RESUMEN

The dynamics of n-dimensional Lotka--Volterra equations having an invariant hyperplane is investigated herein. There exists a fundamental difference between the even- and the odd-dimensional case. For even dimensions the dynamics are generated by a Hamiltonian system with respect to an appropriately chosen Poisson structure. For odd dimensions the dynamics are Hamiltonian if there is a continuum of interior fixed points, and gradient-like if the interior fixed point is unique. In this case the invariant hyperplane is part of the $\omega$-limit set, and the dynamics restricted to this set are those of a Hamiltonian system. Furthermore, an example of a completely integrable four-dimensional Hamiltonian Lotka--Volterra system is presented.

RESUMEN

Slowly varying conservative systems are analyzed in the case of a reverse subcritical pitchfork bifurcation in which two saddles and a center coalesce. Before the bifurcation there is a hyperbolic double-homoclinic orbit connecting a linear saddle point. At the bifurcation a double nonhyperbolic homoclinic orbit connects to a nonlinear saddle point. Strongly nonlinear oscillations obtained by the method of averaging are not valid near unperturbed homoclinic orbits. In the case in which the solution slowly passes through the nonhyperbolic homoclinic orbit associated with the subcritical pitchfork bifurcation, the solution consists of a large sequence of nonhyperbolic homoclinic orbits connecting autonomous nonlinear saddle approaches. Solutions are captured into the left and right well. Phase jumps and the boundaries of the basins of attraction are computed. It is shown that the change in action in the slow passage through the nonhyperbolic homoclinic orbits is much larger than the known change in action for the slow crossing of hyperbolic homoclinic orbits. Near the boundary of the basin of attraction, where the energy is particularly small, one of the saddle approaches is governed by the second Painlev\a'e transcendent, which is not autonomous, and the solution may oscillate around the middle center or approach the two saddles created by the subcritical pitchfork bifurcation in addition to oscillating around the left and right wells.