Controlling intermediate dynamics in a family of quadratic maps

Abstract

The intermediate dynamics of composed one-dimensional maps is used to multiply attractors in phase space and create multiple independent bifurcation diagrams which can split apart. Results are shown for the composition of k-paradigmatic quadratic maps with distinct values of parameters generating k-independent bifurcation diagrams with corresponding k orbital points. For specific conditions, the basic mechanism for creating the shifted diagrams is the prohibition of period doubling bifurcations transformed in saddle-node bifurcations.

Proliferation of stability in phase and parameter spaces of nonlinear systems

“…The replication of a shrimp-like ISSs was observed in a continuous oscillator [8], but its origin remained unknown. This work extends previous results for one-dimensional systems [25] to the non-trivial two-dimensional case.…”

“…This explains the origin of the duplication of the PDBs sequence and of the ISSs which contain them. For more simples examples of the origin of shifted bifurcation diagrams via prohibition of PDBs we refer the reader to the one-dimensional case [25].…”

“…Recently it was shown [25] that by controlling the dynamics of composed one-dimensional quadratic maps (QMs), multiple independent attractors and independent shifted bifurcation diagrams can be generated. The appearance of extra stable motion together with the prohibition of period doubling bifurcations (PDBs) is the mechanism which leads to shifted bifurcations diagrams.…”

“…It is clear that for F = 0 no orbit with period p = 1 exists anymore and the PDB at a 1→2 becomes forbidden. In fact, it was shown [25] that this PDB is transformed in a saddle-node bifurcation and two orbits of period p = 2 exist with distinct stabilities. Consequently, for increasing values of a these pair of per-2 orbits suffer PDBs at distinct values of a 2→4 , and subsequently along the whole PDB sequence a 4→8 , a 8→16 , .…”

“…, N , parameters (a, b, F ) and the function g(F, n) with period k which will define the protocol of proliferation. For b = 0, map (1) reduces to the MQM from [25]. The MHM corresponds to the HM with a time dependent parameter a ′ j = [a + g(F, n)].…”