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The topological structure of basin boundaries plays a fundamental role in thesensitivity to the initial conditions in chaotic dynamical systems. Herewith wepresent a study on the dynamics of dissipative systems close to the Hamiltonianlimit, emphasising the increasing number of periodic attractors and on thestructural changes in their basin boundaries as the dissipation approacheszero. We show numerically that a power law with nontrivial exponent describesthe growth of the total number of periodic attractors as the damping isdecreased. We also establish that for small scales the dynamics is governed by\emph{effective} dynamical invariants, whose measure depends not only on theregion of the phase space, but also on the scale under consideration.Therefore, our results show that the concept of effective invariants is alsorelevant for dissipative systems.

Emerging attractors and the transition from dissipative to conservative dynamics