Abstract

 A unified theory of nonlinear dynamical systems is presented. The unification relies on the Quasi-polynomial approach of these systems. The main result of this approach is that most nonlinear dynamical systems can be exactly transformed to a unique format, the Lotka-Volterra system. An abstract Lie algebraic structure underlying most nonlinear dynamical systems is found. This structure, based on two sets of operators obeying specific commutation rules and on a Hamiltonian expressed in terms of these operators, bears a strong similarity with the fundamental algebra of quantum physics. From these properties, two forms of the exact general solution can be constructed for all Lotka-Volterra systems. One of them corresponds to a Taylor series in power of time. In contrast with other Taylor series solutions methods for nonlinear dynamical systems, our approach provides the exact analytic form of the general coefficient of that series. The second form of the solution is given in terms of a path integral. These solutions can be transformed back to solutions of the general nonlinear dynamical systems.
Highlights
This series of two lectures was dedicated to the Quasipolynomial (QP) approach to the theory of Nonlinear Dynamical Systems, i.e. systems of nonlinear ordinary differential equations that frequently appear in mathematical models in Physics, Chemistry, Biology and other scientific disciplines
Though a great diversity of dynamical systems are used in scientific modelling, most of them can be cast in the form of systems of ordinary differential equations with polynomial or, more generally, quasi-polynomial nonlinearities
The Quasi-Polynomial approach relies on a special notation for writing the quasi-polynomials that appear in the vector field of a given dynamical system
Summary
This series of two lectures was dedicated to the Quasipolynomial (QP) approach to the theory of Nonlinear Dynamical Systems, i.e. systems of nonlinear ordinary differential equations that frequently appear in mathematical models in Physics, Chemistry, Biology and other scientific disciplines. Though a great diversity of dynamical systems are used in scientific modelling, most of them can be cast in the form of systems of ordinary differential equations with polynomial or, more generally, quasi-polynomial nonlinearities. The latter are linear superposition of monomials with exponents that can be non-integers, real numbers. The Hamiltonian is expressed in terms of creation-destruction operators but is not Hermitian Another realization leads to the Liouville partial derivative equation that represents the conservation of the probability density norm. We show that two forms of the exact general solution can be constructed for LV systems They are based on two different realizations of the abstract LV algebra. Two other realizations of the abstract algebra: creation-destruction operators and Liouville
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