Nonlinear dynamical systems may be exposed to tipping points, critical thresholds at which small changes in the external inputs or in the system’s parameters abruptly shift the system to an alternative state with a contrasting dynamical behavior. While tipping in a fold bifurcation of an equilibrium is well understood, much less is known about tipping of oscillations (limit cycles) though this dynamics are the typical response of many natural systems to a periodic external forcing, like e.g. seasonal forcing in ecology and climate sciences. We provide a detailed analysis of tipping phenomena in periodically forced systems and show that, when limit cycles are considered, a transient structure, so-called channel, plays a fundamental role in the transition. Specifically, we demonstrate that trajectories crossing such channel conserve, for a characteristic time, the twisting behavior of the stable limit cycle destroyed in the fold bifurcation of cycles. As a consequence, this channel acts like a “ghost” of the limit cycle destroyed in the critical transition and instead of the expected abrupt transition we find a smooth one. This smoothness is also the reason that it is difficult to precisely determine the transition point employing the usual indicators of tipping points, like critical slowing down and flickering.

In the bi-dimensional parameter space of an impact-pair system, shrimp-shaped periodic windows are embedded in chaotic regions. We show that a weak periodic forcing generates new periodic windows near the unperturbed one with its shape and periodicity. Thus, the new periodic windows are parameter range extensions for which the controlled periodic oscillations substitute the chaotic oscillations. We identify periodic and chaotic attractors by their largest Lyapunov exponents.

In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have periodicity and pattern similar to stable and unstable periodic orbits already existent for the unperturbed oscillator. These features indicate that the reported replicate periodic windows are associated with chaos control of the considered oscillators.

In parameter space of nonlinear dynamical systems, windows of periodic states are aligned following routes of period-adding configuring periodic window sequences. In state space of driven nonlinear oscillators, we determine the torsion associated with the periodic states and identify regions of uniform torsion in the window sequences. Moreover, we find that the measured of torsion differs by a constant between successive windows in periodic window sequences. We call this phenomenon as torsion-adding. Finally, combining the torsion and the period adding rules, we deduce a general rule to obtain the asymptotic winding number in the accumulation limit of such periodic window sequences.A conspicuous characteristic in parameter space of dissipative nonlinear dynamical systems is the appearance of periodic states for parameter sets immersed in parameter regions correspondent to chaotic states. In the literature, much attention has been devoted to establish connections between these periodic states. For example, a successive constant increment on the period of oscillation of such states (period-adding phenomenon) [1] have been experimental and numerically observed in several real-world systems such as neuronal activities [2, 3], electronic circuits [4], bubble formation [5], semiconductor device [6], and chemical reaction [7]. The periodadding phenomenon has been also observed for sequences of shrimp-shaped periodic windows accumulating in specific parameter space regions [8][9][10][11][12][13]. Once nonlinear dynamical systems can exhibit many different kinds of motion, knowing adding rules, such as the period-adding and further information about the accumulating parameter regions, is very advantageous, specially, for predicting periodic states for different parameter sets in real-world applications.Furthermore, besides the intrinsic period of oscillations, in dissipative systems, periodic states have other interesting convergence properties. For instance, for driven nonlinear oscillators, the torsion number n is defined as number of twists that local flow perform around a given periodic solution during a dynamical period m, and the winding number defined as w = n/m [14-18]. However, besides the existence of such convergence properties, additional connecting rules between periodic states and accumulating regions characteristic have not yet been discovered.Our aim here is to investigate the convergence characteristics, namely, the torsion and winding number of periodic states within complex periodic windows, in periodadding sequences in the parameter space of driven nonlinear oscillators. A torsion-adding formulation between such periodic states are proposed here. Combining both additive sequences properties, the torsion and the period adding, we describe a generic periodic window in a se-quence in terms of its winding number. The asymptotic limit of such description gives a general rule to determine the winding number for any accumulation of periodadding sequences.Generally, the driven nonlinear oscillator is described...

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