Periodic orbits of competitive and cooperative systems

Three-Dimensional Competitive Lotka-Volterra Systems with no Periodic Orbits

The following conjecture of M. L. Zeeman is proved. If three interacting species modeled by a competitive Lotka–Volterra system can each resist invasion at carrying capacity, then there can be no coexistence of the species. Indeed, two of the species are driven to extinction. It is also proved that in the other extreme, if none of the species can resist invasion from either of the others, then ...

Periodic Tridiagonal Systems Modeling Competitive-cooperative Ecological Interactions†

The dynamics of the Poincaré map, associated with a periodic tridiagonal system modeling cooperative-competitive ecological interactions, has been investigated. It is shown that the limit-set is either a fixed point or is contained in the boundary of the positive cone and itself contains a cycle of fixed points. Furthermore, the dynamics is trivial if the number of interactive species is not gr...

Periodic Orbits of Hamiltonian Systems

5 The Variational principles and periodic orbits 21 5.1 Lagrangian view point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.2 Hamiltonian view point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.3 Fixed energy problem, the Hill’s region . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.4 Continuation of periodic orbits as critical p...

Relative Periodic Orbits of Symmetric Lagrangian Systems

Optimal periodic orbits of chaotic systems.

Invariant sets embedded in a chaotic attractor can generate time averages that diier from the average generated by typical orbits on the attractor. Motivated by two diierent topics (namely, controlling chaos and riddled basins of attraction), we consider the question of which invariant set yields the largest (optimal) value of an average of a given smooth function of the system state. We presen...