Flaviano Battelli

Heteroclinic orbits in singular systems: A unifying approach

AbstractWe consider singularly perturbed systems

Chaos in singular impulsive O.D.E.

\xi = f(\xi ,\eta ,\varepsilon ),\dot \eta = \varepsilon g(\xi ,\eta ,\varepsilon )

PERTURBING TWO-DIMENSIONAL MAPS HAVING CRITICAL HOMOCLINIC ORBITS

, such thatξ=f(ξ, αo, 0). αo∃ℝm, has a heteroclinic orbitu(t). We construct a bifurcation functionG(α, ɛ) such that the singular system has a heteroclinic orbit if and only ifG(α, ɛ)=0 has a solutionα=α(ɛ). We also apply this result to recover some theorems that have been proved using different approaches.

Heteroclinic period blow-up in certain symmetric ordinary differential equations

A diffeomorphism on a

Dynamics of generalized PT-symmetric dimers with time-periodic gain–loss

C^1

Melnikov theory for weakly coupled nonlinear RLC circuits

-smooth manifold is studied possessing a hyperbolic fixed point. If the stable and unstable manifolds of the hyperbolic fixed point have a nontrivial local topological crossing then a chaotic behaviour of the diffeomorphism is shown. A perturbed problem is also studied by showing the relationship between a corresponding Melnikov function and the nontriviality of a local topological crossing of invariant manifolds for the perturbed diffeomorphism.

Periodic Solutions of Symmetric Elliptic Singular Systems: the Higher Codimension Case

It is well-known [l] that the existence of a transversal homoclinic orbit of a diffeomorphism implies chaos, i.e. the diffeomorphism possesses a compact domain which is invariant for some of its iterates and such that the dynamics of such an iterate on it is homeomorphic to the Bernoulli shift on a finite number of symbols. However, the location of transversal homoclinic points is generally difficult. One of the most common approaches is a perturbation method based on the derivation of the so-called Melnikov functions [2], i.e. we look for a transversal homoclinic point of a perturbed diffeomorphism when the unperturbed diffeomorphism has a known dynamics. Analytical methods, based on the Lyapunov-Schmidt procedure, are presented in [3-51 for periodically, regularly perturbed autonomous ordinary differential equations (o.d.e.), and in [6-81 for diffeomorphisms. Regularly perturbed impulsive o.d.e. are studied in [8]. Singular perturbation problems are studied analytically in [9, lo] for o.d.e., and in [l 11 for impulsive o.d.e. In all these papers it is assumed that the so-called reduced o.d.e. has a homoclinic orbit. In particular, the following impulsive o.d.e. is studied in [ll]

Periodic Solutions of Symmetric Elliptic Singular Systems

AbstractThe problem of existence of aglobal center manifold for a system of O.D.E. like(*)