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We have found a way for penetrating the space of the dynamical systemstowards systems of arbitrary dimension exhibiting the nonlinear mixing of alarge number of oscillation modes through which extraordinarily complex timeevolutions arise. The system design is based on assuring the occurrence of anumber of Hopf bifurcations in a set of fixed points of a relatively genericsystem of ordinary differential equations, in which the main peculiarity isthat the nonlinearities appear through functions of a linear combination of thesystem variables. The paper presents the design procedure and a selection ofnumerical simulations with a variety of designed systems whose dynamicalbehaviors are really rich and full of unknown features. For concreteness, thepresentation is focused to illustrating the oscillatory mixing effects on theperiodic orbits, through which the harmonic oscillation born in a Hopfbifurcation becomes successively enriched with the intermittent incorporationof other oscillation modes of higher frequencies while the orbit remainsperiodic and without necessity of bifurcating instabilities. Even in theabsence of a proper mathematical theory covering the nonlinear mixingmechanisms we find enough evidence to expect that the oscillatory scenario betruly scalable concerning the phase space dimension, the multiplicity ofinvolved fixed points and the range of time scales, so that extremely complexbut ordered dynamical behaviors could be sustained through it.

Designable Dynamical Systems for the Generalized Landau Scenario and the Nonlinear Complexification of Periodic Orbits