Abstract
A system of impulsive integro-differential equations representing a hybrid counterpart of the renown Goodwin oscillator is considered. The continuous part of the system possesses a cascade structure and contains a distributed bounded delay. The impulses impacting the continuous part are modulated in amplitude and frequency by the continuous output thus implementing an impulsive feedback. This kind of mathematical models appears in mathematical biology and computational medicine. Applying a version of the linear chain trick, it is demonstrated that a discrete-time (Poincaré) map can be constructed to capture the main dynamical properties of the system in hand.
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A. N. Churilov acknowledges Russian Foundation for Basic Research for Grant 14-01-00107-a and Saint Petersburg State University for a research Grant 6.38.230.2015. A. Medevedev was in part financed by the European Research Council, Advanced Grant 247035 (SysTEAM) and Grant 2012-3153 from the Swedish Research Council.
Proof of Theorem 1
Proof of Theorem 1
The following lemma plays a key role in the proof. Let \(\widetilde{K}(t)\) be a matrix valued function of a scalar argument t:
Lemma 2
Let \(t_n<t<t_{n+1}\) where \(n\ge 2\). Then any solution x(t) of (18), (19) satisfies the ordinary differential equation
where the vector-valued function \(\eta (t)\) is defined as follows:
Case (i) \(T_n>h\), \(T_{n-1}>h\). Then
Case (ii) \(T_n<h\), \(T_{n-1}>h\). Then
Case (iii) \(T_n<h\), \(T_{n-1}<h\). Then
Case (iv) \(T_n>h\), \(T_{n-1}<h\). Then
Proof of Lemma 2
From (13), one gets
Now (6) can be rewritten as
Without loss of generality (cf. Lemma 1), let the continuous part of (18) be readily given in the form of (14). Let also the matrix B be partitioned as
where the dimensions of the vectors \(B_1\), \(B_2\) correspond to those of u, v, respectively. The matrix D can be represented as (16) and
It follows from (14) that
where asterisks stand for any matrix blocks of appropriate size. Hence (36) and (37) yield
where
\(u_d(t)\) is defined by (15) and \(*\) is any column of a suitable size. Equation (38) holds for any time interval where the function x(t) is smooth, i.e., in any interval that does not contain the points \(t_0,t_1,\ldots \).
Define the time intervals
For brevity, denote \(u_k^- = u(t_k^-)\), \(u_k^+ = u(t_k^+)\). One has \(u_k^+= u_k^- +\lambda _k B_1\) and
Suppose \(t\in L_n\), \(n\ge 2\). To prove Lemma 2, it suffices to show that \(\eta (t)\) can be chosen as
(a) Let \(t-h\in L_n\), \(t-\tau \in L_n\). Since \(t\in L_n\), (39) implies \(u(t) = \mathrm {e}^{U(t-t_n)} u_n^+\). Moreover, since \(t-s\in L_n\) for \(\tau \le s\le h\) and from (39), one obtains
Hence x(t) is a solution of (8) and \(\eta (t)\equiv 0\) is justified.
(b) Let \(t-h\in L_{n-1}\), \(t-\tau \in L_{n-1}\). Hence \(t-s\in L_{n-1}\) for \(\tau \le s\le h\) and making use (39) yields
Since \(u_n^- = u_n^+ -\lambda _n B_1\), one gets
Thus
Since
where asterisks stand for some blocks, \(\eta (t)= \lambda _n \mathrm {e}^{A_0(t-t_n)} B\) can be taken in (38).
(c) Let \(t-h\in L_{n-1}\), \(t-\tau \in L_n\). Then \(t-h<t_n<t-\tau \), so \(t-s\in L_n\) if \(\tau <s<t-t_n\) and \(t-s\in L_{n-1}\) if \(t-t_n<s<h\). Hence
Then
Using (40), rewrite the previous line as
leading to
Thus \(\eta (t)\) in (38) can be chosen as
(d) Let \(t-h\in L_{n-2}\), \(t-\tau \in L_{n-1}\). Hence \(t-h<t_{n-1}<t-\tau \), so \(t-s\in L_{n-1}\) if \(\tau <s<t-t_{n-1}\) and \(t-s\in L_{n-2}\) if \(t-t_{n-1}<s<h\). Hence
Since
it can be concluded that
and hence
It is deduced from (40) that
Thus
can be taken in (38),
The proof of Lemma 2 is complete. The corollary below follows immediately. \(\square \)
Corollary 1
The function \(\eta (t)\) in Lemma 2 is continuous in \((t_n,t_{n+1})\) and piecewise smooth. For \(n\ge 2\) in its intervals of smoothness \(\eta (t)\) satisfies the differential equation
where H(t) is defined as follows
Case (i) \(T_n>h\), \(T_{n-1}>h\). Then
Case (ii) \(T_n<h\), \(T_{n-1}>h\). Then
Case (iii) \(T_n<h\), \(T_{n-1}<h\). Then
Case (iv) \(T_n>h\), \(T_{n-1}<h\). Then
Now all the prerequisites for the proof of Theorem 2 are in place.
For \(t_n<t< t_{n+1}\), \(n\ge 2\), introduce the function \(z(t)=x(t)-\eta (t)\), where x(t) is a solution of (18), (19). From (35) and Corollary 1, on those subintervals of \((t_n,t_{n+1})\), where \(\eta (t)\) is smooth, it applies that
Since \(x(t^-_{n+1})=\bar{x}_{n+1}\), \(x(t^+_n)=\bar{x}_n+\lambda _n B\), from (41) it follows
Consider four previously defined cases (i)–(iv).
Case (i) From Lemma 2, the following boundary conditions are obtained
Since
(42) takes the form
This implies the claim of Theorem 2 for case (i).
Case (ii) The boundary conditions are
Then (42) takes the form
Since
one finally arrives at
where G(t) is defined by (22). Since \(T_n={\varPhi }(C\bar{x}_n)\) and \(\lambda _n=F(C\bar{x}_n)\),
Case (iii) The boundary conditions are
Since
and
equality (42) takes the form
This implies
Case (iv) The boundary conditions are
Hence
The proof of Theorem 2 is complete.
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Churilov, A.N., Medvedev, A. Discrete-time map for an impulsive Goodwin oscillator with a distributed delay. Math. Control Signals Syst. 28, 9 (2016). https://doi.org/10.1007/s00498-016-0160-y
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DOI: https://doi.org/10.1007/s00498-016-0160-y