Chaos Driven by Interfering Memory

The transmission of information can couple two entities of very different nature, one of them serving as a memory for the other. Here we study the situation in which information is stored in a wave field and serves as a memory that pilots the dynamics of a particle. Such a system can be implemented by a bouncing drop generating surface waves sustained by a parametric forcing. The motion of the resulting “walker” when confined in a harmonic potential well is generally disordered. Here we show that these trajectories correspond to chaotic regimes characterized by intermittent transitions between a discrete set of states. At any given time, the system is in one of these states characterized by a double quantization of size and angular momentum. A low dimensional intermittency determines their respective probabilities. They thus form an eigenstate basis of decomposition for what would be observed as a superposition of states if all measurements were intrusive.

Figure 1

(a) Sketch of the experiment. The droplet loaded with ferrofluid is located in an axisymmetric spatially varying magnetic field ${\mathbf{B}}_d(r)$ and thus trapped in a two-dimensional attractive harmonic potential well. (b) The eigenmodes defined by a plot of their mean nondimensional spatial extension $\overline{R}$ versus their mean nondimensional angular momentum $\overline{L}$. (c) When the control parameter $\mathrm{\Lambda }$ changes, the stable modes ($n$, $m$) are observed in narrow ranges of $\mathrm{\Lambda }$ shown in grey. (d) A highly intermittent trajectory of a drop of velocity $<V>=8.1\mathrm{mm}{\mathrm{s}}^{-1}$ at $\mathrm{Me}\approx 200$ for $\mathrm{\Lambda }=0.83$ and $S_{\mathrm{Me}}/2\pi \mathrm{\Lambda }=1.6$. The selected sections show the coexistence of orbits (1, $\pm 1$), ovals (2, $\pm 2$), lemniscates (2,0), and loops (3, $\pm 1$).

Figure 2

(a) A complex trajectory obtained for a walker of velocity $<V>=9.7\mathrm{mm}{\mathrm{s}}^{-1}$ and memory $\mathrm{Me}=63$, $S_{\mathrm{Me}}/2\pi \mathrm{\Lambda }=1$ and for $\mathrm{\Lambda }=0.49$, a value located in the range ${\mathrm{\Lambda }}_{1,1}^{+}<\mathrm{\Lambda }<{\mathrm{\Lambda }}_{2,0}^{-}$. (b) A transition from a wobbling orbit (1,1) to a lemniscate (2,0). (c) A short sample of the time recording of the nondimensional angular momentum $L$. The wobbling of increasing amplitude of a circular orbit (1,1) leads to a reversal (1, $-1$). These transitions are mediated by unstable lemniscates. (d) Map of first return of the local maximum $R_{k+1}$ as a function of the previous one $R_k$ for a recording lasting 40 min. The light grey points are all the iterates, the colored ones are averaged. The red dots correspond to increasing values of $R_k$ while the black ones are obtained for decreasing $R_k$.

Figure 3

Analysis of the evolution of the multistability obtained for increasing values of the control parameter in a range where the three eigenstates corresponding to $n=2$ are lemniscates (2,0) and oval shaped orbits ($2,\pm 2$). (a)–(d) Four time recordings of the normalized angular momentum $L$ obtained for a walker of velocity $<V>=9.7\mathrm{mm}{\mathrm{s}}^{-1}$ for $\mathrm{Me}=60$ are shown corresponding to $\mathrm{\Lambda }=0.74$, $\mathrm{\Lambda }=0.807$, $\mathrm{\Lambda }=0.828$, $\mathrm{\Lambda }=0.875$, respectively. The typical intermittency time can be large, e.g., $\sim 150$ orbital periods in (d). (e) Evolution of the probability $p_{2,0}$ and $p_{2,\pm 2}$ of being in lemniscate (square) and oval-shaped states (o) as a function of $\mathrm{\Lambda }$ obtained with 12 different drops for $\mathrm{Me}=60\pm 10$. The sum of the two probabilities $p_{2,\pm 2}+p_{2,0}$ is close to 1.