Infinite-memory classical wave-particle entities, attractor-driven active particles, and the diffusionless Lorenz equations

Active particles are self-propelled entities that extract energy from their surroundings and convert it into directed motion. Active entities can be found at all scales in nature, for example, macroscopic living organisms, such as humans, birds, and fish or microorganisms, such as sperm cells, bacteria, and algae. They also arise in artificial systems, such as active colloidal particles¹ and microrobots.² In some active particle systems, the particle motion is guided by interaction with an environment that is itself created by the particle. For example, the motion of autophoretic microswimmers is powered by chemical activity at the particle’s surface, which generates long-lived chemical gradients.^3–5 This self-generated chemical environment, in turn, guides the motion of the microswimmer. A curious hydrodynamic system of active entities that are driven by self-generated dynamic environments is walking and superwalking droplets.^6,7 In this system, millimeter-sized droplets of oil walk horizontally while periodically bouncing on the free surface of a vertically vibrating bath of the same liquid. Each bounce of the droplet excites a spatially localized standing wave that slowly decays in time. The droplet then interacts with these waves on subsequent bounces to propel itself horizontally, giving rise to a classical, active wave-particle entity (WPE) on the liquid surface.

In the high-memory regime of WPEs, the waves generated by the droplet on each impact decay very slowly in time, hence the droplet’s walking dynamics are not only influenced by the recent waves generated by the droplet but also by the waves generated in the distant past along its trajectory. This gives rise to path memory in the system and makes the dynamics non-Markovian. Remarkably, in the high-memory regime, WPEs have been shown to exhibit hydrodynamic analogs of various quantum systems.^8,9

In the absence of obstacles and other droplets, a WPE typically moves steadily along a straight line. However, in the high-memory regime, it has been observed in experiments that this steady motion of WPEs can become unstable and one observes speed oscillations.¹⁰ To capture the experimentally observed walking dynamics of a WPE, many theoretical models have been developed over the years.^11,12 One such routinely used model with intermediate complexity is the stroboscopic model of Oza et al.¹³ This model provides a trajectory equation for the two-dimensional horizontal walking dynamics in the form of an integrodifferential equation of motion. With the aid of these models, WPE motion can be explored in the very high-memory regime that is currently not achievable in experiments. Simulations of an individual WPE in this regime have shown the emergence of rich dynamical behaviors, such as a run-and-tumble-like diffusive motion.^14,15

To explore the dynamics of WPEs and their hydrodynamic quantum analogs beyond the restricted parameter space of experiments, Bush¹⁶ proposed the framework of generalized pilot-wave dynamics. It is a theoretical abstraction rooted in the walking-droplet system that has allowed for the exploration of a broader class of dynamical systems and the discovery of new quantum analogs.⁹ The generalized pilot-wave framework has motivated the exploration of idealized pilot-wave systems that consider the dynamics of WPEs in one horizontal dimension.^17–24 Typically, in the stroboscopic model of Oza et al.,¹³ one uses a Bessel function wave form to reasonably capture the experimentally observed waves generated by the droplet. This wave form has two key features: spatial oscillations and spatial decay. However, choosing a simple sinusoidal wave form that only captures spatial oscillations for a single one-dimensional WPE, results in the reduction of the infinite-dimensional dynamical system generated by the integrodifferential equation^13,18 to a low-dimensional system of ODEs that can be mapped onto the classic Lorenz system.^19,20 Moreover, Valani²⁵ showed a general transformation for a $1$D WPE that can map the infinite-dimensional integrodifferential equation to low-dimensional Lorenz-like equations for certain choices of wave forms.

One of the key features associated with many of the hydrodynamic quantum analogs of WPEs is the emergence of wave-like statistics from underlying chaotic dynamics in the high-memory regime.^18,26–30 However, the non-Markovian nature of the system in this regime makes the integrodifferential dynamical equation analytically intractable, hence making it difficult to comprehensively explore the underlying chaos that gives rise to emergent wave-like statistics. Motivated by this, in this paper, we consider the infinite-memory limit in the Lorenz model of a $1$D WPE that employs a sinusoidal wave form.²⁵ Contrary to intuition, we show that this infinite-memory limit reduces the system dynamics to one of the algebraically simplest chaotic systems, the diffusionless Lorenz equations (DLE).³¹ We explore this single-parameter dynamical system in detail and connect the geometry, dynamics, and bifurcations of phase-space attractors to the dynamical and statistical features of WPE motion.

As we will show at the end of Sec. II, our infinite-memory WPE dynamical system may also be interpreted as an attractor-driven active particle,³² i.e., an overdamped active particle driven by internal low-dimensional chaotic DLE system, as opposed to constant self-propulsion and stochastic noise that is generally considered in modeling traditional active particles.^33,34 In addition to the WPE system, there are examples of active particles in nature where signatures of low-dimensional chaos have been observed in the motility of organisms. Examples include movement patterns of ants,³⁵ mud snails,³⁶ amoebas,³⁷ and worms.^38,39 In light of these examples, our dynamical system may also be viewed as describing the dynamics of a simple active particle driven by internal complexity, which is modeled by a low-dimensional chaotic system of DLE.³² 

This paper is organized as follows. In Sec. II, we consider the stroboscopic model of Oza et al.¹³ and derive the diffusionless Lorenz equations (DLE) that govern the dynamics of our system, In Sec. III, we consider the steady states of the particle governed by DLE and determine their stability by performing linear stability analysis. In Sec. IV, we numerically explore the unsteady dynamical states in the parameter space of the dynamical system linking the phase-space features of the DLE system with particle’s dynamical and statistical features. We conclude in Sec. V.

Thus, the pair of equilibrium points in Eq. (12) are unstable, specifically a pair of saddle-focus,⁵⁴ implying that the steady motion of WPE at infinite memory is always unstable to small perturbations. Moreover, the presence of a complex conjugate pair of eigenvalues hints at oscillatory instability of the steady WPE. From the viewpoint of an attractor-driven particle, the instability of the steady state implies that the internal state $(X,Y,Z)$ never settles onto a fixed point and is always changing with time. We now proceed to numerically explore the various unsteady dynamical behaviors arising in the parameter space of the system.

We start by presenting the main types of unsteady dynamical behaviors observed in the motion of the WPE as a function of the dimensionless wave-amplitude parameter $R$⁠. At first, it may appear that in the infinite-memory limit, the non-decaying nature of the waves can potentially lead to an ever increasing amplitude of the underlying wave field via constructive interference of individual waves, i.e., $|Z | → ∞$⁠, and, consequently, an ever increasing magnitude of wave-memory force, $|Y | → ∞$⁠, and particle speed, $|X | → ∞$⁠. However, the divergence of the dynamical flow, $∇⋅( X ˙ , Y ˙ , Z ˙ )=−1$⁠, is negative, hence phase-space volume elements contract with time resulting in the $(X,Y,Z)$ phase-space trajectories settling onto an attractor of the system and remaining bounded. Since the steady states of the system are always unstable, we only find unsteady behaviors of the WPE that result in limit cycle attractors and strange attractors in phase-space.

Figure 2(a) shows a bifurcation diagram as a function of the parameter $R$⁠, which plots the wave-memory force on the particle, $Y n=Y( t n)$⁠, sampled at times $t n$ that correspond to the particle’s instantaneous velocity being zero, i.e., $X( t n)=0$⁠. The corresponding maximal Lyapunov exponent (MLE) as a function of $R$ is shown in Fig. 2(b). A positive MLE hints at the presence of chaos. The bifurcation diagram is colored based on the following three distinct types of unsteady WPE dynamics: (i) self-trapped oscillating WPE (red), where the WPE undergoes back-and-forth oscillations about a fixed point, (ii) runaway oscillating WPE (yellow), where the WPE undergoes inline oscillations along with a net drift, and (iii) irregular WPE (purple) that exhibit chaotic walks. We can associate these motions of WPE to the corresponding dynamics and geometry in the $(X,Y,Z)$ phase-space of DLE [see Fig. 2(c)]. The self-trapped oscillating WPE corresponds to a limit cycle in phase-space with a symmetric geometry while the runaway oscillating WPE corresponds to a limit cycle in phase-space with asymmetric geometry. Runaway oscillating WPE occurs in pairs that correspond to the net drift of the WPE in the positive or negative direction. Irregular chaotic WPE corresponds to dynamics on a strange attractor in phase space. We now proceed to explore the dynamics as a function of $R$⁠. We have organized the remaining manuscript by dividing the parameter space into three regions: (i) small wave-amplitude regime $( R < 0.2 )$⁠, (ii) intermediate wave-amplitude regime $( 0.2 < R < 2 )$⁠, and (iii) large wave-amplitude regime $(R>2)$⁠. These ranges approximately correspond to qualitatively different types of WPE dynamics observed in each regime.

We start by exploring in detail the regime of small wave-amplitude parameter $R$⁠. Figure 3(a) shows a detailed bifurcation diagram, similar to the one in Fig. 2(a), but focused on the region $0.02<R<0.2$⁠. We find that this regime is dominated by intermittent dynamics, where the trajectory of the WPE alternates between long stationary phases and short walking phases. Three different types of motion described at the start of Sec. IV are realized in this regime with intermittent dynamics: (i) self-trapped intermittent WPE as shown in Fig. 3(b), (ii) runaway intermittent WPE as shown in Fig. 3(c), and (iii) irregular intermittent WPE as shown in Fig. 3(d). As shown in the inset of Fig. 3(b), two symmetric pairs of solutions exist for runaway intermittent WPE that correspond to the net drift of the particle in the positive (yellow) and negative (black) direction. We see from the crossing of the two asymmetric limit cycles that they are topologically linked in phase-space, i.e., the yellow and black limit cycles pass through each other in phase-space.

From the bifurcation diagram in Fig. 3(a), we can see that the three types of intermittent motion are intricately interwoven in what appears to be a self-similar period-doubling bifurcation structure with an increasing number of bifurcations squeezed into an infinitely thin region as $R → 0$⁠. We refer the interested reader to the work of van der Schrier and Maas,³¹ who derived analytical approximations of such self-similar bifurcations arising in this regime. We further observe in Fig. 3(a) that the periodic attractors for these self-similar bifurcations alternate between symmetric (red) and asymmetric (yellow) limit cycles. Self-similar bifurcations have also been recently reported in certain regimes of the classic Lorenz system;⁵⁵ this system is connected to $1$D WPE dynamics at finite memory.^20,25 The bifurcation diagram in Fig. 3(a) also reveals multistability in this system, as evident from the presence of multiple attractors (multiple colors, e.g., yellow/red/purple) at the same $R$ value. We will discuss this aspect of multistability and the corresponding basin of attraction in some detail in Sec. IV D.

We have observed similarities between the intermittent dynamics described here and the stop-and-go motion of superwalking droplets.^7,56 Superwalkers^7,42 are bigger and faster walking droplets that emerge when the bath is vibrated at two frequencies simultaneously, namely, a particular frequency and half of that frequency, along with a constant phase difference. By detuning the two driving frequencies by a small amount, one can get the phase difference to drift slowly in time. This detuned two-frequency driving results in a novel walking motion for superwalkers known as stop-and-go motion.^7,56 The stop-and-go motion of droplets, enabled by the varying phase difference, results in periodic traversals of the stationary and walking regimes in the parameter space of the physical system. In their simulations of stop-and-go motion, Valani et al.⁵⁶ reported three different types: (i) back-and-forth, (ii) forth-and-forth, and (iii) irregular. The particle trajectories of these stop-and-go motions and the three intermittent motions observed in our system (self-trapped intermittent, runaway intermittent, and irregular intermittent) are very similar despite being different systems. The stop-and-go motion of superwalkers is a complex nonlinear phenomenon with multiple time scales coming into play, such as the bouncing time scale of the droplet, the memory time scale associated with the decay of droplet-generated waves, the even longer time scale introduced by the detuning, and the time scale of the inertial response of the droplet. Conversely, the mechanism for these intermittent motions for our WPE does not require any external parametric driving of the system between stationary and walking states, as done in stop-and-go motion. The intermittent motion of our WPE at infinite memory is an emergent phenomenon arising from a combination of a slow instability of the stationary state and the WPE trapping itself in a nearby trough when walking.

We note that Durey et al.¹⁸ also reported intermittent motion of the WPE (called jittering modes in their paper) in their numerical simulations of the integrodifferential equation of motion with a Bessel wave form in the high-memory regime. They rationalized the mechanics of growth-relaxation process of intermittent motion in terms of a linearized integrodifferential equation model during the growth stage of the wave field and an overdamped particle moving in a static potential during the relaxation stage. The similarities in intermittent dynamics suggest that our simple WPE with sinusoidal wave form and infinite memory can successfully capture at least qualitatively these intricate states observed with a more realistic Bessel wave form. Moreover, our simple ODE model sheds light on the dynamical processes in $3$D phase-space that result in intermittent trajectories of the WPE in physical space–time. It further suggests that a low-dimensional attractor might be governing the dynamics of the WPE with Bessel wave form in this regime. Durey et al.¹⁸ further analyzed the irregular intermittent WPE motion by modeling it as a stochastic, discrete-time, and Markovian jump process, where the particle can move left or right with equal probability and a fluctuating step length. They showed that this model showed consistent results with their simulations with multimodal statistics at intermediate timescale and Gaussian distribution at long times in the particle’s position distribution. In the remainder of this section, we explore chaotic and statistical properties of irregular intermittent WPEs using our ODE model and compare some of these results with that of Durey et al.¹⁸ obtained for the Bessel function wave form.

We start by exploring the chaotic aspects of irregular intermittent WPE. A typical trajectory is shown in Fig. 5(a) for $R=0.11$⁠. To understand the chaotic behavior, we plot $1$D return maps of different quantities associated with the trajectory of irregular intermittent WPE. A $1$D return map that plots the location of particle $x n$ in the $n$th stationary phase vs the (⁠ $n+1$⁠)th stationary phase is shown in Fig. 5(b). One observes two parallel line structures in this map indicating that the map is multi-valued. This is consistent with the intermittent irregular trajectory since at a given location in the stationary phase, $x n$⁠, the particle can unpredictably either take the next step, $x n + 1$⁠, to its left or to its right. We note that the structure of this return map and the intermittent irregular WPE trajectories are reminiscent of pseudolaminar chaotic diffusion,⁵⁷ where a time series with constant-value laminar phases is periodically interrupted by chaotic bursts. This is different from laminar chaos arising in time-delay systems with periodically varying delay, where also a similar time series is encountered but in addition, the levels of laminar phases in the time series are related by a simple and robust one-dimensional map.^58,59 Next, we analyze the return map of the step length $L n$ (scaled by the wavelength $2π$⁠) of consecutive steps resulting in Fig. 5(c). We make two observations: (i) all the steps are nearly constant and slightly bigger than half the wavelength and (ii) the variations in the step length are well captured by this $1$D return map with a well-defined structure. Hence, we see evidence of low-dimensional chaos in the variations in step length during intermittent irregular trajectories. However, instead of the length of the step, if one plots the return map of consecutive durations $T n$ of time spent in the stationary phase, then one gets a more complicated multi-valued map as shown in Fig. 5(d). It would be interesting to compute these maps for more complete WPE models, such as the Bessel wave form model of Durey et al.¹⁸ to see if they also show signatures of low-dimensional chaos for these quantities.

We now explore the statistical features of these nearly constant step-length chaotic walks. We have calculated the position distribution of the particle in this irregular intermittent regime as a function of time as shown in Fig. 6(a). This was done by initiating $3000$ WPEs at $x d(0)=0$ with a uniform velocity (⁠ $X$⁠) distribution in the range $[−0.5,0.5]$⁠. We note that only those trajectories that displayed intermittent irregular behavior were included in the position distribution since the system exhibits multistability [see Fig. 9(a)]. From Fig. 6(a), we observe a wave-like distribution that persists for a long time. The long persistence of the wave-like distribution can be attributed to the coherence of the sinusoidal waves. Since each step taken by the WPE is nearly of half the wavelength, i.e., $π$⁠, with a narrow distribution in uncertainty [see Fig. 5(c)], it may take a very long time for these small differences in step-length uncertainties to accumulate and for the sharp peaks to diffuse. However, as it can be seen in bottom panels (A)–(E) of Fig. 6, the envelope of the distribution diffuses with time and the sharp wave-like features decay. Thus, we observe that wave-like features in the distribution persist for long time but the spreading of the overall envelope due to the diffusive nature of trajectories results in these wave-like features diminishing with time. For the WPE dynamics in this regime with a Bessel wave form studied by Durey et al.,¹⁸ they observed that wave-like features diffused relatively quickly into a Gaussian-like distribution which then spreads over space. The relatively early suppression of wave-like features in the position distribution with Bessel wave form might be due to phase shifts between consecutive peaks of the Bessel function (as compared to sinusoidal waves) in combination with spatial decay. Such features may result in larger fluctuations in the step length of intermittent irregular WPEs and, hence, the distribution transitions early from multimodal to Gaussian compared to our sinusoidal wave form. We also observe another feature from Fig. 6(a) that this probability distribution oscillates at small and intermediate timescales, i.e., at a fixed location $x$⁠, the distribution is oscillating with time. This is due to the discrete nature of the intermittent trajectories where a given location in space is occupied and unoccupied by different intermittent irregular walkers of nearly constant step length.

The diffusive behavior of intermittent irregular WPEs can be characterized by calculating how the mean squared displacement (MSD) scales with time, i.e., $MSD=⟨ ( x d ( t ) − x d ( 0 ) ) 2⟩∼ t α$ with $α$ being the diffusion exponent. To quantify this, we define a time-dependent diffusion exponent $α(t)=d(log(MSD))/d(log(t))$ and plot it as a function of time as shown in Fig. 6(b). We observe subdiffusion (⁠ $0<α<1$⁠) for intermediate time scales and the WPE appears to be approaching normal diffusion, i.e., $α → 1$ asymptotically. This was also observed by Durey et al.¹⁸ for intermittent irregular WPE dynamics with Bessel wave form.

From the viewpoint of an attractor-driven particle, the appearance of such wave-like statistical features that persist for long is not common in traditional $1$D active particles. A commonly used minimal model for a $1$D active particle is a run-and-tumble particle (RTP). It is an overdamped particle that moves with a constant self-propulsion speed and flips its direction of motion following a constant-rate Poisson process.⁶⁰ Sometimes Gaussian white noise is also added as an additional stochastic force.⁶¹ Such RTPs show bimodal position distribution at very short timescale due to their ballistic motion while at long time scales the distribution approaches a Gaussian. Due to the intermittent nature of the trajectory of our attractor-driven particle, we obtain persistent spatial oscillations and also temporal oscillations at short and intermediate times. Thus, the rich dynamical and statistical features arising from our attractor-driven particle can motivate the modeling of new classes of active particles.

We now turn to explore the intermediate wave-amplitude regime $0.2<R<2$⁠. This is also the regime of $R$ parameter that would typically correspond to experiments with walkers and superwalkers^7,41 if one can achieve this regime of very high memory and confine the droplet motion to one-dimension, e.g., by restricting its motion to a thin annular region.^23,62,63 However, we do not expect the dynamics observed here to quantitatively match experiments since we are using an idealized model, but qualitative similarities in trajectories may be realized.

A detailed bifurcation diagram of this regime is shown in Fig. 7(a) and the corresponding MLEs are shown in Fig. 7(b). We find that this regime mainly comprises of irregular WPEs with small regions of self-trapped WPEs and runaway WPEs. A multi-stable region is observed near $R≈0.2$ with the coexistence of irregular and self-trapped WPEs followed by a region of runaway WPEs near $R≈0.3$⁠. These runaway WPEs bifurcate into irregular WPEs near $R≈0.35$⁠. A trajectory of an irregular WPE just after this transition is shown in Fig. 7(c). Here, we find that the WPE shows subdiffusive behavior for a large range of intermediate timescales and very slowly seems to be approaching asymptotic diffusion [see inset of Fig. 7(c)]. Further increase in $R$ leads to increasing complexity as well as enlarging of the chaotic attractor in phase-space as depicted in Figs. 7(c)–7(e). The increasing physical extent of the attractor is also reflected in the widening of the envelope in the bifurcation diagram in Fig. 7(a) with increasing $R$⁠. For these trajectories, we also find asymptotic diffusion [see inset of Figs. 7(d) and 7(e)] with the diffusion constant typically larger for larger $R$⁠. The MLE typically also increases with an increase in the $R$ value in this regime [see Fig. 7(b)].

The increasing complexity of the DLE strange attractor also provides a rich set of statistical features for our attractor-driven particle that can be tuned by the control parameter $R$⁠. By varying $R$⁠, one can induce desired transport properties, i.e., trapping from self-trapped WPEs, or ballistic motion from runaway WPEs, or subdiffusion and normal diffusion with diffusion coefficients that can be tuned by varying $R$⁠.

We now turn toward the large wave-amplitude regime that corresponds to $R>2$⁠. A detailed bifurcation diagram in this regime is shown for $2<R<20$ in Fig. 8(a). We see that as $R$ increases in this regime, the irregular WPEs cease near $R=5$⁠. This happens via a period halving bifurcation where irregular WPEs bifurcate into runaway WPEs. These runaway oscillating WPEs further bifurcate into self-trapped oscillating WPEs near $R≈7$⁠. The self-trapped oscillating WPEs persist as $R → ∞$⁠.

To explore this regime further, we show plots of the space–time trajectories (left) and phase-space attractors (right) for increasing $R$ values in Figs. 8(b)–8(i). At $R=4.6$⁠, a strange attractor exists in phase space with a symmetric structure [Fig. 8(b)]. The symmetry of the attractor implies that the irregular WPE on average has no net displacement. As $R$ increases to $4.8$ [Fig. 8(c)], a dynamical symmetry breaking⁵⁴ takes place forming a pair of strange attractors [purple and black in the inset of Fig. 8(c)]. The two attractors correspond to a net drift in the positive (black) and negative (purple) direction. This asymmetric strange attractor causes irregular modulations in the oscillations of runaway WPEs. We further observe from the inset of Fig. 8(c) that the two strange attractors are topologically linked. Further increasing $R$⁠, we obtain a runaway oscillating WPE at $R=5.2$ [Fig. 8(d)], which undergoes a period halving as $R$ increases to $5.5$ [Fig. 8(e)]. For these runaway oscillating walkers, the link between the two attractors is preserved as shown in the insets of Figs. 8(d) and 8(e), respectively. On further increasing $R$ to $7$ [Fig. 8(f)], we see that the phase-space limit cycle of the runaway oscillating WPE becomes less asymmetric corresponding to a smaller drift speed of the WPE. This state eventually transitions to a symmetric limit cycle and one obtains self-trapped oscillating WPE as shown in Fig. 8(g) for $R=10$⁠.

From hereon, further increasing $R$ to $20$ [Fig. 8(h)] and $100$ [Fig. 8(i)] we find that the extent of the limit cycle in phase space keeps on increasing while the amplitude of oscillations of the WPE in physical space remains constant and the frequency of oscillations increases. This can be understood as follows (see also Video 1 in the supplementary material): At large $R$⁠, the particle performs self-trapped oscillations between two consecutive peaks of its own wave field. When the particle is near the trough between the two peaks, the particle-generated waves will interfere destructively with the built-up wave field. This will decrease the amplitude of the overall wave field. Conversely, when the particle is near the peaks of the wave field, the particle-generated waves will interfere constructively with the built-up wave field and the overall wave field amplitude will increase. Now, the turning points of the particle’s oscillations occur just below peaks and the particle spends a long time there, whereas near the trough, the particle is moving fast and spends little time. Thus, the particle can never lower its wave field enough to escape to neighboring minima of its wave field and its motion is always confined between two consecutive peaks of its wave field. Moreover, since the amplitude of the particle-generated wave at each instant scales with $R$⁠, the corresponding height of the wave field $Z$⁠, its wave gradient $Y$ and the particle velocity $X$ also increase with $R$ resulting in an increase in size of the limit cycle with $R$ in phase space. However, since the particle can never escape its peaks, its motion remains bounded between two consecutive peaks, and, hence, the amplitude of oscillations in particle position remains fixed.

In this limit of large $R$⁠, surprisingly, the system becomes integrable and reduces to solving second Painlevé transcendent that behaves asymptotically like elliptic functions.³¹ There have been few studies that have explored this regime of DLE in detail^31,64–66 and we refer the interested reader to these papers.

We have observed multistability in this system where phase-space attractors that correspond to different types of WPE motion coexist at the same $R$ value in the small, intermediate, as well as large wave-amplitude regime. Some examples of coexisting attractors and their basin of attraction are shown in Figs. 9(a)–9(d). Figures 9(a) and 9(b) show multistability in the small wave-amplitude regime for $R=0.06$ and $R=0.11$⁠, respectively. For $R=0.06$⁠, we observe the coexistence of self-trapped intermittent WPEs and runaway intermittent WPEs. The basin of attraction reveals a complex structure with the basins of left and right moving runaway oscillating WPEs intricately intertwined in a sea of self-trapped oscillating WPEs. From the phase-space trajectories, we see that the two asymmetric limit cycles for runaway oscillating WPEs seem to have multiple links. For $R=0.11$⁠, we find the coexistence of irregular intermittent WPEs and runaway intermittent WPEs. Here also, we find an intricate basin of attraction for the two runaway oscillating WPE attractors embedded in a sea of irregular WPE attractors. However, the structure appears to be less complex as compared to the basin of attraction of $R=0.06$⁠. This relatively low complexity is also reflected in the phase-space trajectory of the two asymmetric limit cycles for runaway WPEs, which have one simple link. Figure 9(c) shows multistability for $R=0.22$⁠, where irregular WPEs co-exist with self-trapped oscillating WPEs. Here, the basin boundary is even more smooth and the two asymmetric runaway oscillating WPE attractors are replaced by a single symmetric self-trapped oscillating WPE attractor. Figure 9(d) shows multistability in the large wave-amplitude regime for $R=6$⁠, where now two different kinds of runaway oscillating WPEs co-exist and we find a fractal basin of attraction. Different types of coexisting runaway oscillating WPEs were also observed in the high-memory regime by Durey et al.¹⁸ using their Bessel wave form.

From the viewpoint of a WPE, a typical experimental initial condition would correspond to the particle initially at rest with no wave-memory, i.e., $(X(0),Y(0),Z(0))=(0,0,0)$⁠. These lie right inside the fractal structure seen in these basins of attractions. Hence, if this regime can be realized in experiments with walking/superwalking droplets, then one might expect extreme sensitivity to initial conditions since typical initial conditions for WPEs in experiments are likely to fall in the intricate structure.

This aspect of multistability also enables an easy way to access different dynamical states from the viewpoint of an attractor-driven particle. By adding a small amount of noise in the internal state dynamics of the DLE for fixed $R$⁠, the attractor-driven particle can transition from irregular motion to self-trapped motion or runaway motion as the internal state-space system switches between different types of phase-space attractors. Of course, this may also be achieved by tuning the control parameter $R$ in the appropriate regime.

In this paper, we have explored the rich dynamical behaviors of a classical active WPE in the limit of infinite wave-memory. We showed that the system reduced to one of the algebraically simplest chaotic systems, the diffusionless Lorenz equations (DLEs), with a single parameter $R$ representing the dimensionless wave amplitude. The algebraic simplicity of ODEs is deceiving, and the system exhibits rich dynamics and bifurcation structure, which we have explored in the context of WPE motion and an attractor-driven particle.

The rich dynamical behaviors observed for the WPE as a function of $R$ were classified into three distinct types: self-trapped oscillating WPE, runaway oscillating WPE, and irregular WPE. In the small $R$ regime, these three types of dynamical behaviors were realized with intermittent dynamics, where the WPE spends a long time in a stationary state while it is building/erasing its wave field and then swiftly takes a step of nearly half the wavelength. We linked this mechanism of intermittent motion to the corresponding dynamics taking place in the phase space of the system, where each step in the intermittent motion of WPE is related to an orbit around one wing of the corresponding phase-space attractor. Durey et al.¹⁸ in their infinite-dimensional integrodifferential equation model for WPE dynamics with a Bessel wave form found similar trajectories. We find that our simple model that reduces to a system of three nonlinear ODEs captures qualitative features of the more complete model. The bifurcation diagram in the small $R$ regime showed a self-similar period-doubling structure, where all three types of motion exist. We explored chaotic aspects of irregular intermittent WPE, where the return map of the step length showed a low-dimensional structure and the trajectory showed similarities with pseudolaminar chaotic diffusion. We also explored the statistical properties by investigating the position distribution of particles and found wave-like statistics that persist for long times. Moreover, time-periodic fluctuations were observed in the position distribution at short and intermediate time scales. In the intermediate $R$ regime, the system exhibited mainly chaotic dynamics with the extent and complexity of the phase-space attractors increasing with $R$⁠. In the large $R$ regime, a period halving bifurcation ceases chaos and one eventually gets symmetric limit cycles corresponding to self-trapped oscillations with the size of the limit cycle in phase-space increasing with increasing $R$ but the particle motion confined between two consecutive peaks of its wave field. We also showed multistability in the system where different types of motion coexist at the same $R$⁠, and they are intricately interwoven in the basin of attraction.

The rich set of dynamical behaviors exhibited by the DLE also gives our attractor-driven particle a diverse array of features that are not typically observed in traditional active particles. The single parameter $R$ provides a convenient way to assign different dynamical states to the attractor-driven particles and the presence of multistability further enhances this richness and provides ways to access different dynamical states at the same $R$ value. When the DLE system exhibits chaos on a strange attractor with the complexity of the attractor varying with $R$⁠, this provides a way to tune the transport properties of the attractor-driven particle. Moreover, the intermittent motion for small $R$⁠, gives rich statistical features to the attractor-driven particle such as spatial and temporal oscillations. This specific example of attractor-driven particle explored in this paper shows the richness of the framework of attractor-driven matter.³² 

The ODE framework of our simple Lorenz-like system enables a detailed exploration of three-dimensional phase-space attractors and their bifurcations allowing us to link the dynamics and geometry in phase-space to the motion and trajectories of the WPE or the attractor-driven particle. Even these deceptively simple looking Lorenz-like systems exhibit a complex array of behaviors in phase-space that have not been completely uncovered, and research is still in progress to understand the interplay between geometry, dynamics, and topology.^67–73 A comprehensive understanding of phase-space behaviors associated with the underlying attractors of WPE systems may lead to new perspectives in rationalizing quantum-like statistics in hydrodynamic quantum analogs of walking droplets and also new advances in active particle modeling using attractor-driven particles.

Video 1 in the supplementary material shows self-trapped oscillating trajectory of the WPE for R = 10. MATLAB code shows a template code that simulates the nonlinear ODEs of the system explored in this manuscript using MATLAB ode45 solver.

I would like to thank David Paganin for helpful comments and discussions. R.V. was supported by Australian Research Council (ARC) Discovery Project DP200100834 during the course of the work. Some of the numerical results were computed using supercomputing resources provided by the Phoenix HPC service at the University of Adelaide.

The authors have no conflicts to disclose.

R. N. Valani: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Resources (lead); Software (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.