Lorenz-like systems emerging from an integro-differential trajectory equation of a one-dimensional wave-particle entity

A millimetric self-propelled wave-particle entity can emerge in the form of a walking droplet when a liquid bath is vibrated vertically Couder et al. (2005a, b); Valani, Slim, and Simula (2019). The walking droplet, also known as a walker, on each bounce generates a localized standing wave on the liquid surface that decays in time. The droplet interacts with these self-generated waves on subsequent bounces to propel itself horizontally. The droplet and its underlying wave field coexist as a wave-particle entity; the droplet generates the wave field which in turn guides the motion of the droplet. At large amplitudes of vertical bath vibrations, the waves created by the walker decay very slowly in time and the walker’s motion is not only influenced by the wave created on its most recent bounce, but also by the waves generated in the distant past, giving rise to memory in this hydrodynamic system. In the high-memory regime, walkers have been shown to mimic several features that are typically associated with quantum systems. These include orbital quantization in rotating frames (Fort et al., 2010; Harris and Bush, 2014; Oza et al., 2014a) and confining potentials (Perrard et al., 2014a, b; Labousse et al., 2016a; Montes, Revuelta, and Borondo, 2021), Zeeman splitting in rotating frames (Eddi et al., 2012; Oza, Rosales, and Bush, 2018a), wave-like statistical behavior in confined geometries (Harris et al., 2013; Gilet, 2016; Sáenz, Cristea-Platon, and Bush, 2018; Cristea-Platon, Sáenz, and Bush, 2018; Durey, Milewski, and Wang, 2020) as well as in an open system (Sáenz, Cristea-Platon, and Bush, 2020), tunnelling across submerged barriers (Eddi et al., 2009; Nachbin, Milewski, and Bush, 2017; Tadrist et al., 2020) and macroscopic analog of spin systems (Sáenz et al., 2021). Walkers have also been predicted to show anomalous two-droplet correlations (Valani, Slim, and Simula, 2018; Nachbin, 2018). Recently, efforts have also been made to develop a hydrodynamic quantum field theory for the walking-droplet system (Dagan and Bush, 2020; Durey and Bush, 2020). A detailed review of hydrodynamic quantum analogues for walking droplets is provided by Bush (2015) and Bush and Oza (2020).

Several theoretical models have been developed over the years that capture the walker’s dynamics (Turton, Couchman, and Bush, 2018; Rahman and Blackmore, 2020). An analytically tractable integro-differential equation of motion for the walker that captures the essential features of the horizontal walking dynamics was developed by Oza, Rosales, and Bush (2013). This stroboscopic model averages over the droplet’s vertical periodic bouncing motion and provides a trajectory equation for its two-dimensional horizontal walking motion by taking into account two key horizontal forces acting on the walker: (i) the horizontal wave force proportional to the gradient of the underlying wave field generated by the walker, and (ii) an effective horizontal drag force composed of aerodynamic drag and momentum loss during impact with the fluid surface. This stroboscopic model rationalizes several hydrodynamic quantum analogs (Oza et al., 2014a; Harris and Bush, 2014; Labousse et al., 2016b; Cristea-Platon, Sáenz, and Bush, 2018; Oza, Rosales, and Bush, 2018b; Kurianski, Oza, and Bush, 2017; Tambasco and Bush, 2018; Valani, Slim, and Simula, 2018; Sáenz et al., 2021) and also results in rich dynamical behaviors for walkers (Valani et al., 2021a; Durey, 2020; Valani and Slim, 2018; Oza et al., 2014b; Tambasco et al., 2016; Arbelaiz, Oza, and Bush, 2018; Oza et al., 2017).

The dynamics of a walker emerging from a one-dimensional reduction of the stroboscopic model were explored by Durey (2020) and Valani et al. (2021a) by employing a simple sinusoidal form for the waves generated by the walker. They uncovered a variety of unsteady motions for a walker in addition to the constant velocity walking state. These include oscillating walkers, self-trapped oscillations and irregular walking. Both of these studies highlighted similarities between the walker’s dynamical system and the Lorenz system (Lorenz, 1963). Moreover, Valani et al. (2021a) showed an exact correspondence between the walker’s equation of motion and the Lorenz system.

As a paradigm of chaos, the Lorenz system and other Lorenz-like systems that exhibit chaos or hyperchaos are being studied widely. Such systems are interesting to study not only because of their rich dynamical behaviors, but also due to the various applications of their chaotic behaviors in areas such as image encryption Zou et al. (2020); Wang et al. (2018); Kaur and Kumar (2018), secure communication (Feki, 2003; Pan, Ding, and Du, 2012), electronic oscillators and nonlinear circuits (Li et al., 2009; Yu et al., 2010; Blakely, Eskridge, and Corron, 2007), robotics (Zang et al., 2016) and chemical reactions (Poland, 1993).

Building on the equivalence between the walker’s trajectory equation and the Lorenz equation as demonstrated by Valani et al. (2021a) for a one-dimensional sinusoidal wave form, in this paper, we present this connection in a generalized framework. By transforming the one-dimensional trajectory equation for a wave-particle entity with an arbitrary spatial wave form into a system of ODEs, we explore different wave forms that result in Lorenz-like dynamical systems. The paper is organized as follows. We start by presenting the integro-differential trajectory equation for the wave-particle entity in Sec. II. We then in Sec. III present a transformation to convert the integro-differential equation of motion into a system of infinite ODEs having Lorenz-like structure. Then, we consider different wave forms in Secs. IV-VII and show that they can be reduced to a finite system of Lorenz-like ODEs and present the resulting dynamics. In Sec. IV, we start by exploring the Bessel wave form that serves as a good approximation of the wave form for the experimentally realized walking droplet. After that, we consider other wave forms that result in Lorenz-like dynamical systems. Such wave forms are currently not realized in walking-droplet experiments but they may provide to be useful in exploration of hydrodynamic quantum analogs in a generalized framework. In Sec. V, we consider wave forms whose gradients are solutions of constant coefficient ODEs and explore the Lorenz-like systems of a pure sinusoidal wave form as well as sinusoidal wave form with an exponential envelope. In Sec. VI, we consider polynomial wave form and explore the Lorenz-like system emerging for a double-well wave form. In Sec. VII, we consider periodic wave forms and explore the Lorenz-like system for a wave form composed of two frequencies. Finally, we conclude in Sec. VIII.

As shown schematically in Fig. 1, consider a particle located at position $x_{d}$ and moving horizontally with velocity $\dot{x}_{d}$ while continuously generating waves with prescribed spatial structure $W(x)$ that decay exponentially in time. The dimensionless equation of motion governing the horizontal dynamics of the particle is given by Oza, Rosales, and Bush (2013),

The left-hand-side of Eq. (1) is composed of an inertial term $\kappa\ddot{x}_{d}$ and an effective drag term $\dot{x}_{d}$, where the overdot denotes differentiation with respect to time $t$. The first term on the right-hand-side of the equation captures the forcing on the droplet by the underlying wave field $h(x,t)$. This force is proportional to the gradient of the underlying wave field. The second term is an external force $\mathcal{F}$ that may act on the particle. This external force is generic and may depend on the position of the particle $\mathcal{F}(x_{d})$ or its velocity $\mathcal{F}(\dot{x}_{d})$ or more generally depend on time $\mathcal{F}(t)$. The shape of the wave field $h(x,t)$ is calculated through integration of the individual wave forms $W(x)$ that are continuously generated by the particle along its trajectory. This gives

Combining Eqs. (1) and (2), we obtain the integro-differential equation

where $f(x)=-W^{\prime}(x)$ is the negative gradient of the wave form and the prime denotes differentiation with respect to the argument $x$. This integro-differential trajectory equation was derived by Oza, Rosales, and Bush (2013) to describe the horizontal dynamics of the walking droplet by employing a Bessel function of the first kind and zeroth order, $W(x)={J}_{0}(x)$, wave form for the individual waves generated by the droplet on each bounce. The two parameters in this dimensionless equation of motion, $\kappa>0$ and $\beta>0$, follow directly from Oza, Rosales, and Bush (2013) and may be usefully interpreted as the ratio of inertia to drag and the ratio of wave forcing to drag respectively. We note that $\kappa\sim 1/\text{Me}$ and $\beta\sim\text{Me}^{2}$, where Me is the memory parameter that governs the rate of temporal decay of the underlying waves with a larger value of Me indicating slower temporal decay (Oza, Rosales, and Bush, 2013). Thus, the high-memory regime where the walker typically exhibits hydrodynamic quantum analogs, corresponds to the region of small $\kappa$ and large $\beta$ in the parameter space of the model (Valani, Slim, and Simula, 2018; Valani et al., 2021a).

We start by transforming the integro-differential equation presented in Eq. (3) into a infinite system of ODEs. Let the wave-memory force term in the trajectory equation be,

Differentiating this with respect to time and using the Leibniz integration rule we get,

where the prime denotes the derivative of the function with respect to its argument, and

Similarly if we differentiate $M_{1}(t)$ with respect to time we get,

where,

In this way, the integro-differential equation of motion can be changed into the following infinite system of ODEs:

where $n=0,1,2,...$ and

with the superscript ‘$(n)$’ representing the $n$th derivative of the function with respect to its argument. We note that the system of ODEs in Eq. (4) include multiplicative nonlinear terms of the form $vM_{n+1}$ which are similar in form the the nonlinear terms encountered in Lorenz-like systems.

We start by considering a typical wave form that has been used to model the waves generated by the walker in experiments. This is the Bessel function of the first kind and zeroth order. This functional form serves a good approximation to the droplet-generated waves in the vicinity of the droplet Moláček and Bush (2013); Oza, Rosales, and Bush (2013). Choosing a Bessel function wave form $W(x)=J_{0}(x)$ results in the wave form gradient $f(x)=J_{1}(x)$. Substituting in Eq. (5) gives,

This gives us an infinite system of ODEs to solve for the dynamics of the wave-particle entity with a Bessel wave form. However, one can get an approximate finite system of ODEs from this infinite system by neglecting terms with higher order Bessel functions. Two different approximate forms of finite Lorenz-like system obtained in this manner and consisting of four and six ODEs respectively are given by (see Appendix A for derivation)

and

Here $X=v$, $Y=\beta M_{0}$, $Z=\beta(1/2-M_{1})$, $W=\beta N_{3}$ and $U=\beta N_{2}$, where

and $\sigma=1/\kappa$, $r=\beta/2$ and $b=1$. We also assume no external force on the walker i.e. $\mathcal{F}=0$. A comparison of the numerical solution of these approximate Lorenz-like systems with the solution of the full Bessel wave form system for a typical parameter value in the chaotic regime of the walker’s dynamics is shown in Fig. 2. The numerical solution for the complete Bessel wave form was obtained by solving the walker’s integro-differential equation using a semi-implicit Euler method as described in Valani et al. (2021a). The finite Lorenz-like system of ODEs, here and in the remainder of the paper, were solved using the $\mathtt{MATLAB}$ inbuilt ode45 solver. We see from Fig. 2(a)-(c) that the chaotic attractor of all three solutions have similar morphology with the “double wing" structure that is typical of Lorenz strange attractor. We note that the simpler four ODEs Lorenz-like system only differs from the original Lorenz system (Lorenz, 1963) by the coefficient $3/4$ in the last equation of Eq. (6). Observing the probability distribution of the particle’s velocity $X$, we find a qualitative difference between the four ODEs and the six ODEs Lorenz-like systems with the probability distribution of the six ODEs Lorenz-like system having a closer resemblance to that of the full Bessel wave form system (see Figs. 2(d)-(f)). Examining the 1D return map of the maxima in time series of particle’s speed $|X|$ (see Fig. 2(g)-(i)) reveals cusp-like structures that are different for the three systems. Specifically, we find that cusps from the full Bessel wave form solution and the simpler four ODEs Lorenz-like system have a well-defined structure compared to the six ODEs Lorenz-like system. However, the cusp structure of the six ODEs Lorenz-like system has a closer resemblance to the cusp structure of the full Bessel wave solution.

In the remainder of the paper, we will consider different functional forms of the wave form gradient $f(x)$ that exactly reduce to Lorenz-like systems with finite ODEs. Such wave forms at present are not realized in the experimental hydrodynamic system of walking droplets, but nevertheless they may provide to be useful in exploration of hydrodynamic quantum analogs in a generalized pilot-wave framework.

We consider spatial wave forms for the wave-particle entity whose wave form gradient function, $f(x)$, is a solution of an $n$th order constant coefficient linear homogeneous ODE of the form:

This form of the gradient function allows us to reduce the infinite system of ODEs in Eq. (4) to a finite system by writing the $n$th memory term in Eq. (5) as

Substituting this in Eq. (4) results in a finite system of ODEs consisting of $n+2$ equations as follows:

Thus, the integro-differential trajectory equation of a one-dimensional wave-particle entity whose wave form gradient obeys a constant coefficient linear homogeneous ODE, can be reduced to a finite system of nonlinear ODEs with the nonlinearity being similar in form to the Lorenz-like systems. We now consider two examples of such wave form gradients in Secs. V.1 and V.2.

We now consider the wave form gradient to be an $n$th degree polynomial as follows:

It readily follows that $f^{(n)}(x)=n!\,b_{n}$ and $M_{n}=n!\,b_{n}$. Hence, the infinite system of ODEs in Eq. (4) reduces to the following system of $n+2$ ODEs:

In this section we consider a periodic function for the wave form gradient $f(x)$ with period $L$. Using a Fourier series, we can expand this periodic function in terms of the trigonometric basis functions as follows:

where

Now, using this Fourier expansion we can rewrite the memory force term as

where,

Substituting this in Eq. (4) results in the following system of infinite ODEs,

Hence, when the integro-differential equation of motion for the wave-particle entity having a periodic wave-form is transformed to this system of ODEs, each mode of the Fourier series of the periodic wave form corresponds to a Lorenz-like system with all the different modes coupled to each other in the $\dot{v}$ equation. Moreover, if the Fourier series of the periodic wave form is reduced to a finite number of terms, then Eq. (18) results in a finite Lorenz-like system.

In this paper, we presented a transformation to convert the integro-differential trajectory equation of a one-dimensional wave-particle entity into a system of ODEs which has Lorenz-like structure. By considering different spatial forms of the waves generated by the particle, Lorenz-like systems were obtained that govern the dynamics of the wave-particle entity. For the Bessel wave form, an approximate Lorenz-like system of six ODEs and four ODEs were obtained that reasonably captures the chaotic attractor. For wave form gradients that (i) are solutions of linear constant coefficient homogeneous ODEs or (ii) have a polynomial form or (iii) are periodic and can be approximated by a finite number of terms in Fourier series, exact Lorenz-like systems of finite ODEs were obtained. We presented an example for each of these types of wave form gradients and the corresponding Lorenz-like systems that emerge.

Lorenz-like systems are studied widely by the dynamical systems community due to their various applications. The results in this paper provide an alternate physical interpretation of some Lorenz-like systems in terms of the dynamics of a wave-particle entity. Moreover, the formalism presented in this paper may be useful in constructing new chaotic Lorenz-like systems and understanding their dynamical behaviors in terms of the dynamics of a wave-particle entity.

In the hydrodynamic quantum analogs of walking droplets, the emergent quantum-like statistics emerge from the underlying chaotic dynamics (Bush, 2015; Bush and Oza, 2020). Such emergence has been successfully captured by using the stroboscopic model with the experimentally realized Bessel function wave form. It would be interesting to explore the emergent statistics of a one-dimensional wave-particle entity in confining potentials using a simpler sinusoidal wave form. Investigation of such a simple system may allow us to rationalize the emergent quantum-like statistics from the chaos arising in well-studied Lorenz-like systems. Moreover, the formalism presented in this paper can be easily extended to taken into account interactions of multiple wave-particle entities. Investigations of many interacting wave-particle entities using the framework presented in this work may allow us to understand their collective behavior in terms of coupled Lorenz-like systems.