The Painlevé property and singularity analysis of integrable and non-integrable systems

The methods of singularity analysis are applied to several third order non-Hamiltonian systems of physical significance including the Lotka-Volterra equations, the three-wave interaction and the Rikitake dynamo model. Complete integrability is defined and new completely integrable systems are discovered by means of the Painleve property. In all these cases we obtain integrals, which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation (ODE) solved by Painleve transcendents. Relaxing the Painleve property we find many partially integrable cases whose movable singularities are poles at leading order, with In( t - t 0 ) terms entering at higher orders. In an N th order, generalized Rossler model a precise relation is established between the partial fulfillment of the Painleve conditions and the existence of N - 2 integrals of the motion.

On the complete and partial integrability of non-Hamiltonian systems

A new class of ‘‘solvable’’ nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations (ODE’s) describing each member of this class possess nonlinear superposition principles. These systems of ODE’s are generally not derived from a Hamiltonian and are classified by associated pairs of Lie algebras of vector fields. In this paper, all such systems of n≤3 ODE’s are integrated in a unified way by finding explicit integrals for them and relating them all to a ‘‘pivotal’’ member of their class: the projective Riccati equations. Moreover, by perturbing two parametrically driven projective Riccati equations (thus making them nonsolvable in the above sense) evidence is discovered of chaotic behavior on the Poincare surface of section—in the form of sensitive dependence on initial conditions—near a boundary separating bounded from unbounded motion.

On the integrability of systems of nonlinear ordinary differential equations with superposition principles

The Hamiltonian formulation of N=3 systems is considered in general. The most general solution of the Jacobi equation in R3 is proposed. The form of the solution is shown to be valid also in the neighborhood of some irregular points. Compatible Poisson structures and corresponding bi-Hamiltonian systems are also discussed. Hamiltonian structures, the classification of irregular points and the corresponding reduced first order differential equations of several examples are given.

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Hamiltonian equations in R3

The generalized Hamiltonian structures of several three-dimensional dynamical systems of interest in physical applications are considered. In general, Hamiltonians exist only for systems that possess at least one time-independent constant of motion. Systems with only time-dependent constants of motion may sometimes be rescaled and their constant of motion made time-independent. When this is possible, the transformed system may be cast in a generalized Hamiltonian formalism with non-canonical structure functions.

https://iopscience.iop.org/article/10.1088/0305-4470/27/19/020/pdf

Jaap Hijmans

Papers

On the integrability of some generalized Lotka-Volterra systems