A contribution to the theory of Lyapunov exponents for linear systems of differential equations

Sergeev, N. N.

doi:10.1007/bf01084752

Cited by 8 publications

(11 citation statements)

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“…Remark 2. The corresponding assertion for minimal exponents [4] was proved in [10]. In other words, Theorem 2 claims that the estimate (2) is uniform both on solutions in the corresponding subspaces and on systems in some ball ε-neighborhood of the system A in the metric .…”

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confidence: 89%

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Some properties of majorants of Lyapunov exponents for systems with unbounded coefficients

Bykov

2014

Diff Equat

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A number of well-known results on majorants of Lyapunov exponents of linear differential systems with bounded coefficients on the half-line (such as their simultaneous attainability in the class of infinitesimal perturbations, belonging to the second Baire class in the compact-open topology, etc.) are generalized to systems with unbounded coefficients.For n ∈ N, consider the set M n of linear differential systemṡwith continuous (not necessarily bounded) operator functions A : R + → End R n (which are identified with the corresponding systems, and we write A ∈ M n ). We equip the set M n with the structure of a linear space over R with the natural operations of multiplication by a number and addition and with the uniform topology (the topology of uniform convergence of the coefficients of the systems on the half-line) defined by the metric (A, B) = sup t∈R + min{|A(t) − B(t)|, 1}, where we have set |Y | = sup{|Y x| : |x| = 1}, Y ∈ End R n , and |x| = x 2 1 + · · · + x 2 n , x = (x 1 , . . . , x n ) T . This topological space is denoted by M n U . By M n we denote the subset of M n that consists of all systems with bounded coefficients on the half-line. Definition 1. The Lyapunov exponents of system (1) are the numbers [1]where G i (R n ) is the set of i-dimensional subspaces of the space R n , the mapping Y | L is the restriction of the mapping Y to the set L, and X A (· , ·) is the Cauchy operator of system (1).In our notation, unlike [1], the Lyapunov exponents are numbered in nondescending order. Since we do not assume that the coefficients of the systems are bounded on the half-line, it follows that their Lyapunov exponents are points of the extended numerical line R ≡ R {−∞, ∞}. (As usual, we assume that −∞ < r < ∞ for each r ∈ R and −∞ < ∞.) If all Lyapunov exponents of system (1) are finite, then they coincide with the numbers defined in [2, p. 63].1279

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confidence: 89%

“…Dropping the requirement of boundedness of the coefficients requires using different methods than in [2][3][4].…”

mentioning

confidence: 99%

Some properties of majorants of Lyapunov exponents for systems with unbounded coefficients

Bykov

2014

Diff Equat

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“…Since the sequence {ψ m } is bounded, we can extract a convergent subsequence (see, e.g., [50]). In what follows, without loss of generality we assume that {ψ m } itself is convergent.…”

Section: Lemmamentioning

confidence: 99%

“…Recall (see, e.g., [44,50]) that the partition of the solution space S(C) of Eq. (22) into a direct sum of subspaces…”

Section: Necessary and Sufficient Conditions Of Decomposability Of Spmentioning

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Decomposition of models of special classes of complex dynamical systems

Makarov¹,

Ахрем²,

Rakhmankulov³

2011

Comput Math Model

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Mobility limits of the Lyapunov exponents of linear systems under small average perturbations

Сергеев¹

1989

J Math Sci

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A contribution to the theory of Lyapunov exponents for linear systems of differential equations

Cited by 8 publications

References 3 publications

Some properties of majorants of Lyapunov exponents for systems with unbounded coefficients

Some properties of majorants of Lyapunov exponents for systems with unbounded coefficients

Decomposition of models of special classes of complex dynamical systems

Mobility limits of the Lyapunov exponents of linear systems under small average perturbations

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