Abstract
We study two coupled linear delay differential equations (DDEs) with additive impulses at regular time intervals. The equations are transformed to a DDE coupled to an ODE. Conditions are found for positive periodic solutions, and some examples are given for periodic solutions and for non-periodic solutions.
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1 Introduction
Periodic solutions to delay differential equations (DDE) have been studied by analogy to Floquet theory of ODE [1], by lower and upper solutions [2], by Lyapunov’s second method and the contraction mapping principle [3], or by fixed point arguments [4,5,6]. In this work, we use the results of [4] to investigate the conditions for periodic solutions for the following linear DDE with impulses:
where the constant time delay satisfies: \( r> 0 \), and \(t_0 \) is related to the initial time value \(t_{in} \) by \( t_0 - r \ge t_{in} \). The impulses are assumed to be additive, as in [6]. The arrays in (1) are defined as follows:
Here, \( b_1 \) is a constant, and the time interval T is the common period of the functions h(t), \(a_1 (t)\), \(a_2(t)\).
This is given together with the initial condition
for \( \quad t_0 - r< t < t_0 \quad \), with \( x_{1}( t_0 ) = m_1 \qquad x_{2}( t_0 ) = m_2 \).
2 Solutions for the DDE
In order to simplify the treatment of the coupled equations presented above, we define the transformation
Then Eq. (1) leads to
There are still two coupled functions, but now only one function, \( y_1 (t) \), satisfies a DDE, whereas \( y_2 (t) \) satisfies an ODE. Impulses can be considered for each of these functions. The initial conditions for the two functions are
for \( \quad t_0 - r< t < t_0 \quad \), with \( y_{1}( t_0 ) = \frac{1}{2} \left( m_1 + m_2 \right) \qquad y_{2}( t_0 ) = \frac{1}{2} \left( m_1 - m_2 \right) \)
Using the notation
the function \( y_2 (t)\) is calculated (for \( t_0 < t \)) as
Note that if the initial conditions include \( m_1 = m_2 \), then \( y_2 (t_0) =0 \) so the function remains zero for all times. We shall assume here that \( m_1 \ne m_2 \) so that \( y_2 (t) \) is not trivial.
Proposition 1
For the solution of Eq. (9), if
then the solution is periodic. Otherwise, if the solution is modified by adding for each \( t_k \) ( k = 1, 2, ...) the impulse
the resulting modified solution is periodic. If the function a(t) is continuous in the interval \( [ t_k , t_k + T ] \), then the solution y(t) is bounded.
The periodicity is checked by the evolution of the solution between \( t_k \) and \( t_{k+1} = t_k + T \):
In the trivial case where
the solution for \( y_2 (t) \) is already periodic, without any need for impulses. If
and no impulses are applied, then the solution tends to zero for \( t \rightarrow \infty \), so that the zero solution is stable, but there is no periodicity. If
and no impulses are applied, the solution diverges as \( t \rightarrow \infty \). In the last two cases, if the additive impulse
is applied at the times \( t_k = k \cdot T \), i.e.,
then \( y_2 (t) \) is periodic. If the function a(t) is continuous in each interval \( [ t_k , t_k + T ] \), then \( y_2 (t) \) is bounded.
Equation (8) for \( y_1 (t) \) will be re-written as
where \( b(t) \equiv 2 \cdot b_1 \cdot h(t) \). In [4], Schauder’s fixed point theorem is used in order to prove that if there exists a continuous function w(t) such that
and also
then there is a positive periodic solution to the homogeneous part of Eq. (15). A way to construct the solution is given in [4]. If this periodic solution is denoted by \( y_0 (t) \), then the solution to full Eq. (15) is
where X(t, s) is the fundamental solution to Eq. (15) [7]. This solution evolves over one period of \( y_0 (t) \) as
Proposition 2
For the equation as Eq. (15) above, if
then the solution to the equation is positive and periodic. If the functions b(t) and a(t) are continuous in each interval \( [ t_k , t_k + T ] \) (with at most a finite number of finite discontinuities), then the solution is bounded.
Note: If the integral in Eq. (19) is not zero, stability for Eq. (15) can hold if: (a) the integral tends to zero as \( t \rightarrow \infty \) and (b) the equation for \(y_0 (t) \) is stable. The stability of \(y_0 (t) \) can be checked as in [8]. However, if the integral in Eq. (18) diverges for \( t \rightarrow \infty \), then the equation for \( y_1 (t) \) is not stable, even if the equation for \( y_0 (t) \) is stable.
The original Equation (1) is solved (for \( t_k < t \le t_k + T \)) by
where the properties of the individual terms ( \(y_2 \) and \(y_1 \) ) determine the properties of the original variables \( x_1 (t) , x_2 (t) \).
3 Examples
3.1 Example 1
Consider a delay of \(\qquad r = 6 \pi \qquad \) and the following functions:
\( h(t) = \mathrm{cos}(t) \)
\( a_1 (t) = c_0 + c_1 \cdot \mathrm{cos}(t), \qquad a_2 (t) = c_2 \cdot \mathrm{cos}(t) \) where \( c_0 , c_1 , c_2 \) are constants. For initial conditions, let us choose \( m_1 \ne m_2 \), so that \( y_2 ( t_0 ) \ne 0 \) and take \( t_0 = 0 \). Then for \( 0 < t \),
As for \(y_1 (t) \), the solution for the homogenous equation of Eq. (15) can be obtained by choosing \( w(t) = 1 \), and then Eqs. (16) and (17) become
The solution for the homogeneous equation of \( y_1 (t) \) is
so that the fundamental solution is
The integral in Eq. (18) is
3.1.1 Case 1.A
If \( \qquad c_1 = c_2 \), the only contribution to this integral in the \( a_1 - a_2 \) term is from \( c_0 \).
Substituting in Eq. (18), one gets
The integral term \( J \equiv y_{1} (t) - y_0 (t) \) is equal (for \( t_0 = 0 \)) to
The result of the integral is a non-periodic function, so calculating the integral between the limits:\( \qquad t_k \) and \( t_k + T \qquad \) will not give zero. In the special case
\(b_1 = c_1 \), the integral term is much simpler, but still the result is not periodic.Thus, the function \( y_1 (t) \) is not periodic, unlike \( y_0 (t) \). Then the original variables \( x_1 (t) \) and \( x_2 (t) \) are a combination of a periodic part (that of \( y_2 (t) \) and \( y_0 (t) \)) and a non-periodic part (J). If one adds an impulse to \( y_1 (t) \):
this will correct the value of the function only for a single time point. Due to the dependence on the time delay, the behavior of the function for the next time interval will in general be different from that in the previous interval, so \( y_1 (t) \) will remain non-periodic. Therefore, in both cases, \(b_1 \ne c_1 \) and \(b_1 = c_1 \), the solution diverges for \( t \rightarrow \infty \).
3.1.2 Case 1.B
Now assume \( c_1 \ne c_2 \) and \( c_0 = 0 \). Now the integral term is equal to
If \( \qquad 2 b_1 \ne c_1 + c_2 \), then the expression above is equal to
and this is a periodic function, so that also \(y_1 (t) \) is periodic. If \( \qquad 2 b_1 = c_1 + c_2 \), then the expression is
which is also periodic. Thus, regardless of the value of \( b_1 \), the solution is periodic, both for \(x_1 (t) \) and for \( x_2 (t) \).
3.2 Example 2
With the time delay: \( r = \frac{\pi }{2} \), consider the following functions:
and \( a_1 (t) , a_2 (t) \) as in the previous example. Then \( y_2 (t) \) is the same as above, and for \( y_1 (t) \), we define the function \( w(t) = exp \{ 2 b_1 \cdot \left( \mathrm{cos}(t) - \mathrm{sin}(t) \right) \} \).
Then Eqs. (16) and (17) become
Now the solution for the homogenous equation of Eq. (15) is
so that the fundamental solution is
3.2.1 Case 2.A
If \( c_1 = c_2 \), the only contribution to this integral is from \( c_0 \).
Substituting in Eq. (17), one gets
The integral term \( J \equiv y_{1} (t) - y_0 (t) \) is equal to
The result of the integral is a non-periodic function, so calculating the integral between the limits:\( \quad t_k \) and \( t_k + T \) will not give zero. In fact, for this case,
where \( I_0 (x) \) is the modified Bessel function of order zero.
Thus, the function \( y_1 (t) \) is not periodic, unlike \( y_0 (t) \). Then the original variables \( x_1 (t) \) and \( x_2 (t) \) are a combination of a periodic part (that of \( y_2 (t) \) and \( y_0 (t) \)) and a non-periodic part (J).
3.2.2 Case 2.B
Now assume \( c_1 \ne c_2 \) and \( c_0 = 0 \). Now the integral term is equal to
The result of the integration is not a periodic function. Also, for the special case \( \left( c_1 + c_2 \right) = 0 \), the result is not periodic, and in that case,
where \( I_1 (x) \) is the modified Bessel function of order one. Thus, regardless of the value of \( b_1 \), the solution is not periodic, both for \(x_1 (t) \) and for \( x_2 (t) \) . The solutions diverge for \( t \rightarrow \infty \).
3.3 Example 3
With the time delay: \( r = \frac{\pi }{2} \) consider the following functions:
and
\( a_1 (t) = c_0 + c_1 \cdot \mathrm{sin}(t) , \qquad a_2 (t) = c_2 \cdot \mathrm{sin}(t) \) where \( c_0 , c_1 , c_2 \) are constants. Then for \( 0 < t \),
and for \( y_1 (t) \), we define the function \( w(t) = exp \{ 2 b_1 \cdot \left( \mathrm{cos}(t) - \mathrm{sin}(t) \right) \} \).
Then Eqs. (16) and (17) are the same as in Example 2 above,
and also the solution for the homogenous equation of Eq. (15) and consequently the fundamental solution are the same as in Example 2 above.
3.3.1 Case 3.A
If \( c_1 = c_2 \), the only contribution to this integral is from \( c_0 \).
Substituting in Eq. (17), one gets
The integral term \( J \equiv y_{1} (t) - y_0 (t) \) is equal to
Thus, the function \( y_1 (t) \) is not periodic, unlike \( y_0 (t) \).
3.3.2 Case 3.B
Now assume \( c_1 \ne c_2 \) and \( c_0 = 0 \). Now the integral term is equal to
If \( \qquad 2 b_1 + c_1 + c_2 \ne 0 \), then the expression above is equal to
and this is a periodic function, so that also \(y_1 (t) \) is periodic. If \( \qquad 2 b_1 = c_1 + c_2 \), then the expression is
which is also periodic. Thus, regardless of the value of \( b_1 \), the solution is periodic, both for \(x_1 (t) \) and for \( x_2 (t) \).
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Gamliel, D. (2021). Periodic Solutions for a Class of Impulsive Delay Differential Equations. In: Domoshnitsky, A., Rasin, A., Padhi, S. (eds) Functional Differential Equations and Applications. FDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 379. Springer, Singapore. https://doi.org/10.1007/978-981-16-6297-3_20
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