A Fractional Model for the Single Stokes Pulse from the Nonlinear Optics ()
1. Introduction
The theory of fractional differential equations (i.e. fractional calculus) and its applications are based on non-integer order of derivatives and integrals [1] [2].
The use of fractional models has received a great degree of interest in a series of works due to its applications in different fields of science and engineering. For example, these models played an important role in applied mathematics [3], mathematical physics [4], theoretical and applied physics [5], study of biological systems [6], control processing [7], chaos synchronization [8] [9] and so on. The dynamics of fractional-order systems associated to dynamical systems (in particular, Hamilton-Poisson systems) have been studied by many researchers in the recent decades [10] [11]. Another series of works deals with the study of dynamical behaviors of classical and fractional differential systems on Lie groups, Lie algebroids and Leibniz algebroids [12] [13] [14].
In this paper we consider the single Stokes pulse system [15]. It is described by the following differential equations on
:
(1.1)
where
are state variables,
,
for
are parameters and t is the time.
The Hamilton-Poisson system (1.1) has been studied from mechanical geometry point of view [16]. It is associated to this system, the general fractional Stokes pulse system. The aim of our paper is focused on the study of a certain type of the fractional Stokes pulse system.
This paper is structured as follows. The Stokes pulse system (2.6) is described in Section 2. In Section 3 we define the fractional Stokes pulse system (3.1). The existence and uniqueness of solutions of initial value problem for the fractional model (3.1) are discussed. Also, are proposed four types of fractional Stokes pulse systems which are physically inequivalent. From the four types of fractional Stokes pulse systems, we choose a subcase of the second type, called the special fractional Stokes pulse system (3.3). The Section 4 is dedicated to analyzing of asymptotic stability of equilibrium states for the fractional model (3.3). For stabilization problem of the system (3.3), we associate the fractional Stokes pulse system with controls, denoted by (4.2). In Propositions (4.3) - (4.6) are established sufficient conditions on parameters k and k1 to control the chaos in the fractional system (4.2). Using the fractional Euler’s method, the numerical integration of the system (4.2) is presented in Section 5.
2. The Single Stokes Pulse as Hamilton-Poisson System
For details on Hamiltonian dynamics, see e.g. [17] [18] [19].
The equations of motion for the Stokes polarization parameters of a single optical beam propagating as a traveling wave in a nonlinear medium (the single Stokes pulse) from the nonlinear optics are described by using the Stokes vector u and a Hamiltonian function H.
The Stokes vector is defined using the Pauli spin matrices and is called the polarization parameters of the single Stokes pulse. The vector
is assumed to be expressed in a linear polarization basis [15]. Let be A the transition matrix from the canonical basis of
to the polarization basis. Since A is symmetric, then it one can always transform to a polarization basis in which A has the diagonal form
, where
are the eigenvalues of A.
The Hamiltonian function H is determined by the Stokes vector u, the diagonal matrix W, and the constant vectors
and
. Using the vectors a and c, we define the vector
by:
(2.1)
The matrix W describes the self-induced ellipse rotation. The vectors a and c describes the effects of linear and nonlinear anisotropy, respectively.
In terms of the Stokes parameters (the components of the vector u), the Hamiltonian function
,
, is defined by [15]:
(2.2)
where
and
.
The diagonal matrix W and the choice of the vectors a and c generates the dynamics of the Stokes vector u with the frequence b.
In the coordinate system
, the Hamiltonian function H defined by (2.2), is written as:
(2.3)
The dynamics of a single Stokes pulse is written as Hamilton-Poisson system. More precisely, the Stokes pulse system is defined on the Lie-Poisson manifold
the dual of Lie algebra
with the following bracket:
(2.4)
and the Hamiltonian function
given by (2.3).
The dynamical system defined on
with Poisson bracket {., .} given by (2.4), enabling the equations of motion to be expressed in Hamiltonian form:
(2.5)
where
, t is the time and H is the Hamiltonian function [20].
We determine the equations
of the system (2.5). We have:
Then, the first equation of the system (2.5) is
.
Finally one obtains the following differential system on
:
(2.6)
where the parameters
for
are connected with the nature of the material and the medium.
The system (2.6) is called theStokes pulse dynamical system.
Acording to [20] there are six types of Equation (2.6) which are physically inequivalent and which correspond to different types of optical media [16].
Proposition 2.1. The functions H given by (2.4) and
, defined by:
are constants of the motion (first integrals)for the dynamics (2.6).
Proof. Indeed, we have
Also, we have
Remark 2.1. By Proposition 2.1, it follows thatthe trajectories of motion of Stokes pulse dynamical system (2.6) are intersections of the surfaces:
and
.
3. The Fractional Stokes Pulse System
For basic knowledge on fractional calculus, one may refer to [21] [22].
In this paper we consider the fractional derivative operator
with
to be Caputo’s derivative. This fractional derivative operator is often used in concrete applications.
Let
and
. The q-order Caputo differential operator [21], is described by
, where
represents the m-order derivative of the function f,
is an integer such that
and
is
,
,
where
is the Euler Gamma function. If
, then
.
The Hamilton-Poisson system (2.6) is modeled by the following fractional differential equations:
(3.1)
where
are the Stokes polarization parameters and
for
.
The system (3.1) is called the fractional Stokes pulse system associated to (2.6).
The initial value problem of fractional model (3.1) can be represented in the following matrix form:
(3.2)
where
,
,
and
Proposition 3.1.The initial value problem of the fractional Stokes pulse system (3.1)has a unique solution.
Proof. Let
. It is obviously continuous and bounded on
for any
. We have
, where
,
and
,
Then
(1)
, where
and denote
matrix norm and vector norm, respectively.
It is easy to see that
. Then
(2)
.
Similarly, we prove that
(3)
.
(4)
.
According to (2)-(4), the relation (1) becomes
(5)
.
We have
,
,
,
. If
, then
for
. From the relation (5) we deduce that
(6)
.
Given that inequalities
,
, are valid, the relation (6) becomes
(7)
,
where
.
The inequality (7) shows that
satisfies a Lipschitz condition. Using Theorems 1 and 2 in [23], it follows that the system (3.1) has a unique solution.
As with the nonlinear dynamics generated by the Stokes pulse system (2.6) there are six types of fractional Equation (3.1) which are physically inequivalent and which correspond to different types of optical media.
In this section we will refer to the types of fractional Stokes pulse systems for which the parameters
meet the following condition
.
In this context there are the following four types of fractional Stokes pulse systems:
Type 1.
and
;
Type 2.
,
and
;
Type 3.
,
and
;
Type 4.
,
and
.
As an example, the fractional system corresponding to type 2, is given by:
(3.3)
where
and
such that
.
The system (3.3) is called the special fractional Stokes pulse system. It is determined by a single nonzero component of vector b and contains the three nonlinear terms of the system (3.1) (given in general form).
For the system (3.3) we introduce the following notations:
(3.4)
Proposition 3.2. The equilibrium states of the special fractional Stokes pulse system (3.3) are given as the union of the following three families:
Proof. The equilibrium states are solutions of the equations
, where
are given by (3.4).
Remark 3.1. If in the fractional model (3.3) we take
, then one obtains the system for integer-order derivative which corresponds to type 4 $ of the dynamics (1.1). For this dynamical system, the nonlinear stability and the problem of existence of periodic solutions are studied, see Theorems 2.4, 3.5-3.7 [16].
4. Asymptotic Stability of the Special Fractional Stokes Pulse System (3.3)
Let us we present the study of asymptotic stability of equilibria for the fractional system (3.3). Finally, we will discuss how to stabilize the unstable equilibrium states of the system (3.3) via fractional order derivative. For this study we apply the Matignon’s test [7].
In the follows we will use the notations:
(4.1)
With the notations (4.1), the Jacobian matrix associated to system (3.3) is:
Proposition 4.1. ( [7]) Let
be an equilibrium state of system (3.3) and
be the Jacobian matrix
evaluated at
.
(i)
is locally asymptotically stable, iff all eigenvalues of the matrix
satisfy:
(ii)
is locally stable, iff either it is asymptotically stable, or the critical eigenvalues of
which satisfy
have geometric multiplicity one.
Proposition 4.2. The equilibrium states
are unstable
.
Proof. The characteristic polynomial of the matrix
is
. For
, the characteristic polynomials of the matrix
is
.
The characteristic polynomials of matrices
and
are the following:
and
.
The equations
and
have the root
. Since
for all
, by Proposition 4.1 follows that the equilibrium states
and
are unstable for all
.
In the case when
is a unstable equilibrium state of the fractional system (3.3), we associate to (3.3) a new fractional system, called the special fractional Stokes pulse system with (external)controls and given by:
(4.2)
where
are given in (4.1) and
are controls.
If one selects the parameters
which then make the eigenvalues of the Jacobian matrix of fractional model (3.3) satisfy one of the conditions from Proposition 3.1, then its trajectories asymptotically approaches the unstable equilibrium state
in the sense that
, where
is the Euclidean norm.
The Jacobian matrix of the fractional model (4.2) with the controls
is
Proposition 4.3. Let be the fractional Stokes pulse system (4.2)with the controls
.
(i) If
, then
is asymptotically stable
;
(ii) If
and
, then:
(1)
is asymptotically stable
and it is stable for
.
(2)
is unstable
.
(iii) If
and
, then
is unstable
.
Proof. The characteristic polynomial of the Jacobian matrix
is
. The roots of the equation
are
,
.
(i) We suppose
and
. In this case we have
for
. Since
for all
, by Proposition 4.1(i), it implies that
is asymptotically stable for all
.
(ii) We suppose
and
. In this case we have
and
. Applying Proposition 4.1(i),
is locally asymptotically stable, for
, where
. If
,
is stable. For
,
is unstable
. Hence, the assertion (ii) holds.
(iii) We suppose
and
. Since
has at least a positive eigenvalue, it follows that
is unstable. Hence, (iii) holds,
.
Proposition 4.4. Let be the fractional Stokes pulse system (4.2)with the controls
,
and
.
1. Let
and
.
(i) If
and
, then
is asymptotically stable.
(ii) Let
and
.
(1) If
and
, then
is asymptotically stable
, stable for
and unstable
.
(2) If
and
, then
is asymptotically stable
, stable for
and unstable
.
(iii) If
and
, then
is unstable
.
2. Let
and
.
(i) If
and
.
(1) If
and
, then
is asymptotically stable.
(2) If
and
, then
is asymptotically stable.
(ii) Let
and
. If
,
or
,
, then
is unstable.
Proof. The characteristic polynomial of (4.2) at
is
whose characteristic polynomial is
. The roots of the characteristic equation
are
,
, where
.
1. Case
and
. We have the following two situations:
(1) if
, then
for all
;
(2) if
, then
for all
.
In this case,
,
.
(i) We suppose
and
. In this case we have
and
. Since
for all
, by Proposition 4.1(i), it implies that
is locally asymptotically stable for all
.
(ii) (1)-(2). For
and
, we have
and
. Applying Proposition 4.1(i),
is asymptotically stable, for
, where
. If
, then
is stable. For
,
is unstable.
(iii) Let
and
. Since
,
has at least a positive eigenvalue and so
is unstable. Hence, the assertions (i)-(iii) hold.
2. Case
and
. Then
,
. We have the following two situations:
(1) if
, then
for all
;
(2) if
, then
for all
.
(i)-(ii) The eigenvalues
are all negative if and only if
and
. In these hypotheses it folows that
is asymptotically stable. Also, if
, then
is unstable. Therefore, the assertions (i)-(ii) hold.
Proposition 4.5. Let be the fractional Stokes pulse system (4.2)with the controls
,
and
.
1. Let
and
.
(i) Let
and
.
(1) If
and
, then
is asymptotically stable
.
(2) If
and
, then
is asymptotically stable
.
(3) If
,
and
, then
is asymptotically stable
, stable for
and unstable
.
(ii) Let
and
.
(1) If
and
, then
is asymptotically stable
.
(2) If
and
, then
is asymptotically stable
.
(3) If
or
,
, then
is unstable.
2. Let
,
and
.
(i) Let
and
.
(1) If
,
, then
is asymptotically stable.
(2) If
,
, then
is asymptotically stable.
(ii) Let
or
. If
,
or
, then
is unstable.
Proof. The Jacobian matrix of (4.2) at
is
whose characteristic polynomial is
. The roots of the equation
are
,
, where
.
1. Case
.
(i) Case
and
. Then
,
. We have
if and only if
when
or
when
.
(1) We suppose
. In this case we have
and
. Since
for all
, by Proposition 4.1(i), it implies that
is locally asymptotically stable for all
.
(2) We suppose
and
. In this case we have
and
. Applying the same reasoning as in the case (i)(1), one obtains that
is locally asymptotically stable for all
.
(3) We suppose
and
. In this case,
if and only if
. Then
and
. Applying Proposition 4.1(i),
is locally asymptotically stable, for
, where
. If
, then
is stable. For
,
is unstable.
(ii) Case
and
. Then
,
. We have
if and only if
when
or
when
.
(1) and (2). In these cases, if
and
, then
and
. It follows
. Then,
is locally asymptotically stable.
(3) Let
and
or
and
. Then
has at least a positive eigenvalue and so
is unstable. Hence, the assertion (ii) holds.
2. Case
,
and
. In this case
.
(i) We suppose
. We have
and
if and only if
and
. Then
and
. It follows
for all
such that
. Hence,
is locally asymptotically stable.
(ii) We suppose
or
. Then,
has at least a positive eigenvalue and so
is unstable. Therefore, the assertion (ii) holds.
Proposition 4.6. Let be the fractional Stokes pulse system (4.2)with the controls
,
and
.
1. Let
and
.
(i) Let
and
.
(1) If
, then
is asymptotically stable
.
(2) If
and
, then
is asymptotically stable
.
(3) If
,
and
, then
is asymptotically stable
, stable for
and unstable
.
(ii) Let
and
.
(1) If
and
, then
is asymptotically stable
.
(2) If
and
, then
is asymptotically stable
.
(3) If
or
,
, then
is unstable.
2. Let
,
and
.
(i) Let
and
.
(1) If
,
, then
is asymptotically stable.
(2) If
,
, then
is asymptotically stable.
(ii) Let
or
. If
,
or
,
, then
is unstable.
Proof. The Jacobian matrix of (4.2) at
is
Whose characteristic polynomial is
. The roots of the equation
are
,
, where
.
1. Case
.
(i) Case
and
. Then
,
. We have
if and only if
when
or
when
.
(1) We suppose
. In this case we have
and
. Since
for all
, by Proposition 4.1(i), it implies that
is locally asymptotically stable for all
.
(2) We suppose
and
. In this case we have
and
. Applying the same reasoning as in the case (i)(1), one obtains that
is locally asymptotically stable for all
.
(3) We suppose
and
. In this case,
if and only if
. Then
and
. Applying Proposition 4.1(i),
is locally asymptotically stable, for
, where
. If
, then
is stable. For
,
is unstable.
(ii) Case
and
. Then
,
. We have
if and only if
when
or
when
.
(1) and (2). In these cases, if
and
, then
and
. It follows
. Then,
is locally asymptotically stable.
(3) Let
and
or
and
. Then
has at least a positive eigenvalue and so
is unstable. Hence, the assertion (ii) holds.
2. Case
,
and
. In this case
.
(i) We suppose
. We have
and
if and only if
and
. Then
and
. It follows
for all
such that
. Hence,
is locally asymptotically stable.
(ii) We suppose
or
. Then,
has at least a positive eigenvalue and so
is unstable. Therefore, the assertion (ii) holds.
Example 4.1. (i) Let be the special fractional Stokes pulse system (4.2). We select
and
. Then
. We have
and
.
(i) Chosing
and
, it follows that
and
. According to Proposition 4.4, 2.(i)(1) it follows that the equilibrium state
is asymptotically stable for
.
(ii) For
and
, follows t
and
.
The conditions of Proposition 4.4, 2.(i)(2) are achieved. Then
is unstable for
.
Using Matlab, in Table 1 we give a set of values for the parameters
, the equilibrium states and corresponding eigenvalues of special fractional Stokes pulse system (4.2).
Table 1. The controls
, equilibrium states
and corresponding eigenvavues.
5. Numerical Integration of the Special Fractional Stokes Pulse System (4.2)
In this section we start with some mathematical preliminaries of the fractional Euler’s method for solving initial value problem for fractional differential equations.
Consider the following general form of the initial value problem (IVP) with Caputo derivative:
(5.1)
where
is a continuous nonlinear function and
, represents the order of the derivative.
The right-hand side of the IVP (5.1) in considered examples is Lipschitz functions and the numerical method used in this works to integrate system (5.1) is the Fractional Euler’s method.
Since f is assumed to be continuous function, every solution of the initial value problem given by (5.1) is also a solution of the following Volterra fractional integral equation:
(5.2)
where
is the q-order Riemann-Liouville integral operator, which is expressed by:
(5.3)
Moreover, every solution of (5.2) is a solution of the (IVP) (5.1).
To integrate the fractional Equation (5.1), means to find the solution of (5.2) over the interval
. In this context, a set of points
are produced which are used as approximated values. In order to achieve this approximation, the interval
is partitioned into n subintervals
each equal width
for
.
For the fractional-order q and
, it computes an approximation denoted as
for
.
The general formula of the fractional Euler’s method for to compute the elements
, is:
(5.4)
For more details, see [24] [25].
For the numerical integration of the special fractional Stokes pulse system (4.2), we apply the fractional Euler method (FEM). For this, consider the following fractional differential equations:
(5.5)
where
(5.6)
where
and
such that
.
Since the functions
are continuous, then the initial value problem (5.5) is equivalent to system of Volterra integral equations, which is given as follows:
(5.7)
The system (5.7) is called the Volterra integral equations associated to special Stokes pulse system (4.2).
The problem for solving the system (5.5) is reduced to one of solving a sequence of systems of fractional equations in increasing dimension on successive intervals
.
For the numerical integration of the system (5.6) one can use the fractional Euler method (the formula (5.4), which is expressed as follows:
(5.8)
where
,
,
,
.
More precisely, the numerical integration of the fractional system (5.5) is given by:
(5.9)
where
,
,
.
Using [21] [24], we have that the numerical algorithm given by (5.9) is convergent.
Example 5.1. Let us we present the numerical integration of the special fractional Stokes pulse system with controls which has considered in Example 4.1(i). For this we apply the algorithm (5.9) and software Maple. Then, in (5.9) we take:
,
,
,
,
, and
. It is known that the equilibrium state
is asymptotically stable.
For the numerical simulation of solutions of the above fractional model we use the rutine Maple. spec-fract-Stokes-pulse-system-with-controls, denoted by [sp-fr.Stokes-pulse syst]. Applying this program for
,
,
,
,
,
,
, one obtain the orbits
,
and
, for
.
Finally, we present the rutine [sp-fr.Stokes pulse syst]:
# Fractional equations associated to Stokes pulse system for
Du1/dt=(l2-l3)*u2*u3 + b2*u3 + k1* u1;
Du2/dt=(l3-l1)*u1*u3 + k* u2;
Du3/dt=(l1-l2)*u1*u2 - b2*u1 + k1* u3;
> with (plots):
> l1:=1.; l2:=0.5; l3:=1.5; alpha:=l2-l3; beta:=l3-l1; gamma:=l1-l2; b2:=1.; k:=-0.15; k1:= -0.4; q:=0.8; u1e:=0.; u2e:=1.6; u3e:=0.;
> with (stats):
> h:=0.01; epsilon:=0.01; n:=100:t:=n+2; u1:= array (0 .. n): u2:= array (0 .. n): u3:= array (0 .. n): u1[0]:=epsilon + u1e; u2[0]:=epsilon + u2e; u3[0]:=epsilon + u3e;
> for j from 1 by 1 to n do
> u1[j]:= u1[j-1] + h Ùq *(alpha* u2[j-1]*u3[j-1] + b2*u3[j-1] + k1* u1[j-1])/GAMMA(q+1);
u2[j]:= u2[j-1] + h Ùq *(beta* u1[j-1]*u3[j-1] + k* u2[j-1])/GAMMA(q+1);
u3[j]:= u3[j-1] + h Ùq *(gamma* u1[j-1]*u2[j-1] - b2*u1[j-1] + k1* u3[j-1])/GAMMA(q+1);
od:
> plot (seq([j,u1[j]], j = 0 .. n), style = point, symbol = point, scaling = UNCONSTRAINED);
plot (seq([j,u2[j]], j = 0 .. n), style = point, symbol = point, scaling = UNCONSTRAINED);
plot (seq([j,u3[j]], j = 0 .. n), style = point, symbol = point, scaling = UNCONSTRAINED);
pointplot 3d ( {seq([u1[j], [u2[j], [u3[j]], j = 0 .. n)}, style = point, symbol = point, scaling =
UNCONSTRAINED, color = red);
Remark 5.1. Appyling (5.9) and Maple for the numerical simulation of solutions of fractional model (4.2) for each set of values for parameters
and
, given in the Table 1, it will be found that the results obtained are valid.
Conclusions. This paper presents the fractional Stokes pulse system (3.1) associated to system (2.6). The special fractional Stokes pulse system (3.3) was studied from fractional differential equations theory point of view: asymptotic stability, determining of sufficient conditions on parameters
to control the chaos in the proposed fractional system and numerical integration of the fractional model (4.2). By choosing the right parameters
and
in the fractional model (4.2), this work offers a series of chaotic fractional differential systems. The other types of systems mentioned in the four types of fractional models will be studied in future works.
Acknowledgments
The author has very grateful to be reviewers for their comments and suggestions.