On a Smale theorem and nonhomogeneous equilibria in cooperative systems

Enciso, Germán

doi:10.1090/s0002-9939-08-09346-5

Cited by 3 publications

(2 citation statements)

References 15 publications

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“…It states in this context that any compactly supported, ðn 2 1Þ-dimensional, C 1 dynamical system defined on H ¼ {x [ R n jx 1 þ · · · þ x n ¼ 0} can be embedded into some cooperative C 1 system (2). Equivalently, the dynamics of cooperative systems can be completely arbitrary on unordered hyperplanes such as H. See also [5], where the cooperative system (2) is shown to have bounded solutions and has only two equilibria outside of H.…”

Section: The Smale and Hirsch Theoremsmentioning

confidence: 99%

Analogues of the Smale and Hirsch theorems for cooperative Boolean and other discrete systems

Enciso

Just

2012

Journal of Difference Equations and Applications

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Dedicated to Avner Friedman, on the occasion of his 75th birthday Discrete dynamical systems defined on the state space P ¼ {0; 1; . . . ; p 2 1} n have been used in multiple applications, most recently for the modelling of gene and protein networks. In this paper, we study to what extent well-known theorems by Smale and Hirsch, which form part of the theory of (continuous) monotone dynamical systems, generalize or fail to do so in the discrete case.We show that arbitrary m-dimensional systems cannot necessarily be embedded into n-dimensional cooperative systems for n ¼ m þ 1, as in the Smale theorem for the continuous case, but we show that this is possible for n ¼ m þ 2 as long as p is sufficiently large.We also prove that strict cooperativity, a natural weakening of the notion of strong cooperativity, implies non-trivial bounds on the lengths of periodic orbits in discrete systems and imposes a condition akin to Lyapunov stability on all attractors. Finally, we explore several natural candidates for definitions of irreducibility of a discrete system. While some of these notions imply the strict cooperativity of a given cooperative system and impose even tighter bounds on the lengths of periodic orbits than strict cooperativity alone, other plausible definitions allow the existence of exponentially long periodic orbits.

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Section: The Smale and Hirsch Theoremsmentioning

confidence: 99%

Analogues of the Smale and Hirsch theorems for cooperative Boolean and other discrete systems

Enciso

Just

2012

Journal of Difference Equations and Applications

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“…+ x n = 0} can be embedded into some cooperative C 1 system (2). Equivalently, the dynamics of cooperative systems can be completely arbitrary on unordered hyperplanes such as H. See also [5], where the cooperative system (2) is shown to have bounded solutions and only two equilibria outside of H.…”

Section: Introductionmentioning

confidence: 99%

Analogues of the Smale and Hirsch Theorems for Cooperative Boolean and Other Discrete Systems

Just,

Enciso

2007

Preprint

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Asymptotic stability criterion for nonlinear monotonic systems and its applications (review)

Martynyuk

2011

Int Appl Mech

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Introduction. The comparison method in the qualitative theory of equations is of paramount importance in studies of large-scale systems. This method is based on differential inequalities such as the Chaplygin-Wazewski inequality and Lyapunov functions (scalar, vector, or matrix-valued), which play the role of nonlinear transformation of the original system to an equation (a system or a matrix system) of lower dimension. The comparison method states that if for the system in question, there exists a Lyapunov function (scalar, vector, or matrix-valued) satisfying appropriate conditions, then various dynamic properties of the solutions of the original system follow from the respective dynamic properties of the comparison system.Thus, the stability analysis of a dynamic system involves setting up a Lyapunov function and establishing stability criteria for the zero solution of the comparison system.The present review discusses some approaches to solving such problems, establishes a uniform asymptotic stability criterion for a nonlinear comparison system, and exemplifies its applications in modern nonlinear dynamics.1. Problem Formulation. Let x t ( )be an n-dimensional state vector of some mechanical system with a finite number of degrees of freedom. Its behavior is described by the perturbed equations of motionwhere a, b are positive constants and j( ) t is a bounded nonnegative function. This equation cannot be integrated analytically, but using it leads to a number of interesting results for both system (8) and uncertain systems (see [30] and the references therein).3. System of Comparison with Vector Lyapunov Function. The curse of dimensionality in the sense of Bellman (see [8]) in the qualitative theory of equations makes it impossible to set up a scalar Lyapunov function for a large-scale system. In these conditions, Kron developed diakoptics [22] for the analysis of complex systems by parts. This method is based on the concept of a vector Lyapunov function and corresponding comparison system.Recall the following concept from the theory of monotonic systems (see [98]).

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On a Smale theorem and nonhomogeneous equilibria in cooperative systems

Cited by 3 publications

References 15 publications

Analogues of the Smale and Hirsch theorems for cooperative Boolean and other discrete systems

Analogues of the Smale and Hirsch theorems for cooperative Boolean and other discrete systems

Analogues of the Smale and Hirsch Theorems for Cooperative Boolean and Other Discrete Systems

Asymptotic stability criterion for nonlinear monotonic systems and its applications (review)

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