Multiply periodic states appear in a wide variety of physical contexts, such as the Rayleigh-Bénard convection, Faraday waves, liquid crystals and skyrmion crystals recently observed in chiral ...magnets. Here we study the phase diagram of an anisotropic frustrated magnet which contains five different multiply periodic states including the skyrmion crystal. We clarify the mechanism for stabilization of these states and discuss how they can be observed in magnetic resonance and electric polarization measurements. We also find stable isolated skyrmions with topological charge 1 and 2. Their spin structure, interactions and dynamics are more complex than those in chiral magnets. In particular, magnetic resonance in the skyrmion crystal should be accompanied by oscillations of the electric polarization with a frequency depending on the amplitude of the a.c. magnetic field. These results show that skyrmion materials with rich physical properties can be found among frustrated magnets. We formulate rules to help the search.
Magnetic skyrmions are particle-like topological excitations recently discovered in chiral magnets. Their small size, topological protection and the ease with which they can be manipulated by ...electric currents generated much interest in using skyrmions for information storage and processing. Recently, it was suggested that skyrmions with additional degrees of freedom can exist in magnetically frustrated materials. Here, we show that dynamics of skyrmions and antiskyrmions in nanostripes of frustrated magnets is strongly affected by complex spin states formed at the stripe edges. These states create multiple edge channels which guide the skyrmion motion. Non-trivial topology of edge states gives rise to complex current-induced dynamics, such as emission of skyrmion-antiskyrmion pairs. The edge-state topology can be controlled with an electric current through the exchange of skyrmions and antiskyrmions between the edges of a magnetic nanostructure.
The general method for proving the existence of homoclinic trajectories in dissipative systems is developed. The applications of this method to Lorenz-like systems: Lorenz, Shimizu–Morioka, Lu and ...Chen systems are demonstrated. A criterion for the existence of a homoclinic trajectory within a given family of differential equations (Fishing principle) is presented. New numerical algorithm for the approximation of a homoclinic point in parameters space is constructed. The comparison with Kaplan–Yorke and Shilnikov results is made.
► General method for proof of existence of homoclinic trajectories is developed. ► This method applied to Lorenz, Shimizu–Morioka, Lu and Chen systems. ► New existence conditions of homoclinic trajectories for these systems are obtained. ► New numerical algorithm of homoclinic point approximation is given.
The hidden oscillations (a basin of attraction of which does not contain neighborhoods of equilibria) have been obtained first in the 50–60s of the 20th century in automatic control systems with ...scalar piecewise-linear nonlinearity. This brings up the question about the excitation nature of hidden oscillations.
In the present paper it is shown that hidden oscillations can exist not only in systems with piecewise-linear nonlinearity but also in smooth systems. Here the possibility of the existence of a hidden chaotic attractor in a modified Chua’s system with a smooth characteristic of nonlinear element is demonstrated.
► There are hidden oscillations: basin does not contain neighborhoods of equilibria. ► Firstly hidden oscillations were found in systems with piecewise-linear nonlinearity. ► We show that hidden attractors can exist in systems with smooth nonlinearity. ► We suggest localization methods and find hidden attractors in a smooth Chua’s system.
The classical attractors of Lorenz, Rossler, Chua, Chen, and other widely-known attractors are those excited from unstable equilibria. From computational point of view this allows one to use ...numerical method, in which after transient process a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another type:
hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria. In the present Letter for localization of hidden attractors of Chuaʼs circuit it is suggested to use a special analytical–numerical algorithm.
► There are hidden attractors: basin doesnʼt contain neighborhoods of equilibria. ► Hidden attractors cannot be reached by trajectory from neighborhoods of equilibria. ► We suggested special procedure for localization of hidden attractors. ► We discovered hidden attractor in Chuaʼs system, L. Chua in his work didnʼt expect this.
•The Lorenz-like system, describing rotating fluid convection is considered.•The scenario of transition to chaos in this system is described.•Analytical-numerical localization of hidden attractor in ...this system is demonstrated.
In this paper a Lorenz-like system, describing convective fluid motion in rotating cavity, is considered. It is shown numerically that this system, like the classical Lorenz system, possesses a homoclinic trajectory and a chaotic self-excited attractor. However, for the considered system, unlike the classical Lorenz system, along with self-excited attractor a hidden attractor can be localized. Analytical-numerical localization of hidden attractor is demonstrated.
The papers in this edited volume aim to provide a better understanding of the dynamics and control of a large class of hybrid dynamical systems that are described by different models in different ...state space domains. They not only cover important aspects and tools for hybrid systems analysis and control, but also a number of experimental realizations. Special attention is given to synchronization — a universal phenomenon in nonlinear science that gained tremendous significance since its discovery by Huygens in the 17th century. Possible applications of the results introduced in the book include control of mobile robots, control of CD/DVD players, flexible manufacturing lines, and complex networks of interacting agents.
Axisymmetric solitonic states (chiral skyrmions) were first predicted theoretically more than two decades ago. However, until recently they have been observed in a form of skyrmionic condensates ...(hexagonal lattices and other mesophases). In this paper we report experimental and theoretical investigations of isolated chiral skyrmions discovered in PdFe/Ir(111) bilayers two years ago by Romming et al (2013 Science 341 636). The results of spin-polarized scanning tunneling microscopy analyzed within the continuum and discrete models provide a consistent description of isolated skyrmions in thin layers. The existence region of chiral skyrmions is restricted by strip-out instabilities at low fields and a collapse at high fields. We demonstrate that the same equations describe axisymmetric localized states in all condensed matter systems with broken mirror symmetry, and thus our findings establish basic properties of isolated skyrmions common for chiral liquid crystals, different classes of noncentrosymmetric magnets, ferroelectrics, and multiferroics.
Currently it is being actively discussed the question of the equivalence of various Lorenz-like systems and the possibility of universal consideration of their behavior (Algaba et al., 2013a,b, ...2014b,c; Chen, 2013; Chen and Yang, 2013; Leonov, 2013a), in view of the possibility of reduction of such systems to the same form with the help of various transformations. In the present paper the differences and similarities in the analysis of the Lorenz, the Chen and the Lu systems are discussed. It is shown that the Chen and the Lu systems stimulate the development of new methods for the analysis of chaotic systems. Open problems are discussed.
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