Attractor selection in a modulated laser and in the Lorenz circuit

Abstract

By tuning a control parameter, a chaotic system can either display two or more attractors (generalized multistability) or exhibit an interior crisis, whereby a chaotic attractor suddenly expands to include the region of an unstable orbit (bursting regime).Recently, control of multistability and bursting have been experimentally proved in a modulated class B laser by means of a feedback method. In a bistable regime, the method relies on the knowledge of the frequency components of the two attractors. Near an interior crisis, the method requires retrieval of the unstable orbit colliding with the chaotic attractor.We also show that a suitable parameter modulation is able to control bistability in the Lorenz system. We observe that, for every given modulation frequency, the chaotic attractor is destroyed under a boundary crisis. The threshold control amplitude depends on the control frequency and the location of the operating point in the bistable regime. Beyond the boundary crisis, the system remains in the steady state even if the control is switched off, demonstrating control of bistability.

Influence of Spatial Inhomogeneity on the Formation of Chaotic Modes at the Self-Organization Process

“…Let the initial conditions for the system of equations (5)-(7) be equal to 0 , 0 , and 0 , respectively. This, according to relations (4), gives the initial conditions 0 = 0 √︀ / , 0 = 0 √︀ / ,and 0 = − 0 for the system of equations (1)- (3). In order to provide a full consistency of the results of a numerical simulation obtained when solving the initial, Eqs.…”

“…We are faced with the self-organization processes in our daily life, both in animate and inanimate nature. Bright examples of self-organization include the ordering of electrons at low temperatures (superconductivity) [2], coherent laser radiation [1,3], the emergence of ordered vortex fluxes in the heated liquid, self-organization in nanomaterials (allotropic forms of carbon) [4], some chemical reactions (e.g., the Belousov-Zhabotinsky reaction) [5], the formation of ordered structures in viscous liquids [6], and so forth. In biological systems, the self-organization reveals itself as the coordinated behavior of bacteria, formation of ocean pyrosomes (these are large colonies of tiny organisms that look like hollow tubes closed at their one end), grouping of birds and fishes into flocks, and so on.…”

“…Olemskoi, when describing the nonequilibrium noise-induced phase transitions [10], self-organization in quasiequilibrium condensation processes [11], kinetics of the first-and second-order phase transitions [12], and mechanisms of explosive crystallization [13] and self-organized criticality [14]. Furthermore, the Lorentz system is used to describe the chaotic radiation emission in lasers [3], fluxes in fluids, effects in electrical circuits in the presence of nonlinear elements (like Gunn diodes), chemical reactions [5], and so forth. When developing the relevant theories, the Lorentz system has been thoroughly analyzed with regard for the terms that are responsible for additive and multiplicative noises, nonlinear relaxation terms, a periodic external influence, and other factors.…”

Hidden chaotic attractors and chaos suppression in an impulsive discrete economical supply and demand dynamical system

“…The existence of hidden attractors, in the case of the SD model, might be related to a multistability-like, i.e. starting from an assigned initial condition, the system remains on one or other attractor, depending on external perturbations [15,16,17,18].…”

Modular, general purpose ODE integration package to solve large number of independent ODE systems on GPUs

“…The size of such systems (number of the involved equations) can vary between orders of magnitude. Among the simplest models (second or third order systems), one can find the classic Duffing [3][4][5][6][7][8][9], Morse [10,11], Toda [12][13][14][15] and Lorentz [16][17][18] equations which are extensively studied for many decades from nonlinear dynamical point of view to establish bifurcation theories. Examples for medium-sized systems are the complex model of a pressure relief vale [19,20] (order greater than 5); single bubble dynamical model including partial differential equations or chemistry [21][22][23][24][25][26] (order greater than 20); globally [27][28][29][30][31][32] or diffusionally [33][34][35][36][37] coupled low order identical models in which a complete system can include several hundreds or even thousands of equations.…”