Tassos Bountis
Publications:
Efthymiopoulos C., Bountis T., Manos T.
Explicit construction of first integrals with quasimonomial terms from the Painlevé series
2004, vol. 9, no. 3, pp. 385398
Abstract
The Painlevé and weak Painlevé conjectures have been used widely to identify new integrable nonlinear dynamical systems. For a system which passes the Painlevé test, the calculation of the integrals relies on a variety of methods which are independent from Painlevé analysis. The present paper proposes an explicit algorithm to build first integrals of a dynamical system, expressed as "quasipolynomial" functions, from the information provided solely by the Painlevé–Laurent series solutions of a system of ODEs. Restrictions on the number and form of quasimonomial terms appearing in a quasipolynomial integral are obtained by an application of a theorem by Yoshida (1983). The integrals are obtained by a proper balancing of the coefficients in a quasipolynomial function selected as initial ansatz for the integral, so that all dependence on powers of the time $\tau = t – t_0$ is eliminated. Both right and left Painlevé series are useful in the method. Alternatively, the method can be used to show the nonexistence of a quasipolynomial first integral. Examples from specific dynamical systems are given.

Marinakis V., Bountis T., Abenda S.
Finitely and Infinitely Sheeted Solutions in Some Classes of Nonlinear ODEs
1998, vol. 3, no. 4, pp. 6373
Abstract
In this paper we examine an integrable and a nonintegrable class of the first order nonlinear ordinary differential equations of the type $\dot{x}=x  x^n + \varepsilon g(t)$, $x \in \mathbb{C}$, $n \in \mathbb{N}$. We exploit, using the analysis proposed in [1], the asymptotic formulas which give the location of the singularities in the complex plane and show that there is an essential difference regarding the formation and the density of the singularities between the cases $g(t)=1$ and $g(t)=t$. Our analytical results are combined with a numerical study of the solutions in the complex time plane.

Rothos V. M., Bountis T.
The Second Order Mel'nikov Vector
1997, vol. 2, no. 1, pp. 2635
Abstract
Mel'nikov's perturbation method for showing the existence of transversal intersections between invariant manifolds of saddle fixed points of dynamical systems is extended here to second order in a small parameter $\epsilon$. More specifically, we follow an approach due to Wiggins and derive a formula for the second order Mel'nikov vector of a class of periodically perturbed $n$degree of freedom Hamiltonian systems. Based on the simple zero of this vector, we prove an $O(\epsilon^2)$ sufficient condition for the existence of isolated homoclinic (or heteroclinic) orbits, in the case that the first order Mel'nikov vector vanishes identically. Our result is applied to a damped, periodically driven $1$degreeoffreedom Hamiltonian and good agreement is obtained between theory and experiment, concerning the threshold of heteroclinic tangency.
