Computer assisted method for proving existence of periodic orbits

Abstract

We introduce a method based on the Conley index theory for proving the existence of a periodic trajectory in a smooth dynamical system in R n where an attracting periodic orbit is numerically observed. We apply this method to prove the existence of a periodic orbit in the Rössler equations, as announced in [9].

Recent development in rigorous computational methods in dynamical systems

“…In this section we describe an approach introduced in [33,34,35] for proving the existence of periodic trajectories.…”

“…Moreover, the following lemma proved in [33] indicates an important feature of isolating neighborhoods for the time-t discretization of a flow: Lemma 3.5. Let N be an isolating neighborhood with respect to a discretization ϕ t of the flow ϕ.…”

“…In [33,35], an algorithm is developed for the construction of a combinatorial index pair (P 1 , P 2 ) such that P 1 \ P 2 contains an apriori given set Q ⊂ H. Given an approximate rough covering Q ⊂ H of a hypothetic periodic trajectory observed in numerical simulations, one can use the algorithm for the construction of a combinatorial index pair containing Q mentioned in Section 3.5 for a representation F of the map ϕ t computed with the use of the software [7] for some apriori chosen t > 0. If the algorithm succeeds then it constructs a combinatorial index pair (P 1 , P 2 ) such that N := |P 1 \ P 2 | is an isolating neighborhood with respect to ϕ t , and the homological Conley index of N with respect to ϕ t can be automatically computed with algorithms available in [8,29,36].…”

“…If this index is the same as the index of some hyperbolic periodic orbit then it only suffices to verify the admittance of a Poincaré section to conclude the existence of a periodic trajectory with respect to ϕ contained in N . The latter stage of computations needs some additional input from the user and is described in [33,34,35].…”

“…The effectiveness of this approach for proving existence of periodic trajectories in ODEs is illustrated in [33,34,35] by several examples, including the well-known Vanderpol equations, Rössler equations, and Lorenz equations (see Figure 5). For the parameter values for which either attracting or unstable periodic trajectories are numerically observed, an index pair is automatically constructed as described above, the homological Conley index of this index pair is computed, and a Poincaré section is defined and verified.…”