Using periodic modulation to control coexisting attractors induced by delayed feedback

Controlling multistability in a vibro-impact capsule system

“…One criterion to choose the type of control strategy can be given in terms of power consumption, calculated via the solution measure defined in (18). In Fig.…”

“…Fluctuational transitions in a discrete dynamical system having two coexisting attractors separated by a fractal basin boundary were studied in [17]. The case of a CO 2 laser model driven by a delayed feedback controller was considered in [18], where the authors were able to lock one of the coexisting attractors and eliminate the others. In [19], the basins of attraction of coexisting solutions were controlled by either a harmonic modulation or a small noise signal applied to a system parameter in a multistable erbium-doped fiber laser.…”

“…The novelty of the position feedback control is that it gives rise to a dynamical scenario with two coexisting solutions, corresponding to forward and backward motion control of the capsule, which cannot be achieved by delayed feedback control [18] [19,20]. The rest of the paper is organized as follows.…”

Exact period-four solutions of a family of n-dimensional quadratic maps via harmonic balance and Gröbner bases

“…The origin of these maps lies in the representation of temporal evolutions that are inherently discrete as well as the analysis of oscillations in continuous dynamical systems via the Poincar e section. Among the extensive list of examples, it can be mentioned the population models in Biology, [1][2][3] the cardiac activity models in Medicine, [4][5][6] the structure markets in Economics, 7,8 the impact systems in Mechanics, 9,10 the modulated lasers in Physics, [11][12][13] the power converters in Electronics, 14,15 etc. In many of these applications, quadratic maps have played an important role in the description of the detected dynamical scenarios.…”

“…Discrete maps can undergo periodic orbits due to existence of period-doubling bifurcations, Neimark-Sacker bifurcations, and even more complicated nonlinear phenomena such as period-doubling cascades, 16,17 weak and strong resonances, 18 bubbles, 19 bistability, [11][12][13] and periodic windows after the onset of chaos. 20,21 Most of these scenarios are related to the existence of saddle-node bifurcations.…”