- Open Access
Symplectic difference systems with periodic coefficients: Krein’s traffic rules for multipliers
© Došlý; licensee Springer 2013
- Received: 20 December 2012
- Accepted: 11 March 2013
- Published: 2 April 2013
We investigate symplectic difference systems with periodic coefficients. We show that the known traffic rules for eigenvalues of the monodromy matrix of linear Hamiltonian differential and difference systems can be extended also to discrete symplectic systems.
- symplectic difference system
- traffic rules
- monodromy matrix
- definite multiplier
In particular, if , then we have . The terminology for symplectic matrices is not unique. Our terminology is taken from . An alternative terminology -unitary, -orthogonal can be found in . Also, complex-valued matrices satisfying (1) are sometimes called complex symplectic.
whose qualitative theory is deeply developed; see, e.g., .
- (A)System (2) is symplectic for , i.e.,
- (B)The matrices are periodic with the period , i.e.,
- (C)The matrices , , are analytic in a neighborhood of , in particular,
the stability theory of periodic Hamiltonian differential systems is deeply developed since the fifties of the last century. In that period, Russian mathematicians Gelfand, Krein, Lidskii, Yakubovich, Starzhinskii and others [2, 8, 9] published fundamental papers on stability of (3), which were summarized in the book .
The following statement plays the crucial role in the later introduced concept of a symplectic system of positive type.
Consequently, each term in the infinite series expressing is a Hermitean matrix, and hence this matrix is Hermitean as well. □
Second-order matrix difference system.
Linear Hamiltonian difference system.
and it is known (see, e.g., [, p.8]) that the logarithm of a symplectic matrix is the Hamiltonian matrix, i.e., .
Symplectic systems with nonlinear dependence on a spectral parameter.
Now we see that the sum of the terms on the first and the last line is a symmetric matrix (since ). Similarly, the sum of the second and last but one line terms is a symmetric matrix, etc. Altogether, the term by , , is a symmetric matrix, which means that the matrix Ψ is really symmetric.
where are the eigenvalues of .
The proof of the following statement (which shows the non-emptiness of the central stability zone under an additional assumption that the eigenvalues of a certain matrix are distinct) is essentially the same as that of [, Theorem 2.2].
Theorem 1 Suppose that the matrix is negative definite and the eigenvalues of are distinct. Then there exists such that the solutions of (2) are bounded on ℤ for real λ satisfying .
(since , see ), and hence and . Now, for , small, the numbers are still different. However, at the same time, for , the matrix is symplectic, i.e., , and hence its eigenvalues are symmetric with respect to the unit circle, and hence are symmetric with respect to the imaginary axis. But this implies that themselves are on the imaginary axis, i.e., . This means, in view of the later given formula (25), that solutions of (2) are bounded on ℝ for , small. □
So far we did not need any assumption on the matrices for . Complex values of λ are needed to get an extension of the previous theorem to the case when the eigenvalues of are not necessarily distinct. In this situation, the stability behavior of solutions of (2) is heavily based on Krein’s traffic rules for the eigenvalues of the monodromy matrix .
A closer examination of the proofs in  reveals that when investigating the behavior of eigenvalues of the monodromy matrix in dependence on the parameter λ, one can forget about the original system from which this monodromy matrix originates (differential system (3) or difference system (2)). The only important fact is that this matrix is symplectic for and -monotonic for . Under assumption (24), the traffic rules of eigenvalues of are the same as in [2, 6].
and generally . Consequently, solutions of (2) are bounded for if and only if all eigenvalues of have modulus one and are of simple type, i.e., multiple eigenvalues have the same algebraic and geometric multiplicity.
Clearly, is the stability point of (2) since all solutions are constant in this case. Theorem 1 states that if the eigenvalues of are distinct, we have a central stability zone for (2). The restriction on distinctness of eigenvalues is removed in  (see also [6, 7] in the discrete case) using a perturbation principle, then Theorem 1 remains true when Krein’s traffic rules for eigenvalues are preserved. For these rules, the crucial concept is the concept of a multiplier of definite type which is defined as follows.
Definition 1 Let be an eigenvalue of the monodromy matrix , and let be its eigenspace. If for every , then the eigenvalue is said to be the multiplier of the first kind and it is called of second kind if for . If there exists such that , the multiplier is of indefinite (mixed) type. If is an eigenvalue of , then it is called of the first kind if and of the second kind if .
The following statement is proved similarly as [, Theorem 4.6].
where are properly taken eigenvectors of corresponding to and are the solutions of (2) with given by the initial condition .
for the first kind multipliers and for the second kind ones. This means that the multipliers resulting from splitting of first kind multipliers move clockwise and second kind multipliers move counterclockwise with increasing , both remain on the unit circle in some neighborhood of . □
has a solution.
Here we have, similarly to , the different situation in comparison with , where the symplectic system corresponding to the second-order system (12) is considered; see the part (i) of Section 2. Since in our case is no longer a polynomial, generally it can happen that the equation has no root, and hence (2) is stable for every λ.
Finally, note that we have made only the first step towards the elaboration of a consistent stability theory of periodic symplectic difference systems in our paper. For example, we have not been able yet to establish a ‘symplectic version’ of the perturbation principle which enables to prove the statement of Theorem 1 without the additional assumption that the eigenvalues of the matrix are distinct. This is one of the problems which are subject of the present investigation, and we hope to solve it, together with other open problems, in subsequent papers.
The author thanks referees (especially one of them) for valuable remarks and suggestions which substantially contributed to the present version of the paper. The research is supported by the grant 201/10/1032 of the Czech Grant Agency (GAČR).
- Dopicio FM, Johnson CR: Complementary bases in symplectic matrices and a proof that their determinant is one. Linear Algebra Appl. 2006, 419: 772–778. 10.1016/j.laa.2006.06.014MathSciNetView ArticleGoogle Scholar
- Krein MG: The basic propositions of the theory of λ -zones of stability of a system of linear differential equations with periodic coefficients. In In Memory of Aleksandr Aleksandrovič Andronov. Izdat. Akad. Nauk. SSSR, Moscow; 1955:413–498. (Russian). English translation: Four papers on ordinary differential equations. Edited by Lev J. Leifman. Transl. Am. Math. Soc., Ser. 2 120, 1–70 (1983). Am. Math. Soc., ProvidenceGoogle Scholar
- Bohner M, Došlý O, Kratz W: Sturmian and spectral theory for discrete symplectic systems. Trans. Am. Math. Soc. 2009, 361: 3109–3123. 10.1090/S0002-9947-08-04692-8MATHView ArticleGoogle Scholar
- Kratz W: Discrete oscillation. J. Differ. Equ. Appl. 2003, 9: 135–147. In honour of Professor Allan Peterson on the occasion of his 60th birthday, Part IIMATHMathSciNetView ArticleGoogle Scholar
- Kelley WG, Peterson AC: Difference Equations: An Introduction with Applications. Academic Press, San Diego; 2001.Google Scholar
- Halanay A, Rasvan V: Stability and BVP’s for discrete-time linear Hamiltonian systems. Dyn. Syst. Appl. 1999, 9: 439–459.MathSciNetGoogle Scholar
- Rasvan V: Stability zones for discrete time Hamiltonian systems. Arch. Math. 2000, 36: 563–573.MATHMathSciNetGoogle Scholar
- Gelfand IM, Lidskii VB: On the structure of regions of stability of linear canonical systems with periodic coefficients. Usp. Mat. Nauk 1955, 10(1(63)):3–40. (Russian). English translation: Transl. Am. Math. Soc. 8, 143–181 (1958)MathSciNetGoogle Scholar
- Krein MG, Yakubovich VA: Hamiltonian systems of linear differential equations with periodic coefficients. In Analytic Methods in the Theory of Non-Linear Vibrations. Izdat. Akad. Nauk Ukrain. SSR, Kiev; 1963:277–305. (Proc. Internat. Sympos. Non-linear Vibrations, vol. I, 1961)Google Scholar
- Yakubovich VA, Starzhinskii VM 1, 2. In Linear Differential Equations with Periodic Coefficients. Israel Program for Scientific Translations, Jerusalem; 1975.Google Scholar
- Rasvan V: On the stability zones for discrete-time periodic linear Hamiltonian systems. Adv. Differ. Equ. 2006., 2006: Article ID 80757Google Scholar
- Ekeland I: Convexity Methods in Hamiltonian Mechanics. Springer, Berlin; 1990.MATHView ArticleGoogle Scholar
- Došlý, O: Symplectic difference systems: natural dependence on a parameter. Adv. Differ. Syst. Appl. (to appear)Google Scholar
- Hilscher RŠ: Oscillation theorems for discrete symplectic with nonlinear dependence in spectral parameter. Linear Algebra Appl. 2012, 437: 2922–2960. 10.1016/j.laa.2012.06.033MATHMathSciNetView ArticleGoogle Scholar
- Gantmacher FR 1, 2. In The Theory of Matrices. AMS Chelsea Publishing, Providence; 1998.Google Scholar
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