Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies

Abstract

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.Newhouse Theorem [33]. Let M 2 be a C ∞ compact 2-dimensional manifold and let r ≥ 2. Assume that f ∈ Diff r (M 2 ) has a hyperbolic set whose stable and unstable

On three types of dynamics and the notion of attractor

“…The same phenomenon takes place at the bifurcations of a symmetric pair of homoclinic tangencies as in Fig. 4d (here g(O) = O and g(W u (O)) = W s (O)), see [20]. We also plan to prove the existence of the absolute Newhouse intervals in one-parameter families of reversible maps, which unfold symmetric quadratic and cubic homoclinic tangencies, as in Figs.…”

“…However, we can always, if necessary, replace the involution h by the involution T • h (identity (15) would not change) and achieve that the involution is identity on I − . After this choice is made, it can be shown (see remark after formula (20) in the proof of Theorem 5 below) that if h = −id on I + , then the symmetric periodic point can be made to disappear by an arbitrarily small perturbation of the map. Therefore, for the generic symmetric periodic orbit we have…”

“…It follows that the only generic case where a symmetric periodic orbit may have multipliers equal to +1 corresponds to h being identity on the u-space. The flow normal form (20) then recasts aṡ z j = iΩ j (Z, w)z j (j = 1, . .…”

“…The existence of elliptic periodic orbits is a characteristic property of reversible maps, for which the dimension of the set Fix(g) of the fixed points of the involution g is half of the dimension of the phase space, or higher. In particular, they emerge in various homoclinic bifurcations [16,2,20,21]. According to the Reversible Mixed Dynamics conjecture of [2], the mixed dynamics emerging at the most typical homoclinic bifurcations of reversible systems must always include a large number of elliptic periodic orbits.…”

On the phenomenon of mixed dynamics in Pikovsky–Topaj system of coupled rotators

“…The Reversible Mixed Dynamics Conjecture (RMD-conjecture) of [2] claims that the same phenomenon should take place for other types of codimension-1 bifurcations of various symmetric homoclinic and heteroclinic cycles in reversible systems. This conjecture is proven for certain basic cases [2,4,17], see Fig. 3, but it remains open in full generality, especially for the multidimensional case.…”