Related papers: A geometrical method towards first integrals for d…
We apply the Darboux theory of integrability to polynomial ODE's of dimension 3. Using this theory and computer algebra, we study the existence of first integrals for the 3-dimensional Lotka-Volterra systems with polynomial invariant…
In this paper we developed an integrating factor matrix method to derive conditions for the existence of first integrals. We use this novel method to obtain first integrals, along with the conditions for their existence, for two and three…
Lotka-Volterra model is one of the most popular in biochemistry. It is used to analyze cooperativity, autocatalysis, synchronization at large scale and especially oscillatory behavior in biomolecular interactions. These phenomena are in…
We consider Lotka-Volterra systems in three dimensions depending on three real parameters. By using elementary algebraic methods we classify the Darboux polynomials (also known as second integrals) for such systems for various values of the…
The Painleve and weak Painleve conjectures have been used widely to identify new integrable nonlinear dynamical systems. The calculation of the integrals relies though on methods quite independent from Painlev\'e analysis. This paper…
Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi--particle dynamical system by finding polynomial solutions of a partial differential equations is…
The one-dimensional system of equations of isentropic gas dynamics is considered. First-order invariants of characteristics of this system are classified. Second-order invariants of characteristics are classified for polytropic processes.…
The physical phenomena are described by physical quantities related by specific physical laws. In the context of a Physical Theory, the physical quantities and the physical laws are described, respectively, by suitable geometrical objects…
The spectral method for building first integrals of ordinary linear differential systems is elaborated. Using this method, we obtain bases of first integrals for linear differential systems with constant coefficients, for linear…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part we discus the main structures…
Here we present/implement an algorithm to find Liouvillian first integrals of dynamical systems in the plane. In \cite{JCAM}, we have introduced the basis for the present implementation. The particular form of such systems allows reducing…
We apply the Darboux integrability method to determine first integrals and Hamiltonian formulations of three dimensional polynomial systems; namely the reduced three-wave interaction problem, the Rabinovich system, the Hindmarsh-Rose model,…
This article proposes a method for forming invariant stochastic differential systems, namely dynamic systems with trajectories belonging to a given smooth manifold. The It\^o or Stratonovich stochastic differential equations with the Wiener…
The main objective of this work is to investigate the integrability and linearizability problems around a singular point at the origin of the family of differential systems Particularly we are interested in the three-dimensional cubic…
In this work we study the integrability of a family of nonlinear oscillators. Dynamical systems from this family appear in different applications from mechanics to chemistry. We propose an approach for finding first integrals and…
In [1], we have presented the theoretical background for finding the Elementary Invariants for a 3D system of first order rational differential equations (1ODEs). We have also provided an algorithm to find such Invariants. Here we introduce…
We propose a method to construct first integrals of a dynamical system, starting with a given set of independent infinitesimal symmetries. In the case of two infinitesimal symmetries, a rank two Poisson structure on the ambient space it is…
A new method is proposed to numerically integrate a dynamical system on a manifold such that the trajectory stably remains on the manifold and preserves first integrals of the system. The idea is that given an initial point in the manifold…
The investigation of nonlinear dynamical systems of the type $\dot{x}=P(x,y,z),\dot{y}=Q(x,y,z),\dot{z}=R(x,y,z)$ by means of reduction to some ordinary differential equations of the second order in the form…
We investigate the dynamical complexity of Cournot oligopoly dynamics of three firms by using the qualitative methods of dynamical systems to study the phase structure of this model. The phase space is organized with one-dimensional and…