1. Introduction
Many applications of fractional calculus and for discontinuous systems can be found in biophysics, quantum mechanics, group theory, robotics or economics. The main reason for the applications of fractional calculus is that fractional-order models have more degrees of freedom and are more flexible than the integer-order ones. On the other hand, discontinuous systems model phenomena with non-smooth forces or dry frictions. As basic sources are books such as [
1,
2,
3,
4,
5,
6], various analytical and numerical methods are applied as they are demonstrated in [
7,
8,
9,
10,
11]. Moreover, the theory and applications of fractional systems are rapidly developing, supported by many recent papers and Special Issues such as [
12,
13,
14,
15,
16], involving stability, asymptotic periodicity, synchronization, memory effect and several other important and interesting behaviours of fractional models.
We work with a perturbed fractional differential equation with globally Lipschitz right-hand sides and which change their forms, i.e., they are discontinuous and transversally cross a discontinuity boundary. We are looking for a solution of a perturbed system with a periodic boundary condition in a neighbourhood of a periodic orbit of an unperturbed system. That means a solution of an unperturbed equation is periodic. We consider Caputo fractional derivatives of an order less than one. Our approach is based on the well-known method of a Poincaré map constructed near an investigated periodic orbit widely applied in dynamical systems as either smooth or non-smooth [
17,
18]. Since our studied problem has a integral-differential structure due of using together integer and Caputo fractional derivatives, the extension of Poincaré method is not elementary, but rather technical. We also believe that it will have a good reason to continue in this study.
The paper is organised as follows. In
Section 2, we introduce our problem and prove the existence of a global solution. In
Section 3, we show the existence of a Poincaré map in a neighbourhood of a periodic orbit of an unperturbed equation, which allows a bifurcation analysis of periodic boundary solutions given in
Section 4.
Section 5 demonstrates our method on a concrete example, and it discusses a possible scenario of qualitative behaviour of our example problem.
Section 6 summarises our achievements and proposes our future task.
3. Poincaré Map
The solution varies depending on . If is non-negative, we work with , otherwise we work with . We will therefore look for times and where is equal to zero, where changes to and vice versa.
Let us discuss a case, where parameters
and
are equal to 0. System (
2) will then be system of ordinary differential equations.
For
-periodic solution of system (
9) is
fulfilled. The fixed
-periodic solution of system (
9) will be denoted by
. The initial condition can be written as
The solution
can be written in the terms of
and
:
The function is of class in all its variables.
Let us suppose the existence of
and
that
and
. At these points, function
changes its form from
to
and vice versa. We work with
, therefore
will be
, which means that
will be of the following form:
It can be seen that
can be identified as a function
that
In the case where parameters
and
are not equal to 0, we obtain for
Similarly, for
where
can be seen as a function
, that
If
and
are not equal to 0, we obtain for
Similarly, for interval
First, let us look at the solution
for
=
=0. The solution can be written in the following way:
It can be easily seen, that
where the last equation is result from
T-periodicity.
The solution for
and
not equal to 0 can be written as
,
and
on corresponding intervals
,
and
, where these solutions are from (
11)–(
13).
It can be seen, that derivatives of function with respect to t are non-zero for and
The solution
can be written similarly:
Let us recall
, a solution to an unperturbed Equation (
1), i.e., for
.
therefore indicates solutions (
14) and (
15). Now, we will look for a
T-periodic boundary solution of system (
2) near
, which means that in system (
2) we take
and
as small, close to 0. To reduce the number of variables, take
and
in the form
and
, while
will be taken small close to 0, and
and
can be any. For simplicity, we will rename the variables
and
to
and
.
The system (
2) can be written in the form
that
,
and
. Now we define Poincaré map. We choose
and
.
is, similarly to
and
, function of variables
and
, that defines as
.
determines the time, when solution meets its initial value, which can be seen in
Figure 1 The Poincaré map is defined as
where
denotes the solution of (
16) with
right-hand side and
the solution of (
16) with
right-hand side. To find periodic boundary solution we solve the equation
Differentiating the Equation (
11) with respect to
at the point
we obtain
In this case, we have
, which gives us
on the right-hand side. Differentiating the Equation (
11) with respect to
at the point
, we have
Denote the first point, where
changes to
as
.
can be obtained by differentiating (
11) at
with respect to
as
Similarly, by differentiating with respect to
, we can express
as
Now, we take time
, which gives us
on the right-hand side. Differentiating (
12) with respect to
at the point
, we obtain
Differentiating (
12) with respect to and
at the point
, we obtain
For
function
changes to
. Denote by
the point, when the change happens. Differentiating (
12) at
with respect to
and
, respectively, we have:
Now, take
, and on the right-hand side of the equation is function
. Differentiating (
13) with respect to
and
at the point
, respectively, we have:
4. Bifurcation Analysis
Poincaré map fulfils:
where
. Denote by
the point, for which the periodic orbit
exists for
. Then, we consider
for
In our particular case, the function
Q equals
, where
is the point, where
Let
. It is easy to see that
solves
hence
By differentiating the equation with respect to
at the point
, two options can occur:
or
In the first option, we obtain a isolated periodic boundary orbit, which means that the equation has a unique solution that fulfils
In the second option, the bifurcation may occur. Let us look at the special, degenerated, case of the second option:
In this case, we obtain several solutions, the one-parameter system of periodic solutions of the unperturbed system. Assume that (
30) is fulfilled. Using Hadamard’s Lemma express
in the form:
The equation
is obtained. After multiplying the equation by
and giving
, we have:
According to the roots of Equation (
31), the existence of periodic boundary orbits can be discussed. If Equation (
31) has a unique non-degenerated solution
for
, we obtain a local unique periodic boundary solution.
If the equation has no solution, a periodic boundary orbit does not exist. If the equation has several solutions, we obtain several periodic boundary orbits. In our particular case of Equation (
1),
can be found by differentiating (
13) with respect to
. We use the notation
and
, as introduced earlier in this chapter. For
:
Since
,
can be expressed by differentiating
with respect to
.
Using (
22), (
26) and (
33) we can express
for
:
5. Example of the Specific Equation
In further research, we will solve the equation, where the degenerated case (
30) occurs. Using methods described in a previous chapter, we will solve specific equations in the form of (
1). Let us solve an Equation (
1) in unperturbed form.
Equation (
35) is equivalent to the system
Let the function
be such that there exists a periodic solution.
It is obvious that the periodic solution of system (
36) exists.
Figure 2 shows that the solution of the system (
36) is periodic. For
positive
, therefore, the solutions above the
axis are circles. The solutions under the
axis are ellipses, and the reason is that
. We can look for the times
and
, in which the function
changes from
to
and vice versa. The solution of the system (
36) for
equals
is the first time in which the solution
Because the time
changes for different values of
and
, we take the initial conditions as we used in the previous chapter, namely
. In this case, the solution will be in the form of:
Time
will be the time when
It is obvious that
. The system has on the interval
initial conditions
a
Therefore, the solution is in the form
Hence, the time
, in which the function
changes to
, is
Let us look for a
T-periodic solution, which means, that the solutions
and
. The solution for
equals
Therefore,
. Using the method described in the previous chapter, it is possible to find a Poincaré map for a perturbed equation
where
is determined by the relation (
37) and
can be chosen as
x, so
. Solution
x can be written as
In this case, we can express
as
where
,
and
are the first, second and third intersection points, respectively.
,
,
,
Times
and
satisfy following equations:
To use (
34) for our specific equation, we have to find functions
and
. Function
was defined (
20), (
24) and (
28) on intervals [0,
],[
,
] and [
], respectively. Knowing that
(
2) (symbol
symbolizes the time derivative), we can formulate ODE system to find functions
and
on each interval using functions (
38), (
39) and (
40) as
and
.
On interval [0,
] = [0,
], we have
with initial conditions
which, using a variation of parameters method, give the solution in the form:
where
and
. Similarly, the ODE system on the second interval [
]:
Its initial conditions can be calculated from Equation (
45) at
. The solution on interval [
] using a variation of parameters method is in the form:
where ,
and .
On the last interval [
] is the ODE system in the form
with initial conditions equal to (
48) at time
The solution on interval [
] is in the form:
where ,
and
. We do not express analytical solutions (
45), (
48) and (
51) in another form, because the given formulas lead to hypergeometric functions. For specific parameter choices, it is possible to find a numerical solution, but we do not go for details. We focus in this paper on the analytical theory of the Poincaré mapping method rather than its numerical investigation, which is interesting but postponed to another of our studies based on [
19,
20,
21,
22].
On the other hand, to understand the dynamics of (
41), we first consider its limit case for
(see [
23]), so we have
We see on
Figure 3,
Figure 4 and
Figure 5 different dynamics depending on
h. The perturbation
on
Figure 3 keeps the symmetry along the origin. It seems that the origin is a global attractor. The perturbation
on
Figure 4 breaks the symmetry along the origin. The perturbation
on
Figure 5 keeps the symmetry along the origin, but it is nonlinear, and the dynamics are more interesting and complex.
Next, we consider a limit case of (
41) for
, so we have
(
53) is no longer an ODE. For instance, if
and
, we obtain
The solutions of (
54) seem to be repelled by
Figure 6. Summarizing, the study of qualitative property of (
41) is challenging by varying
.
6. Conclusions
We work with a fractional differential equation with a discontinuous right-hand side, which is equivalent to the system (
2). Suppose that functions
are globally Lipschitz continuous, which change their form according to the sign of
and transversally cross the discontinuity boundary.
We look for a periodic boundary solution of the system in a neighbourhood of the periodic orbit of an unperturbed system. That means the solution of the unperturbed equation is periodic.
We have shown the existence of the solution of the studied equation and found the corresponding Poincaré map in a neighbourhood of the periodic orbit of the unperturbed equation. We also present a bifurcation analysis of periodic boundary solutions.
We demonstrate how to apply our found formula to a concrete problem.
In the forthcoming work, we intend to generalize this theory to higher dimensions. Some of possible directions are outlined above.