Periodicity characterization of the nonlinear magnetization dynamics

Abstract

In this work, we study numerically the periodicity of regular regions embedded in chaotic states for the case of an anisotropic magnetic particle. The particle is in the monodomain regime and subject to an applied magnetic field that depends on time. The dissipative Landau–Lifshitz–Gilbert equation models the particle. To perform the characterization, we compute several two-dimensional phase diagrams in the parameter space for the Lyapunov exponents and the isospikes. We observe multiple transitions among periodic states, revealing complex topological structures in the parameter space typical of dynamic systems. To show the finer details of the regular structures, iterative zooms are performed. In particular, we find islands of synchronization for the magnetization and the driven field and several shrimp structures with different periods.

Complexity of a peroxidase–oxidase reaction model

“…4-6, where colours indicate phases characterized by periodic oscillations while black denotes phases where no periodic oscillations were detected. Isoperiodic diagrams have been found useful in the study of complex systems [41][42][43][44][45][46] . For a recent survey about the computation of standard Lyapunov stability diagrams and more fruitful alternatives see Ref.…”

Chirality detected in Hartley’s electronic oscillator

“…In experiments, C G D may be simulated using operational amplifiers. The classification is done using isospike stability diagrams [10][11][12][13][14][15][16][17][18], the flow version of the isoperiodic stability diagrams commonly used for maps [19][20][21][22]24]. Briefly, for a given set of parameters, we numerically integrate the equations of motion recording the number of spikes per period for all periodic oscillations.…”

“…To count oscillation spikes is easy to do in a very reliable way. The fruitful isospike technique to classify complex oscillations is discussed in-depth in recent literature [10][11][12][13][14][15][16][17][18].…”

Ubiquity of ring structures in the control space of complex oscillators

“…We computed two types of stability diagrams: the standard Lyapunov diagrams 18 and the so-called isospike diagrams. [19][20][21][22][23][24][25][26][27][28][29] Here, isospike diagrams are constructed by painting each point of the control plane with colors reflecting the number of spikes (local maxima) per period of the periodic oscillations and assigning some specific color to record nonperiodic oscillations. Computationally, isospike diagrams are a much simpler and less costly way to obtain all the information of Lyapunov diagrams, plus a significant enhancement: instead of lumping together all periodic oscillations into a single-phase as Lyapunov diagrams do, isospike diagrams classify oscillations by explicitly displaying the number of spikes per period for every individual oscillation.…”