Fractional Variational Problems Depending on Fractional Derivatives of Differentiable Functions with Application to Nonlinear Chaotic Systems

Abstract

We formulate a necessary condition for functionals with Lagrangians depending on fractional derivatives of differentiable functions to possess an extremum. The Euler-Lagrange equation we obtained generalizes previously known results in the literature and enables us to construct simple Lagrangians for nonlinear systems. As examples of application, we obtain Lagrangians for some chaotic dynamical systems.

1. Introduction

The calculus with fractional derivatives and integrals of noninteger order started more than three centuries ago when Leibniz proposed a derivative of order 1/2 in response to a letter from l’Hôpital [1]. This subject was also considered by several mathematicians as Euler, Fourier, Liouville, Grunwald, Letnikov, Riemann, and others up to nowadays. Although the fractional calculus is almost as old as the usual integer order calculus, only in the last three decades it has gained more attention due to its applications in various fields of science (see [2–7] for a review). Fractional derivatives are generally nonlocal operators and are historically applied to study nonlocal or time-dependent processes, as well as to model phenomena involving coarse-grained and fractal spaces. As an example, applications of fractional calculus in coarse-grained and fractal spaces are found in the framework of anomalous diffusion [8–10] and field theories [11–16].

The fractional calculus of variation was introduced in the context of classical mechanics. Riewe [17, 18] showed that a Lagrangian involving fractional time derivatives leads to an equation of motion with nonconservative forces such as friction. It is a remarkable result since frictional and nonconservative forces are beyond the usual macroscopic variational treatment [19] and consequently, beyond the most advanced methods of classical mechanics. Riewe generalized the usual calculus of variations for a Lagrangian depending on fractional derivatives [17, 18] in order to deal with linear nonconservative forces. Recently, several approaches have been developed to generalize the least action principle and the Euler-Lagrange equations to include fractional derivatives [20–26].

Despite that the Riewe approach has been successfully applied to study open and/or nonconservative linear systems, it cannot be directly applied to nonlinear open systems. The limitation follows from the fact that, in order to obtain a final equation of motion containing only integer order derivatives, the Lagrangian should contain only quadratic terms depending on fractional derivatives. In the present work we formulated a generalization of Riewe fractional action principle by taking advantage of a so-called practical limitation of fractional derivatives, namely, the absence of a simple chain and Leibniz’s rules.

As examples, we applied our generalized fractional variational principle to some nonlinear chaotic third-order dynamical systems, so-called jerk dynamical systems because the derivative of the acceleration with respect to time is referred to as the jerk [27]. These systems are important because they are the simplest ever one-dimensional autonomous ordinary differential equations which display dynamical behaviors including chaotic solutions [28–35]. It is important to mention that jerk dynamical systems describe several phenomena in physics, engineering, and biology, such as electrical circuits, mechanical oscillators, laser physics, and biological systems [28–35].

2. The Riemann-Liouville and Caputo Fractional Calculus

The fractional calculus of derivative and integration of noninteger orders started more than three centuries ago with l’Hôpital and Leibniz when the derivative of order 1/2 was suggested [1]. This subject was also considered by several mathematicians as Euler, Laplace, Liouville, Grunwald, Letnikov, Riemann and others up to nowadays. Although the fractional calculus is almost as old as the usual integer order calculus, only in the last three decades it has gained more attention due to its applications in various fields of science, engineering, economics, biomechanics, and so forth (see [2, 3, 5, 6] for a review). Actually, there are several definitions of fractional order derivatives. These definitions include the Riemann-Liouville, Caputo, Riesz, Weyl, and Grunwald-Letnikov (see [1–7] for a review). In this section we review some definitions and properties of the Riemann-Liouville and Caputo fractional calculi.

Despite haing many different approaches to fractional calculus, several known formulations are somehow connected with the analytical continuation of Cauchy formula for 𝑛-fold integration: ^𝑡_𝑎𝑥𝑡̃𝑑𝑡̃^𝑛=^𝑡_𝑎^𝑡𝑛_𝑎^𝑡𝑛−1_𝑎⋯^𝑡3_𝑎^𝑡2_𝑎𝑥𝑡₁𝑑𝑡₁𝑑𝑡₂⋯𝑑𝑡_𝑛−1𝑑𝑡_𝑛=1Γ(𝑛)^𝑡_𝑎𝑥(𝑢)(𝑡−𝑢)^1−𝑛𝑑𝑢(𝑛∈ℕ), (1) where Γ is the Euler gamma function. The proof of Cauchy formula can be found in several textbooks (e.g., it can be found in [1]). The analytical continuation of (1) gives us a definition for an integration of noninteger (or fractional) order. This fractional order integration is the building block of the Riemann-Liouville and Caputo calculi, the two most popular formulations of fractional calculus, as well as several other approaches to fractional calculus [1–7]. The fractional integration obtained from (1) is historically called Riemann-Liouville left and right fractional integrals.

Definition 1. Let 𝛼∈ℝ₊. The operators _𝑎𝐽^𝛼_𝑡 and _𝑡𝐽^𝛼_𝑏 defined on 𝐿₁[𝑎,𝑏] by _𝑎𝐽^𝛼_𝑡𝑥(𝑡)=1Γ(𝛼)^𝑡_𝑎𝑥(𝑢)(𝑡−𝑢)^1−𝛼𝑑𝑢𝛼∈ℝ₊,_𝑡𝐽^𝛼_𝑏𝑥(𝑡)=1Γ(𝛼)^𝑏_𝑡𝑥(𝑢)(𝑢−𝑡)^1−𝛼𝑑𝑢𝛼∈ℝ₊, (2) with 𝑎<𝑏 and 𝑎,𝑏∈ℝ, are called left and right fractional Riemann-Liouville integrals of order 𝛼∈ℝ, respectively.

For integer 𝛼 the fractional Riemann-Liouville integrals (2) coincide with the usual integer order 𝑛-fold integration (1). Moreover, from definitions (2) it is easy to see that the Riemann-Liouville fractional integrals converge for any integrable function 𝑥 if 𝛼>1. Furthermore, it is possible to prove the convergence of (2) for 𝑥∈𝐿₁[𝑎,𝑏] even when 0<𝛼<1 [4].

The integration operators _𝑎𝐽^𝛼_𝑡 and _𝑡𝐽^𝛼_𝑏 play a fundamental role in the definition of fractional Riemann-Liouville and Caputo calculi. In order to define the Riemann-Liouville derivatives, we recall that for positive integers 𝑛>𝑚 it follows the identity 𝐷^𝑚_𝑡𝑥(𝑡)=𝐷^𝑛_𝑡𝑎𝐽_𝑡^𝑛−𝑚𝑥(𝑡), where 𝐷^𝑚_𝑡 is an ordinary derivative of integer order 𝑚.

Definition 2 (Riemann-Liouville). The left and the right Riemann-Liouville fractional derivatives of order 𝛼>0 (𝛼∈ℝ) are defined, respectively, by _𝑎𝐷^𝛼_𝑡𝑥(𝑡)∶=𝐷^𝑛_𝑡𝑎𝐽_𝑡^𝑛−𝛼𝑥(𝑡) and _𝑡𝐷^𝛼_𝑏𝑥(𝑡)∶=(−1)^𝑛𝐷^𝑛_𝑡𝑡𝐽_𝑏^𝑛−𝛼𝑥(𝑡) with 𝑛=[𝛼]+1, namely, _𝑎𝐷^𝛼_𝑡𝑥(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑛𝑡_𝑎𝑥(𝑢)(𝑡−𝑢)^{1+𝛼−𝑛}𝑑𝑢,_𝑡𝐷^𝛼_𝑏𝑥(𝑡)=(−1)^𝑛Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑛𝑏_𝑡𝑥(𝑢)(𝑢−𝑡)^{1+𝛼−𝑛}𝑑𝑢, (3)(4) where 𝑑^𝑛/𝑑𝑡^𝑛 stands for ordinary derivatives of integer order 𝑛.

On the other hand, the Caputo fractional derivatives are defined by inverting the order between derivatives and integrations.

Definition 3 (Caputo). The left and the right Caputo fractional derivatives of order 𝛼∈ℝ₊ are defined, respectively, by ^𝐶_𝑎𝐷^𝛼_𝑡𝑥(𝑡)∶=_𝑎𝐽_𝑡^𝑛−𝛼𝐷^𝑛_𝑡𝑥(𝑡) and ^𝐶_𝑡𝐷^𝛼_𝑏𝑥(𝑡)∶=(−1)^𝑛_𝑡𝐽_𝑏^𝑛−𝛼𝐷^𝑛_𝑡𝑥(𝑡) with 𝑛=[𝛼]+1; that is, ^𝐶_𝑎𝐷^𝛼_𝑡𝑥(𝑡)1∶=Γ(𝑛−𝛼)^𝑡_𝑎𝑥^(𝑛)(𝑢)(𝑡−𝑢)^{1+𝛼−𝑛}𝑑𝑢,_𝐶𝑡𝐷^𝛼_𝑏𝑥(𝑡)()∶=−1^𝑛Γ(𝑛−𝛼)^𝑏_𝑡𝑥^(𝑛)(𝑢)(𝑢−𝑡)^{1+𝛼−𝑛}𝑑𝑢, (5)(6) where 𝑎≤𝑡≤𝑏 and 𝑥^(𝑛)(𝑢)=𝑑^𝑛𝑥(𝑢)/𝑑𝑢^𝑛∈𝐿₁([𝑎,𝑏]) is the ordinary derivative of integer order 𝑛.

An important consequence of definitions (3)–(6) is that the Riemann-Liouville and Caputo fractional derivatives are nonlocal operators. The left (right) differintegration operator (3) and (5) ((4) and (6)) depends on the values of the function at left (right) of 𝑡, that is, 𝑎≤𝑢≤𝑡 (𝑡≤𝑢≤𝑏). On the other hand, it is important to note that when 𝛼 is an integer, the Riemann-Liouville fractional derivatives (3) and (4) and reduced to ordinary derivatives of order 𝛼. On the other hand, in that case, the Caputo derivatives (5) and (6) differ from integer order ones by a polynomial of order 𝛼−1 [3, 4].

It is important to remark, for the purpose of this work, that the fractional derivatives (3)–(6) do not satisfy a simple generalization of the chain and Leibniz’s rules of classical derivatives [1–7]. In other words, generally we have ^𝐶_𝑎𝐷^𝛼_𝑡𝑥(𝑡)𝑦(𝑡)≠𝑦(𝑡)^𝐶_𝑎𝐷^𝛼_𝑡𝑥(𝑡)+𝑥(𝑡)^𝐶_𝑎𝐷^𝛼_𝑡𝑦(𝑡),_𝐶𝑎𝐷^𝛼_𝑡𝑦(𝑥(𝑡))≠^𝐶_𝑎𝐷^𝛼_𝑢𝑦(𝑢)|^𝐶_{𝑎𝑢=𝑥}𝐷^𝛼_𝑡𝑥(𝑡). (7) The absence of a simple chain and Leibniz’s rules is commonly considered a practical limitation of the fractional derivatives (3)–(6). However, in the present work we take advantage of this limitation in order to formulate a generalized Lagrangians for nonlinear systems.

In addition to the definitions (3)–(6), we make use of the following property in order to obtain a fractional generalization of the Euler-Lagrange condition.

Theorem 4 (integration by parts-see, e.g., [7]). Let 0<𝛼<1 and let 𝑥 be a differentiable function in [𝑎,𝑏] with 𝑥(𝑎)=𝑥(𝑏)=0. For any function 𝑦∈𝐿₁([𝑎,𝑏]) one has ^𝑏_𝑎𝑦(𝑡)^𝐶_𝑎𝐷^𝛼_𝑡𝑥(𝑡)𝑑𝑡=^𝑏_𝑎𝑥(𝑡)_𝑡𝐷^𝛼_𝑏𝑦(𝑡)𝑑𝑡,_𝑏𝑎𝑦(𝑡)^𝐶_𝑡𝐷^𝛼_𝑏𝑥(𝑡)𝑑𝑡=^𝑏_𝑎𝑥(𝑡)_𝑎𝐷^𝛼_𝑡𝑦(𝑡)𝑑𝑡. (8)

It is important to notice that the formulas of integration by parts (8) relate Caputo left (right) derivatives to Riemann-Liouville right (left) derivatives.

Finally, in order to obtain the equation of motion for our examples, we are going to use the following two relations: 𝑖^𝐶_𝑎𝐷_𝑡^1/2𝑥(𝑡)=^𝐶_𝑡𝐷_𝑏^1/2𝑥(𝑡)inthelimit𝑎⟶𝑏,_𝑡𝐷_𝑏^𝐶_𝑡^1/2𝐷_𝑏^1/2𝑥(𝑡)=𝑑𝑑𝑡^𝑡𝐽_𝑏𝑡^1/2𝐽_𝑏^1/2𝑑𝑥(𝑡)=𝑑𝑑𝑡𝑑𝑡^𝑡𝐽¹_𝑏𝑑𝑥(𝑡)=𝑑𝑑𝑡𝑥(𝑡).𝑑𝑡 (9)(10) The proof of (9) can be found in [36], and (10) follows from the general semigroup property _𝑡𝐽^𝛼_𝑏𝑡𝐽^𝛽_𝑏=_𝑡𝐽_𝑏^𝛼+𝛽 (see, e.g., [4, 7]).

3. A Generalized Fractional Lagrangian

In classical calculus of variations it is of no conceptual and practical importance to deal with Lagrangian functions depending on derivatives of nonlinear functions of the unknown function 𝑥. This is due to the fact that in these cases we can always rewrite the Lagrangian 𝐿 as a usual Lagrangian 𝐿 by applying the chain’s rules. As, for example, for a differentiable function 𝑓 we can rewrote 𝐿(𝑡,𝑥,(𝑑/𝑑𝑡)𝑓(𝑥))=𝐿(𝑡,𝑥,(𝑑/𝑑𝑢)𝑓(𝑢)|_𝑢=𝑥𝑥̇)=𝐿(𝑡,𝑥,𝑥̇), where 𝑑𝑥/𝑑𝑡=𝑥̇. However, this simplification for the fractional calculus of variation is not possible due to the absence of a simple chain’s rule for fractional derivatives. It is just this apparent limitation of fractional derivatives that opens the very interesting possibility to investigate new kinds of Lagrangians suitable to study nonlinear systems. In the present work we investigate for the first time these kinds of Lagrangians and we apply them to construct Lagrangians for some Jerk nonlinear dynamical system.

Our main result is the following theorem.

Theorem 5. Let 𝑓,𝑔∶ℝ→ℝ be differentiable functions, and 𝑆 an action of the form 𝑆=^𝑏_𝑎𝐿𝑡,𝑥,𝑥̇,^𝐶_𝑎𝐷^𝛼_𝑡𝑓(𝑥),^𝐶_𝑎𝐷^𝛼_𝑡𝑔(𝑥̇)𝑑𝑡, (11) where ^𝐶_𝑎𝐷^𝛼_𝑡 is a Caputo fractional derivatives of order 0<𝛼<1, and the function 𝑥 satisfies the fixed boundary conditions 𝑥(𝑎)=𝑥_𝑎, 𝑥(𝑏)=𝑥_𝑏, 𝑥̇(𝑎)=𝑥̇_𝑎,  and  𝑥̇(𝑏)=𝑥̇_𝑏. Also let 𝐿∈𝐶²[𝑎,𝑏]×ℝ⁴. Then, the necessary condition for 𝑆 to possess an extremum at 𝑥 is that the function 𝑥 fulfills the following fractional Euler-Lagrange equation: 𝜕𝐿−𝑑𝜕𝑥𝑑𝑡𝜕𝐿𝜕𝑥̇+𝑑𝑓𝑑𝑥^𝑡𝐷^𝛼_𝑏𝜕𝐿𝜕^𝐶_𝑎𝐷^𝛼_𝑡𝑓−𝑑𝑑𝑡𝑑𝑔𝑑𝑥̇^𝑡𝐷^𝛼_𝑏𝜕𝐿𝜕^𝐶_𝑎𝐷^𝛼_𝑡𝑔=0. (12)

Proof. In order to develop the necessary conditions for the extremum of the action (11), we define a family of functions 𝑥 (weak variations) 𝑥=𝑥^∗+𝜀𝜂, (13) where 𝑥^∗ is the desired real function that satisfy the extremum of (11), 𝜀∈ℝ is a constant, and the function 𝜂 defined in [𝑎,𝑏] satisfies the boundary conditions 𝜂(𝑎)=𝜂(𝑏)=0,𝜂̇(𝑎)=𝜂̇(𝑏)=0. (14) The condition for the extremum is obtained when the first Gâteaux variation is zero: 𝛿𝑆=lim_𝜀→0𝑆𝑥^∗+𝜀𝜂−𝑆𝑥^∗𝜀=^𝑏_𝑎𝜂𝜕𝐿𝜕𝑥^∗+𝜂̇𝜕𝐿𝜕(𝑥̇^∗)+^𝐶_𝑎𝐷^𝛼_𝑡𝜂𝑑𝑓𝑑𝑥^∗𝜕𝐿𝜕^𝐶_𝑎𝐷^𝛼_𝑡𝑓+^𝐶_𝑎𝐷^𝛼_𝑡𝜂̇𝑑𝑔𝑑𝑥̇^∗𝜕𝐿𝜕^𝐶_𝑎𝐷^𝛼_𝑡𝑔𝑑𝑡=0. (15) Since the function 𝜂 satisfies both 𝜂(𝑎)=𝜂(𝑏)=0 and 𝜂̇(𝑎)=𝜂̇(𝑏)=0 boundary conditions (14), we can use the fractional integration by parts (8) in (15), obtaining 𝛿𝑆=^𝑏_𝑎𝜂𝜕𝐿𝜕𝑥^∗+𝜂̇𝜕𝐿𝜕(𝑥̇^∗)+𝜂𝑑𝑓𝑑𝑥^∗𝑡𝐷^𝛼_𝑏𝜕𝐿𝜕^𝐶_𝑎𝐷^𝛼_𝑡𝑓+𝜂̇𝑑𝑔𝑑𝑥̇^∗𝑡𝐷^𝛼_𝑏𝜕𝐿𝜕^𝐶_𝑎𝐷^𝛼_𝑡𝑔=𝑑𝑡_𝑏𝑎𝜂𝜕𝐿𝜕𝑥^∗−𝑑𝑑𝑡𝜕𝐿𝜕(𝑥̇^∗)+𝑑𝑓𝑑𝑥^∗𝑡𝐷^𝛼_𝑏𝜕𝐿𝜕^𝐶_𝑎𝐷^𝛼_𝑡𝑓−𝑑𝑑𝑡𝑑𝑔𝑑𝑥̇^∗𝑡𝐷^𝛼_𝑏𝜕𝐿𝜕^𝐶_𝑎𝐷^𝛼_𝑡𝑔𝑑𝑡=0, (16) where an additional usual integration by parts was performed in the terms containing 𝜂̇. Finally, by using the fundamental lemma of the calculus of variations, we obtain the fractional Euler-Lagrange equations (12).

It is important to notice that our theorem can be easily extended for Lagrangians depending on left Caputo derivatives, and Riemann-Liouville fractional derivatives. Actually, it is also easy to generalize in order to include a nonlinear function 𝑔(^𝐶_𝑎𝐷^𝛼_𝑡𝑥) instead of 𝑔(𝑥̇). Finally, it is important to mention that our theorem generalizes [17, 18] and the more general formulation proposed in [20], as well as the Lagrangian formulation for higher order linear open systems [37] (for a review in recent advances in calculus of variations with fractional derivatives see [26]).

Remark 6. For 𝑓(𝑥)=𝑔(𝑥̇)=0 our condition (12), reduced to the ordinary Euler-Lagrange equation, and the boundary conditions 𝑥(𝑎)=𝑥_𝑎, 𝑥(𝑏)=𝑥_𝑏 and 𝑥̇(𝑎)=𝑥̇_𝑎, 𝑥̇(𝑏)=𝑥̇_𝑏 are defined by only two arbitrary parameters. Note that for this particular case the Euler-Lagrange equation is a second-order ordinary differential equation whose solution 𝑥(𝑡) is fixed by two parameters. For example, by imposing the conditions 𝑥(𝑎)=𝑥_𝑎 and 𝑥(𝑏)=𝑥_𝑏 the solution 𝑥(𝑡) is fixed and, consequently, the numbers 𝑥̇_𝑎 and 𝑥̇_𝑏 are automatically fixed as functions of 𝑥_𝑎 and 𝑥_𝑏.

4. Lagrangian for Nonlinear Chaotic Jerk Systems

As an example for application of our generalized Euler-Lagrange equation (12), in this section we obtained Lagrangians for some jerk systems. The first example is the simplest one-dimensional family of jerk systems that displays chaotic solutions [27–35]: 𝑥⃛+𝐴𝑥̈+𝑥̇=𝐺(𝑥), (17) where 𝐴 is a system parameter and 𝐺(𝑥) is a nonlinear function containing one nonlinearity, one system parameter and a constant term. A Lagrangian for this jerk system is given by 𝐿𝑥,^𝐶_𝑡𝐷_𝑏^1/2𝑥,^𝐶_𝑡𝐷_𝑏^1/2𝑥̇𝑖=−2^𝐶𝑎𝐷_𝑡^1/2𝑥̇²−𝐴2(𝑥̇)²+𝑖2^𝐶𝑎𝐷_𝑡^1/2𝑥²−𝐺(𝑥)𝑑𝑥. (18) In order to show that (18) gives us (17), we insert (18) into our generalized Euler-Lagrange equation (12), obtaining 𝑖𝑑𝑑𝑡^𝐶𝑡𝐷_𝑏^𝐶_𝑎^1/2𝐷_𝑡^1/2𝑥̇+𝐴𝑥̈+𝑖^𝐶_𝑡𝐷_𝑏^𝐶_𝑎^1/2𝐷_𝑡^1/2𝑥=𝐺(𝑥), (19) and we follow the procedure introduced in [17, 18] by taking the limit 𝑎→𝑏. Taking the limit in (19) and using (9) and (10) we get (17).

The Lagrangian (18) is equivalent to the introduced by us in [37]. However, it is important to stress that (17) is the only chaotic jerk system containing nonlinearity depending only on 𝑥 [27–35]. For jerk systems with more complex nonlinearities, as for example, 𝑥𝑥̇, 𝑥̇² and 𝑥𝑥̈, it is not possible to formulate a simple Lagrangian, depending only on 𝑥 and its derivatives, by using classical calculus of variation or previous formulations including fractional derivatives. Using our Euler-Lagrange equation (12) we can formulate, by the first time, a Lagrangian for these jerk systems [27–35]: 𝑖𝐿=−2^𝐶𝑎𝐷_𝑡^1/2𝑥̇²−𝐴2(𝑥̇)²+𝑖2^𝐶𝑎𝐷_𝑡^1/2𝑥^2𝐶_𝑎𝐷_𝑡^1/2𝑥̇+𝑥²2⟹𝑥⃛+𝐴𝑥̈−𝑥̇²𝑖+𝑥=0,𝐿=−2^𝐶𝑎𝐷_𝑡^1/2𝑥̇²−𝐴2(𝑥̇)²−𝑖2^𝐶𝑎𝐷_𝑡^1/2𝑥^2𝐶_𝑎𝐷_𝑡^1/2𝑥+𝑥²2⟹⃛̈̇𝑖𝑥+𝐴𝑥−𝑥𝑥+𝑥=0,𝐿=−2^𝐶𝑎𝐷_𝑡^1/2𝑥̇²−𝐴2𝑥(𝑥̇)²+𝑖𝐴+24^𝐶𝑎𝐷_𝑡^1/2𝑥^2𝐶_𝑎𝐷_𝑡^1/2𝑥̇+𝑥²2⟹𝑥⃛+𝐴𝑥𝑥̈−𝑥̇²+𝑥=0. (20)

5. Conclusions

In the present work we obtained an Euler-Lagrange equation for Lagrangians depending on fractional derivatives of nonlinear functions of the unknown function 𝑥. Our formulation enables us to obtain Lagrangians for nonlinear open and dissipative systems, and consequently, it enables us to use the most advanced methods of classical mechanics to study these systems. As examples of applications, we obtained a Lagrangian for some chaotic jerk dynamical system.

Acknowledgments

The author is grateful to the Brazilian foundations FAPERGS, CNPq, and Capes for financial support.