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

Enciso, Germán; Just, Winfried

doi:10.1080/10236190903357535

Cited by 5 publications

(6 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where y y y m = ( , Recall that for the differentiable function g y ( )to quasimonotonically increase, it is necessary and sufficient that ¶ ¶ By virtue of the assumptions, the nonnegative cone ¡ + m is an invariant set for system (61).…”

Section: Stability Of Nonlinear Monotonic Systemsmentioning

confidence: 99%

“…The Martynyuk-Obolenskii (MO) condition is satisfied for system (61) if there exists a positive vector q ÎG such that g ( ) q < 0.…”

Section: Stability Of Nonlinear Monotonic Systemsmentioning

confidence: 99%

“…Let there exists a vector q such that q > 0, || || q < H, g (q) < 0. Then the set K -(q)is a positively invariant set for system (61) and is in the attraction domain of the equilibrium position y = 0. Proof.…”

Section: Theorem 17mentioning

confidence: 99%

“…(61) have the following properties: C1. A unique solution passes through each point of the domain G. C2.…”

mentioning

confidence: 99%

“…Then the zero solution of system(61) is asymptotically stable in a nonnegative cone. Theorem 17 is proved.…”

mentioning

confidence: 99%

See 4 more Smart Citations

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

Martynyuk

2011

Int Appl Mech

View full text Add to dashboard Cite

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]).

show abstract

Section: Stability Of Nonlinear Monotonic Systemsmentioning

confidence: 99%

“…The Martynyuk-Obolenskii (MO) condition is satisfied for system (61) if there exists a positive vector q ÎG such that g ( ) q < 0.…”

Section: Stability Of Nonlinear Monotonic Systemsmentioning

confidence: 99%

Section: Theorem 17mentioning

confidence: 99%

“…(61) have the following properties: C1. A unique solution passes through each point of the domain G. C2.…”

mentioning

confidence: 99%

“…Then the zero solution of system(61) is asymptotically stable in a nonnegative cone. Theorem 17 is proved.…”

mentioning

confidence: 99%

See 3 more Smart Citations

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

Martynyuk

2011

Int Appl Mech

View full text Add to dashboard Cite

show abstract

Cooperative Boolean systems with generically long attractors I

Just

Malicki

2013

Journal of Difference Equations and Applications

View full text Add to dashboard Cite

We study the class of cooperative Boolean networks whose only regulatory functions are COPY, binary AND, and binary OR. We prove that for all sufficiently large N and c < 2 there exist Boolean networks in this class that have an attractor of length > c^N whose basin of attraction comprises an arbitrarily large fraction of the state space. The existence of such networks contrasts with results on various other types of dynamical systems that show nongenericity or absence of non-steady state attractors under the assumption of cooperativity.Comment: This is an Author's Original Manuscript of an article submitted for consideration in the Journal of Difference Equations and Applications [copyright Taylor & Francis]; Journal of Difference Equations and Applications is available online at http://www.tandfonline.com/doi/abs/10.1080/10236198.2012.69116

show abstract

Cooperative Boolean systems with generically long attractors II

Just

Malicki

2013

Adv Differ Equ

View full text Add to dashboard Cite

The purpose of this paper is to investigate to what extent cooperativity, that is, the absence of negative interactions, in Boolean networks with synchronous updating, imposes limits on chaos-like properties that are possible in such systems. Our focus is on notions of sensitive dependence on initial conditions, or a combination of sensitive dependence and large basins of attraction of exponentially long attractors, both of which are well-recognized hallmarks of chaotic dynamics in the Boolean context.We prove that a strong notion of sensitive dependence on initial conditions that formalizes decoherence along the attractor is precluded by cooperativity. Weaker notions of sensitive dependence that formalize decoherence at some time during the trajectory and sensitive dependence of the basin of attraction on initial conditions, respectively, are shown to be consistent with cooperativity, but if each regulatory function is binary AND or binary OR, in N-dimensional networks they impose an upper bound of ≈ √ 3 N on the lengths of attractors that can be reached from a fraction p ≈ 1 of initial conditions. The upper bound is shown to be optimal. These results indicate that the transfer of analogous results for differential equations models crucially depends on the precise conceptualization of chaos in the Boolean context. MSC: 34C12; 39A33; 94C10

show abstract

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

Cited by 5 publications

References 25 publications

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

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

Cooperative Boolean systems with generically long attractors I

Cooperative Boolean systems with generically long attractors II

Contact Info

Product

Resources

About