Dynamics of a driven harmonic oscillator coupled to independent Ising spins in random fields

We aim at an understanding of the dynamical properties of a periodically driven damped harmonic oscillator coupled to a Random Field Ising Model (RFIM) at zero temperature, which is capable of showing complex hysteresis. The system is a combination of a continuous (harmonic oscillator) and a discrete (RFIM) subsystem, which classifies it as a hybrid system. In this paper we focus on the hybrid nature of the system and consider only independent spins in quenched random local fields, which can already lead to complex dynamics such as chaos and multistability. We study the dynamic behavior of this system by using the theory of piecewise-smooth dynamical systems and discontinuity mappings. Specifically, we present bifurcation diagrams and Lyapunov exponents as well as results for the shape and the dimensions of the attractors and the self-averaging behavior of the attractor dimensions and the magnetization. Furthermore we investigate the dynamical behavior of the system for an increasing number of spins and the transition to the thermodynamic limit, where the system behaves like a driven harmonic oscillator with an additional nonlinear smooth external force.

Figure 1

Illustration of the prototypical example of a dynamical system with external force $F_M$ (red curve), because of the interaction between the iron mass (black disk) and the external magnetic field (gray arrows). Here the external force shows non hysteretic behavior because of the neglected nearest-neighbor interaction.

Figure 2

The feedback mechanism of the RFIM coupled to a harmonic oscillator: The actual position $x$ of the oscillator affects the magnetic field $B$ (input) of the RFIM. Therefore the spin configuration and the magnetization $\mathcal{M}$ (output) depend on $x$. Then the magnetization again acts on the oscillator as an additional external force.

Figure 3

An illustration of our piecewise-smooth system. The two smooth regions $S_{i-1}$ and $S_i$ are separated by the boundary ${\mathrm{\Sigma }}_i$. The intersection point of the flow ${\mathit{\Phi }}_i$ with the boundary is called ${\mathit{x}}^{*}$.

Figure 4

Example of a projection of a state space trajectory of the harmonic oscillator coupled to $N=3$ spins. There are four smooth regions $S_i$ with piecewise constant magnetization ($––$) corresponding to four different spin configurations ($\uparrow \downarrow \uparrow$). At each of the three boundaries a jump discontinuity appears in the acceleration $\ddot{q}$. The values of the local disorder are $b_1=-1.0$, $b_2=1.5$, and $b_3=-3.0$.

Figure 5

Illustration of the calculation of the Lyapunov exponent for piecewise-smooth systems. To evolve a small perturbation $\delta \mathit{x}$ in the smooth region, we can use the fundamental solution $\mathbf{Y}(t_2,t_1)$ of the variational equation of the ODE. The effect of the boundaries on $\delta \mathit{x}$ is described by the saltation matrix $\mathbf{X}$.

Figure 6

Dependency of the mean of the minimal value of the distances between two boundaries $\overline{Z_{\text{min}}}$ on the number of spins for different values of randomness $R$. The mean $\overline{Z_{\text{min}}}$ was calculated for 500 different realizations of $b_i$.

Figure 7

The system with one boundary at $x=-0.6$ shows a sensitive dependence of the asymptotic solution on the initial conditions. Black boxes ($\blacksquare$) indicate chaos with a largest Lyapunov exponent greater than zero, while the white boxes ($\square$) correspond to periodic behavior. The parameters are $b_1=0.6$ and $C=1.65$ in Eqs. (13) and (16) for $N=1$.

Figure 8

Bifurcation diagram for the totally symmetric system with $N=1$ and $b_1=0$. The system shows the bifurcation scenarios expected from piecewise-smooth square-root maps, illustrated by the different colored boxes (from left to right): immediate jump to robust chaos ($––$), period-adding with chaos ($––$), and overlapping period-adding cascade ($––$).

Figure 9

Comparison of the bifurcation diagrams for the piecewise-smooth system with $N=20000$ ($\blacksquare$) spins and the system in its thermodynamic limit ($\blacksquare$). It can be seen that the typical grazing scenarios (immediate jump to chaos and period-adding cascades) vanish, whereas the main behavior is pretty similar.

Figure 10

Zoom of the bifurcation diagram from Fig. 9. It can be seen that besides the general similarities between the piecewise-smooth system and the system in its thermodynamic limit there are some differences in the actual behavior of the bifurcations. Mainly this is because of the dependence of the dynamical properties of the system for $N=20000$ on the actual realization of the local disorder $\left\{b_i\right\}$.

Figure 11

Comparison of the chaotic attractor ($––$), Poincaré section ($––$), and magnetization ($––$). (a): The system in its thermodynamic limit $N\to \infty$. (b): The piecewise-smooth system with $N=20000$ spins and one specific disorder realization $\left\{b_i\right\}$. Both systems are evaluated for fixed coupling constant $C=3.5$ and with the randomness set to $R=1.7$.

Figure 12

Phase diagram for the short time chaotic behavior of the nonlinear system in its thermodynamic limit. The shaded areas illustrate the region with a positive short-time Lyapunov exponent at $q^{*}$.

Figure 13

(a) For an increasing number of spins the box counting (red crosses) and the Kaplan-Yorke dimension (blue squares) converge to the corresponding values of the smooth system (dashed lines) in the thermodynamic limit ($N\to \infty$). Here the coupling strength equals $C=3.5$ in Eqs. (16) and (19). (b) There is a linear dependence of the difference between the mean of the fractal dimension and the corresponding limit value for increasing $N$, which can be seen in a semi-log plot. Hence there is an exponential convergence to the limit values.

Figure 14

The self-averaging parameter of the box counting and Kaplan-Yorke dimension in dependence on the number of spins appears to decrease exponentially to zero. Therefore the system shows self-averaging with respect to these dynamical properties for $C=3.5$.

Figure 15

For the attractor at $C=3.5$ the variance of the magnetization goes algebraically to zero with $N^{-1}$ for increasing number of spins similar to Independent and Identically Distributed (iid) input of the RFIM. In contrast, for $C=2.9$ the variance does not vanish for a large number of spins.

Figure 16

(a): For $C=2.9$ and $N=20000$ there exist two different typical attractors of the system illustrated by the red ($––$) and blue ($––$) dots. This explains the non-self-averaging behavior of the magnetization. (b): The non-self-averaging behavior of the magnetization is also reflected in the corresponding histogram by the two main bars at $M=\pm 0.2$.