Discontinuity induced bifurcations of non-hyperbolic cycles in nonsmooth systems

We analyse three codimension-two bifurcations occurring in nonsmooth systems, when a non-hyperbolic cycle (fold, flip, and Neimark-Sacker cases, both in continuous- and discrete-time) interacts with one of the discontinuity boundaries characterising the system's dynamics. Rather than aiming at a complete unfolding of the three cases, which would require specific assumptions on both the class of nonsmooth system and the geometry of the involved boundary, we concentrate on the geometric features that are common to all scenarios. We show that, at a generic intersection between the smooth and discontinuity induced bifurcation curves, a third curve generically emanates tangentially to the former. This is the discontinuity induced bifurcation curve of the secondary invariant set (the other cycle, the double-period cycle, or the torus, respectively) involved in the smooth bifurcation. The result can be explained intuitively, but its validity is proven here rigorously under very general conditions. Three examples from different fields of science and engineering are also reported

Qualitative resonance of feedback-controlled chaotic oscillators

The qualitative resonance of feedback-controlled chaotic oscillators is the ability of the control system to qualitatively synchronize with a reference signal similar to one of the unstable periodic orbits embedded in the open-loop attractor. This property, discovered by O. De Feo (2004a; 2004b) while studying Shilnikov-type attractors, was explained in terms of the random-like rephasing mechanism characterizing the oscillator's dynamics, so to guarantee the eventual in-phase looking with the reference forcing. We experimentally show that the phenomenon works more in general, even in the absence of a rephasing mechanism. Intuitively, the forcing by the target cycle, or by a qualitative approximation of it, is sufficient to bring in the in-phase condition. Our results can make chaos control more practicable than so far imagined, as a qualitative control can be achieved with no a-priori knowledge about the target solution

The First in-silico Model of Leg Movement Activity During Sleep

We developed the first model simulator of leg movements activity during sleep. We designed and calibrated a phenomenological model on control subjects not showing significant periodic leg movements (PLM). To test a single generator hypothesis behind PLM—a single pacemaker possibly resulting from two (or more) interacting spinal/supraspinal generators—we added a periodic excitatory input to the control model. We describe the onset of a movement in one leg as the firing of a neuron integrating physiological excitatory and inhibitory inputs from the central nervous system, while the duration of the movement was drawn in accordance with statistical evidence. The period and the intensity of the periodic input were calibrated on a dataset of subjects showing PLM (mainly restless legs syndrome patients). Despite its many simplifying assumptions—the strongest being the stationarity of the neural processes during night sleep—the model simulations are in remarkable agreement with the polysomnographically recorded data

Conditions on the Energy Market Diversification from Adaptive Dynamics

We study a mathematical model based on ordinary differential equations to describe the dynamic interaction in the market of two types of energy called standard and innovative. The model consists of an adaptation of the generalized Lotka-Volterra system in which the parameters are assumed to depend on a quantitative and continuous attribute characteristic of energy generation. Using the analysis of the model the fitness function for the innovative energy is determined, from which conditions of invasion can be established in a market dominated by the conventional power. The canonical equation of the adaptive dynamics is studied to know the long-Term behavior of the characteristic attribute and its impact on the market. Then we establish conditions under which evolutionary ramifications occur, that is to say, the requirements of coexistence and divergence of the characteristic attributes, whose occurrence leads to the origin of diversity in the energy market

Love stories can be unpredictable: Jules et Jim in the vortex of life

Love stories are dynamic processes that begin, develop, and often stay for a relatively long time in stationary or fluctuating regime, before possibly fading. Although they are, undoubtedly, the most important dynamic process in our life, they have only recently been cast in the formal frame of dynamical systems theory. In particular, why it is so difficult to predict the evolution of sentimental relationships continues to be largely unexplained. A common reason for this is that love stories reflect the turbulence of the surrounding social environment. But we can also imagine that the interplay of the characters involved contributes to make the story unpredictable, that is, chaotic. In other words, we conjecture that sentimental chaos can have a relevant endogenous origin. To support this intriguing conjecture, we mimic a real and well-documented love story with a mathematical model in which the environment is kept constant, and show that the model is chaotic. The case we analyze is the triangle described in Jules et Jim, an autobiographic novel by Henri-Pierr Roche that became famous worldwide after the success of the homonymous film directed by Fran8cois Truffaut