On sharp conditions for the global stability of a difference equation satisfying the Yorke condition

Nenya, O. I.; Тkachenko, V. І.; Trofymchuk, S. I.

doi:10.1007/s11253-008-0043-6

Cited by 2 publications

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References 15 publications

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“…The following theorem is a corollary of Lemma 3.2 and the results of Nenya and Nenya et al [25,26,27] for more general, non-autonomous difference equations with Yorke and generalized Yorke condition.…”

mentioning

confidence: 66%

“…Besides of that from a mathematical point of view, global stability of a unique equilibrium point is always a fundamental topic, in neural networks it is also important in solving optimization and signal processing problems. There is a vast number of papers giving sufficient conditions for global stability of more complicated models of neural networks (see in [4,10,21,34,35] and the references therein) and of more general difference equations (see in [13,25,26,27,19]), without attempting to be comprehensive; but to the best of our knowledge, none of those are claimed to be necessary.…”

Section: Introduction Consider the Difference Equation Given Bymentioning

confidence: 99%

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Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model

Bartha¹,

Garab²

2014

Journal of Computational Dynamics

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We consider the global asymptotic stability of the trivial fixed point of the difference equation x n+1 = mxn − αϕ(x n−1), where (α, m) ∈ R 2 and ϕ is a real function satisfying the discrete Yorke condition: min{0, x} ≤ ϕ(x) ≤ max{0, x} for all x ∈ R. If ϕ is bounded then (α, m) ∈ [|m| − 1, 1] × [−1, 1], (α, m) = (0, −1), (0, 1) is necessary for the global stability of 0. We prove that if ϕ(x) ≡ tanh(x), then this condition is sufficient as well.

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confidence: 66%

Section: Introduction Consider the Difference Equation Given Bymentioning

confidence: 99%

Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model

Bartha¹,

Garab²

2014

Journal of Computational Dynamics

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“…Upper restrictions like q ≤ q k on q are intrinsic to (1) and cannot be omitted for k > 1 since (4) does not guarantee even the local stability for q close to 1. However, as we show below, the values of q k are not optimal.Indeed, it is natural to conjecture (see [2], [3], [6]) that interval (0, q k ] in Theorem 2 can be improved till (0, q * k ], where q * k is the unique positive root of the equation (q * k ) k+1 (q * k + ... + (q * k ) k ) = 1. For comparison, let us show values of q * k : q * 1 = 1, q * 2 = 0.855, q * 3 = 0.831, q * 4 = 0.828, q * 5 = 0.833, q * 6 = 0.839, q * 7 = 0.846, q * 8 = 0.853, q * 9 = 0.859, q * 10 = 0.865, q * 100 = 0.967, q *…”

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confidence: 99%

Global attractivity conditions for a single species model

Тkachenko

Trofimchuk

2007

Proc Appl Math and Mech

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We give sharp conditions for global attractivity of the zero solution of the nonlinear difference equation which arises in many contexts in mathematical biology xn+1 = qxn + fn (xn, ..., x n−k ), n ∈ Z, where q ∈ (0, 1] and functions fn satisfy the generalized Yorke conditions.We study the following difference equationIt is assumed that q ∈ (0, 1] and that f n : R k+1 → R, n ∈ Z, meet the generalized Yorke condition (H):The monotone continuous functional M :Eq.(1) has the unique steady state solution x = 0. The property (H2) is satisfied by f n possessing the negative Schwarzian. If (H2) holds with b = 0, then (H1) is satisfied automatically with ϑ(z) = −aM (−z).The generalized Yorke condition was introduced in [1]. It determines a wide family of scalar functional differential equations (SFDE's) which includes various important models studied in mathematical biology (e.g. let us mention the Wright, Nicholson, and Mackey-Glass equations). In particular, this family includes (1) which can be considered as a SFDE with piecewise-constant argument. In [1] we can find a simple criterion for the global stability of the unique fixed point of SFDEs considered under the generalized Yorke condition. For the particular case of Eq. (1), it reads as: Theorem 1. Suppose that q ∈ (0, 1) and f n satisfy (H). If b = 0 and the conditionholds, then lim n→∞ x n = 0 for every solution {x n } of Eq. (1). However, Theorem 1 is not the best possible global stability result for the family of difference equations satisfying (H) (we will say that Theorem 1 is not sharp, see also Theorem 2). This makes a notable difference with the main statement of [1]. Theorem 2. Assume that b = 0 and that f n , n ∈ Z, satisfy (H). Then there exists q k ∈ (0, 1) such that for q ∈ (0, q k ] the inequality assures the global attractivity of Eq. (1). Condition (4) is sharp within the class of Eqs. (1) determined by (H)and the assumptions b = 0 and q ∈ (0, q k ] : for every triple k ∈ Z + , a < 0, q ∈ (0, q k ], which does not meet (4) there exists a periodic sequence {f n } satisfying (H) and such that the zero solution of the corresponding equation is linearly unstable. Our paper [3] contains formulae from which q k can be found explicitly, let us list some values of q k : q 1 = 0.887; q 2 = 0.796; q 3 = 0.788; q 4 = 0.795; q 5 = 0.805; q 6 = 0.815; q 7 = 0.825; q 8 = 0.834; q 9 = 0.842; q 10 = 0.849; q 100 = 0.965; q 1000 = 0.994. Upper restrictions like q ≤ q k on q are intrinsic to (1) and cannot be omitted for k > 1 since (4) does not guarantee even the local stability for q close to 1. However, as we show below, the values of q k are not optimal.Indeed, it is natural to conjecture (see

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On sharp conditions for the global stability of a difference equation satisfying the Yorke condition

Cited by 2 publications

References 15 publications

Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model

Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model

Global attractivity conditions for a single species model

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