Chaotic dynamics of a magnetic particle at finite temperature

In this work, we study nonlinear aspects of the deterministic spin dynamics of an anisotropic single-domain magnetic particle at finite temperature modeled by the Landau-Lifshitz-Bloch equation. The magnetic field has two components: a constant term and a term involving a harmonic time modulation. The dynamical behavior of the system is characterized with the Lyapunov exponents and by means of bifurcation diagrams and Fourier spectra. In particular, we explore the effects of the magnitude and frequency of the applied magnetic field, finding that the system presents multiple transitions between regular and chaotic states when varying the control parameters. We also address the temperature dependence and evidence that it plays an important role in these transitions, almost suppressing the chaotic behavior close to the Curie temperature. Finally, we find that the system has hyperchaotic states for specific values of field and temperature.

Figure 1

LEs as a function of the field amplitude $h_x$ for three different temperatures: (a) $T=0$, (b) $T=0.6T_C$, and (c) $T=0.9T_C$. The fixed parameters are $\mathrm{\Omega }=0.4, h_z=0.1$, and ${\alpha }_0=0.05$. The resolution in $h_x$ is $\mathrm{\Delta }{h}_x=7.0\times {10}^{-3}$. The physical values of the parameters are $T_C=630$ K, $\omega =1.55$ GHz, and $H_z=1679$ A/m.

Figure 2

Lyapunov exponents as a function of the frequency $\mathrm{\Omega }$. The continuous line depicts the largest Lyapunov exponent, while the dashed lines represent the lower Lyapunov exponents. For the results, we used $T=0.6T_C, h_x=3.0, h_z=0.1$, and ${\alpha }_0=0.05$. The resolution in $\mathrm{\Omega }$ is $\mathrm{\Delta }\mathrm{\Omega }=2.0\times {10}^{-3}$. The physical values of the parameters are $T_C=630$ K, $H_x=52712$ A/m, and $H_z=1679$ A/m.

Figure 3

Frequency thresholds of the region that exhibit a mixed behavior of the Lyapunov exponents, i.e., switching on and switching off of the chaotic states as a function of the the reduced temperature $T/T_C$. The squares and dots denote the lower and higher threshold, respectively. The gray area represents the mixed states from chaotic to regular states. The other fixed parameters are the same as used in Fig. 2.

Figure 4

(a) Lyapunov exponents ${\lambda }_i$ as a function of the reduced temperature $T/T_C$. Bifurcation diagrams as a function of the reduced temperature $T/T_C$ for the angular variables (b) $\theta$, (c) $\varphi$, and (d) the module $\left|\mathbf{m}\right|$. The fixed parameters are $h_z=0.1, h_x=3.0, \mathrm{\Omega }=0.4$, and ${\alpha }_0=0.05$. The resolution in $T/T_C$ is $\mathrm{\Delta }T=1.0\times {10}^{-3}{T}_C$. The physical values of the parameters are $T_C=630$ K, $H_x=52728$ A/m, $H_z=1679$ A/m, and $\omega =1.55$ GHz.

Figure 5

Decay on time of the module $\left|\mathbf{m}\right|$ to the value of equilibrium as a function of the time, for four different temperatures. The fixed parameters are the same as Fig. 4.

Figure 6

Phase diagrams and Fourier power spectra for different temperatures: (a) $T/T_C=0.1$, (b) $T/T_C=0.19$, (c) $T/T_C=0.726$, and (d) $T/T_C=0.98$. These points are denoted on the diagrams of Fig. 4 with a square, circle, triangle, and hexagon, respectively.

Figure 7

Thresholds for switch on and switch off of the chaotic states as a function of the field amplitude, $h_x$, and the reduced temperature, $T/T_C$. The squares and dots denote the lower and higher threshold, respectively. The gray area represents the mixed states from chaotic to regular states. The fixed parameters are $h_z=0.1, \mathrm{\Omega }=0.4$, and $\alpha =0.05$. The physical values of the parameters are $H_z=1679$ A/m and $\omega =1.55$ GHz.

Figure 8

Top: Second-largest Lyapunov exponents (SLLEs) a function of the temperature-reduced temperature, $T/T_C$. Bottom: Phase diagram of the occurrence of SLLEs as a function of $h_x$, and the reduced temperature, $T/T_C$. The fixed parameters are $h_z=0.1, \mathrm{\Omega }=0.4$, and $\alpha =0.005$. The physical values of the parameters are $H_z$ = 1679 A/m and $\omega =1.55$ GHz.