Nonlinear and chaotic magnetization dynamics near bifurcations of the Landau-Lifshitz-Gilbert equation

The behavior of a uniformly magnetized domain of ellipsoidal shape subject to a static external field and oscillatory external driving field is analyzed near bifurcation events. The analysis includes the effects of both linear and circularly polarized driving fields and is performed using numerical simulations of the Landau-Lifshitz-Gilbert (LLG) equation. Under a linearly polarized driving field, the LLG equation is a nonautonomous differential equation which can lead to complex magnetization motions, such as bistability, multiperiodic orbits, quasiperiodicity, and chaos. Under a circularly polarized driving field, the LLG equation can be written in autonomous form by transforming to the frame rotating with the driving field. The autonomous nature allows one to perform a fixed-point analysis of the system for select demagnetization factors. Similarities and differences between the driven systems are highlighted through bifurcation diagrams, phase portraits, basins of attraction, and Lyapunov exponents. Magnetization switching, prolonged transients, quasiperiodicity, and chaos are observed with both linearly and circularly polarized driving fields in the magnetic systems investigated.

Figure 1

Bifurcation diagram of the linearly polarized driven spherical domain (top two panels), average value of $\theta$, and the average largest Lyapunov exponent $〈\lambda 〉$ as a function of $h_d$ from 1238 to 1337 ns. $N_{11}=N_{22}=N_{33}=1/3$, ${\omega }_d=2\pi$ rad/ns, ${\omega }_0=1.84$ rad/ns, $h_s=0.1\mathrm{kOe}$, $\alpha =0.01$, and $({\theta }_0=0,{\varphi }_0=0)$.

Figure 2

Poincaré sections over different time intervals of the linearly polarized driven spherical domain at a critical drive amplitude of $h_d=0.41\mathrm{kOe}$. Cubes: $t=2475.2\text{–}2495.2$ ns; cones: $t=24742.9\text{–}24762.9$ ns; asterisks: $t=123711.5\text{–}123731.5$ ns; spheres (circles) $t=247422.0\text{–}247442.0$ ns. $N_{11}=N_{22}=N_{33}=1/3$, ${\omega }_d=2\pi$ rad/ns, ${\omega }_0=1.84$ rad/ns, $h_s=0.1\mathrm{kOe}$, $\alpha =0.01$, and $({\theta }_0=0,{\varphi }_0=0)$.

Figure 3

Stability diagram for the circularly polarized driven spherical domain in the $(h_d,h_s)$ plane. $N_{11}=N_{22}=N_{33}=1/3$, ${\omega }_d=2\pi$ rad/ns, and $\alpha =0.01$.

Figure 4

Bifurcation diagram of the circularly polarized driven spherical domain and the average largest Lyapunov exponent $〈\lambda 〉$ as a function of $h_d$ from 1238 to 1337 ns. $N_{11}=N_{22}=N_{33}=1/3$, ${\omega }_d=2\pi$ rad/ns, ${\omega }_0=1.84$ rad/ns, $h_s=0.1\mathrm{kOe}$, $\alpha =0.01$, and $({\theta }_0=0,{\varphi }_0=0)$.

Figure 5

Asymptotic behavior of transient lifetime as a function of drive amplitude for both the circularly and linearly polarized driven spherical domains. Circularly polarized: Circles asymptotically approaching the solid line. Linearly polarized: Squares asymptotically approaching the dashed line. $N_{11}=N_{22}=N_{33}=1/3$, ${\omega }_d=2\pi$ rad/ns, ${\omega }_0=1.84$ rad/ns, $h_s=0.1\mathrm{kOe}$, $\alpha =0.01$, and $({\theta }_0=0,{\varphi }_0=0)$.

Figure 6

Bifurcation diagram of the linearly polarized driven cylindrical domain and the average largest Lyapunov exponent $〈\lambda 〉$ as a function of $h_d$ from 1238 to 1337 ns. $N_{11}=N_{22}=1/2$, ${\omega }_d=686.3$ rad/ns, ${\omega }_0=203.9$ rad/ns, $h_s=0.1\mathrm{kOe}$, $\alpha =0.01$, and $({\theta }_0=0,{\varphi }_0=0)$.

Figure 7

Basins of attraction and corresponding attractors for the linearly polarized driven cylindrical domain with different values of the drive amplitude. (a) and (c) are the basins of attraction for the attractors shown in (b) and (d), respectively. (a) $h_d=38.04$ kOe; (b) $h_d=38.04\mathrm{kOe}$; (c) $h_d=47.94$ kOe; and (d) $h_d=47.94$ kOe.

Figure 8

Subsection of Fig. 7. Basins of attraction using nearly 1 million initial conditions for the linearly polarized driven cylindrical domain when $h_d=38.04\mathrm{kOe}$.

Figure 9

Bifurcation diagram of the linearly polarized driven thin-film domain and the average largest Lyapunov exponent $〈\lambda 〉$ as a function of $h_d$ from 1238 to 1337 ns. $N_{22}=1$, ${\omega }_d=93.4$ rad/ns, ${\omega }_0=27.3$ rad/ns, $h_s=0.1\mathrm{kOe}$, $\alpha =0.01$, and $({\theta }_0=0,{\varphi }_0=0)$.

Figure 10

Basins of attraction and corresponding attractors for the linearly polarized driven thin-film domain with different values of the drive amplitude. (a), (c), and (e) are the basins of attraction for the attractors shown in (b), (d), and (f), respectively. (a) $h_d=0.40$ kOe; (b) $h_d=0.40$ kOe; (c) $h_d=0.50$ kOe; (d) $h_d=0.50$ kOe; (e) $h_d=0.67$ kOe; and (f) $h_d=0.67$ kOe.

Figure 11

Zoomed-in view of a small section of Fig. 10 that reveals five different riddled basins of attraction. The white basin corresponds to period-five attractors of various phases, and the remaining basins correspond to the attractors shown in Fig. 10.