Local Stability and Lyapunov Functionals for n-Dimensional Quasipolynomial Conservative Systems

Abstract

We present a method for determining the local stability of equilibrium points of conservative generalizations of the Lotka-Volterra equations. These generalizations incorporate both an arbitrary number of species -including odd-dimensional systems- and nonlinearities of arbitrarily high order in the interspecific interaction terms. The method combines a reformulation of the equations in terms of a Poisson structure and the construction of their Lyapunov functionals via the energy-Casimir method. These Lyapunov functionals are a generalization of those traditionally known for Lotka-Volterra systems. Examples are given.

References in corpus (3)

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• Nonpolynomial vector fields under the Lotka-Volterra normal form
• Hamiltonian structure and Darboux theorem for families of generalized Lotka-Volterra systems

Cited by in corpus (8)

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• Characterization and global analysis of a family of Poisson structures
• Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems
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• Characterization, global analysis and integrability of a family of Poisson structures
• New four-dimensional solutions of the Jacobi equations for Poisson structures
• Quasipolynomial generalization of Lotka-Volterra mappings
• New global solutions of the Jacobi partial differential equations