In the same authors designed an algorithm to explain the construction of these functions. There are sorne systems where the Lyapunov function is defined in a natural way, like in the case of electrical or mechanical systems where energy is often a Lyapunov function.
In mathematical biology, more precisely in population dynamic modeled through the mass action law, the functions of Goh type. Goh Goh, used the function defined in 2 to prove global stability in mutualism models of the form.
In this paper we establish global stability properties for the dynamical system 1 following the same ideas of S. Goh in Goh, Calculus and linear algebra. Theorem 2. Similarly, if Hf x 0 is negative definite, then x0 is a relative maximum of f. Theorem 3.
Stability, Symbolic Dynamics, and Chaos, 2nd Edition
Sylvester's Criterion. A real symmetric matrix is positive definite positive. In this section we establish a test for the asymptotic stability of the system 1 equilibrium when is an open subset of. The following proposition relates the equilibrium stability with the sign of certain determinants.
Proposition 1. Let be an open subset of containing Suppose that the function defined in 1 satisfies and be the determinants defined by. On the other hand, the i-th term of 2 is:. The derivative of g is given by the gradient vector. Therefore is a critical point of g. Since Hg is a symmetric matrix, and assuming that all its principal minors are positive, then from Theorem 3 we have that is a local minimum of g on.
Corollary 4. If the Hessian matrix Hg x defined in 6 evaluated at is positive definite, then is globally asymptotically stable on and unstable when Hg is negative definite. The following theorem summarizes the main result of this work.
Informational structures: A dynamical system approach for integrated information
The novelty of next test consists in replacing the expertise of the authors to find the constants a i defined in 2 for conditions easy to verify. Theorem 5 Stability Test. Let be an equilibrium solution of nonlinear system 1. In consequence, from Corollary 4 we conclude that is globally asymnt.
In this section we will apply the Theorem 5 to prove the asymptotic stability of nontrivial equilibrium of the nonlinear system. Lemma 6. The set 1 defined in 15 is positively invariant for the solutions of the system Then, from 14 we obtain:. Proposition 2. The equilibrium solutions of 14 are given by the solutions of the algebraic system.
Therefore, there are at least two equilibriums in int 1. This completes the proof. Proposition 3. The above implies that the second hypothesis of Theorem 5 is satisfied. Therefore is globally asymptotically stable on interior set of 1. Figure 1. In this sense, the DML is very practical and widely used to analyze the stability of dynamical systems.
In this article we use the DML to establish easier conditions to verify the assurance of global asymptotic stability of the equilibrium solutions of some dynamical systems. The fact that these conditions are defined in terms of suggest the possibility that the stability test Theorem 5 can be used to numerically verify asymptotic stability. We want to thank to anonymous referees and Dr. Esteva for their valuable comments and suggestions that helped us to improve the paper.
Alexandrov, A. Differential Equations, 41 3 , Artstein, Z. Uniform Asymptotic Stability via the Limiting Equations.
Barbashin, E. Lyapunov Functions. Nauka, Moscow.
Escobar, C. Giesl, P. Journal of Mathematical Analysis and Applications , 1 , Goh, B. The American Society of Naturalists, 2 Management and Analysis of Biological Populations. Amsterdam, Netherlands: Elsevier Science. Hirsch, M.
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Hoffman, K. Linear Algebra 2 ed. A Mathematical Model for Cellular Immunology of tuberculosis. Journal of Mathematical Biosciences and Engineering , 8 4 , Khalil, H. Nolinear System 2 ed. London, UK: Prentice-Hall. Li, Y. Computers and Mathematics with Applications , 59 5 , Lyapunov, A. The General Problem of the Stability of Motion. That is, the cause-effect power of each node of the IS determines the cause-effect power of the mechanism in a state represented by that node. The cause-effect power of nodes i. The procedure is analogous for both distributions.
Observe that we do an approach by probability distributions of both the informational structure and the informational field. We know that local dynamics on each node stationary solution in the IS can be described by the associated eigenvalues and eigenvectors when we linearize 7 on nodes [ 57 ]. The exponential of negative eigenvalues gives the the strength of attraction towards the node. The exponentials of positive eigenvalues give the the strength of repulsion from the node, and the repulsion directions are given by the associated eigenvectors of this node.
Heuristically, this functional on positive eigenvalues reflects the exponential divergence of trajectories in the unstable manifolds being pushed out from a neighbourhood of its associated node; the functional on negative eigenvalues of a node describes the exponential convergence of trajectories in the stable manifold being attracted to the node.
Thereby, we reflect the degree to which the saddle fixed points nodes locally attracts, or repels trajectories to create a chain with the other nodes in the global attractor.
Fig 6 shows and example for this calculation on.