveriano M., Lee D., Learning stable dynamical systems using contraction theory, In Ubiquitous Robots and Ambient Intelligence (URAI), 14th International Conference on, 2017, 124-129, IEEE.10.1109/URAI.2017.7992901Search in Google Scholar

[12] Caputo M., Fabrizio M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2015, 1, 73-85.Search in Google Scholar

[13] Losada J., Nieto J.J., Properties of the new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2015, 1, 87-92.Search in Google Scholar

[14] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006Search in Google Scholar

[15] Tarasov V.E., Three-dimensional lattice models with long-range interactions of Grunwald-Letnikov type for fractional generalization of gradient elasticity, Meccanica 2016, 51, 1, 125-138.10.1007/s11012-015-0190-4Search in Google Scholar

[16] Odibat Z.M., Momani S., Application of variational iteration method to nonlinear differential equation of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 2006, 7, 1, 27-34.10.1515/IJNSNS.2006.7.1.27Search in Google Scholar

[17] Baleanu D., Guvenc Z.B., Machado J.A.T., Eds., New Trends in Nanotechnology and Fractional Calculus Applications, Springer Dordrecht Heidelberg, London New York, 201010.1007/978-90-481-3293-5Search in Google Scholar

[18] Atangana A., Koca I., On the new fractional derivative and application to nonlinear Baggs and Freedman Model, J. Nonlinear Sci. Appl., 2016, 9, 2467-2480.10.22436/jnsa.009.05.46Search in Google Scholar

[19] Baskonus H.M., Bulut H., Pandir Y., On The Solution of Nonlinear Time-Fractional Generalized Burgers Equation by Homotopy Analysis Method and Modified Trial Equation Method, International Journal of Modeling and Optimization, 2014, 4, 305-309.10.7763/IJMO.2014.V4.390Search in Google Scholar

[20] Razminia K., Razminia A., Machado J.A.T., Analytical Solution of Fractional Order diffusivity equation with wellbore storage and skin effects, J. Comput. Nonlinear Dynam., 2016, 11, 1, 10.1115/1.4030534Search in Google Scholar

[21] Rostamy D., Alipour M., Jafari H., Baleanu D., Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis, Rom. Rep. Phys., 2013, 65(2), 334-349.Search in Google Scholar

[22] Kumar D., Agarwal R.P., Singh J., A modified numerical scheme and convergence analysis for fractional model of Lienard’s equation, J. Comput. Appl. Math., 201710.1016/j.cam.2017.03.011Search in Google Scholar

[23] Kumar D., Singh J., Baleanu D., Modified Kawahara equation within a fractional derivative with non-singular kernel, Therm. Sci., 2017, 1(2.1), 3.10.2298/TSCI160826008KSearch in Google Scholar

[24] Kumar D., Singh J., Baleanu D., A new fractional model for convective straight fins with temperature-dependent thermal conductivity, Therm. Sci., 2017, 1, 1-12.10.2298/TSCI170129096KSearch in Google Scholar

[25] Kumar D., Singh J., Baleanu D., A new numerical algorithm for fractional Fitzhugh–Nagumo equation arising in transmission of nerve impulses, Nonlinear Dynam., 2018, 91(1), 307-317.10.1007/s11071-017-3870-xSearch in Google Scholar

[26] Inc M., Yusuf A., Isa Aliyu A., Baleanu D., Time-fractional Cahn-Allen and time-fractional Klein-Gordon equations: Lie symmetry analysis, explicit solutions and convergence analysis, Physica A: Statistical Mechanics and its Applications, 2018, 493, 94-106.10.1016/j.physa.2017.10.010Search in Google Scholar

[27] Kumar D., Singh J., Baleanu D., Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel, Physica A: Statistical Mechanics and its Applications, 2018, 492, 155-167.10.1016/j.physa.2017.10.002Search in Google Scholar

[28] Kamal S., Bandyopadhyay B., Spurgeon S., Stabilization of a fractional-order chain of integrators: a contraction-based approach, IMA J. Math. Control I., 2015, 32(2), 291-303.10.1093/imamci/dnt042Search in Google Scholar

[29] Bandyopadhyay B., Kamal S., Stabilization and control of fractional order systems: a sliding mode approach, Springer International Publishing, 201510.1007/978-3-319-08621-7Search in Google Scholar

[30] Diethelm K., The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type, Springer, 201010.1007/978-3-642-14574-2Search in Google Scholar

[31] Liao X., Gao Z., Huang H., Synchronization control of fractional-order discrete-time chaotic systems, Control Conference (ECC), 2013 European, 2013, 2214-2219.10.23919/ECC.2013.6669129Search in Google Scholar

Received: 2018-02-05

Accepted: 2018-04-21

Published Online: 2018-07-17

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/phys-2018-0055

Keywords for this article

Contraction theory; Stability; Convergence; Fractional order systems

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BY-NC-ND 4.0