Orbits In Conservative Systems Research Articles

Conservative generalized bifurcation diagrams

Bifurcation cascades in conservative systems are shown to exhibit a generalized diagram, which contains all relevant informations regarding the location of periodic orbits (resonances), their width (island size), irrational tori and the infinite higher-order resonances, showing the intricate way they are born. Contraction rates for islands sizes, along period-doubling bifurcations, are estimated to be $\alpha_I\sim 3.9$. Results are demonstrated for the standard map and for the continuous H\'enon-Heiles potential. The methods used here are very suitable to find periodic orbits in conservative systems, and to characterize the regular, mixed or chaotic dynamics as the nonlinear parameter is varied.

Circular and elliptic orbits in a feedback-mediated chemostat

A chemostat with two organisms competing for a single growth-limiting nutrient controlled by feedback-mediated dilution rate is analyzed. A specific feedback function is constructed which yields circular and elliptical periodic orbits for the limiting system. A theorem on the stabilization of periodic orbits in conservative systems is developed and for a given elliptical orbit, the result is used to modify the chemostat so that the chosen orbit is asymptotically stable. Finally, the feedback function is modified so that finitely many nested periodic orbits of alternating stability exist.

Bifurcation of degenerate homoclinic orbits in reversible and conservative systems

We study the bifurcation of homoclinic orbits from a degenerate homoclinic orbitγ in autonomous systems which are either conservative or reversible. More precisely we consider degenerate homoclinic orbits where along the orbit the tangent spaces to the stable and unstable manifolds of the equilibrium have an intersection which is two-dimensional. In the reversible case we demand thatγ is symmetric and we distinguish betweenγ being elementary and nonelementary. We show that the existence of elementary degenerate homoclinic orbits is a codimension-two phenomenon, while nonelementary homoclinic orbits and degenerate homoclinic orbits in conservative systems appear generically in one-parameter families. We give a complete description of the set of nearby 1-homoclinic orbits.

Arcs of discrete dynamics and constants of motion

A theorem is proved which may relate the attractors of families of dissipative discrete-time dynamical systems to certain closed orbits of conservative systems. The result is illustrated by an example taken from dynamics defined by \(\mathbb{R}^2 \to \mathbb{R}^2 \) mappings.