Order, chaos, and dimensionality transition in a system of swarmalators

Similarly to sperm, where individuals self-organize in space while also striving for coherence in their tail swinging, several natural and engineered systems exhibit the emergence of swarming and synchronization. The arising and interplay of these phenomena have been captured by collectives of hypothetical particles named swarmalators, each possessing a position and a phase whose dynamics are affected reciprocally and also by the space-phase states of their neighbors. In this work, we introduce a solvable model of swarmalators able to move in two-dimensional spaces. We show that several static and active collective states can emerge and derive necessary conditions for each to show up as the model parameters are varied. These conditions elucidate, in some cases, the displaying of multistability among states. Notably, in the active regime, the system exhibits hyperchaos, maintaining spatial correlation under certain conditions and breaking it under others on what we interpret as a dimensionality transition.

Figure 1

Snapshots of the [(a) and (d)] point synchronous, [(b) and (e)] distributed synchronous, and [(c) and (f)] distributed asynchronous states after a transient. The panels show the spatial distribution of the particles (top row) and the respective correlation scatter plots (bottom row). The parameters $(K,J_A,J_R)$ are set as $(1,1,0.5)$ for the state in (a) and (d), $(1,0.5,1)$ for the state in (b) and (e), and $(0,-0.5,0.5)$ for the state in (c) and (f). See movies S1, S2, and S3 in the Supplemental Material [53].

Figure 2

Snapshots of the static phase-wave states after a transient. The panels show the spatial distribution of the particles (top row) and the respective correlation scatter plots (bottom row). The (a) and (d), (b) and (e), and (e) and (f) pairs correspond to different types of correlation among the $x_i, y_i$, and ${\theta }_i$ states. The parameters $(K,J_A,J_R)$ are set as $(0,0,-0.8)$ for the state in (a) and (d), $(0,0.8,0)$ for the state in [(b) and (e)], and $(0,1,-1)$ for the state in (c) and (f). See movies S4, S5, and S6 in the Supplemental Material [53].

Figure 3

Snapshots of the [(a) and (d)] swirling, [(b) and (e)] butterfly, and [(c) and (f)] bouncing states after ${10}^5$ time steps. The panels show the spatial distribution of the particles (top row) and the respective correlation scatter plots (bottom row). The arrows represent the motion of the particles. The parameters $(K,J_A,J_R)$ are set as $(-1,0,-0.5)$ for the state in (a) and (d), $(-1,-0.5,-0.5)$ for the state in (b) and (e), and $(-1,-1,-0.5)$ for the state in (c) and (f). See movies S7, S8, and S9 in the Supplemental Material [53].

Figure 4

Temporal behavior of single particle trajectories along the $x$ axis with nonperturbed (lavender) and perturbed (red) initial conditions (top row) and maximum Lyapunov exponents for different values of $J_A$ (bottom row). Parameters $(K,J_R)$ are set as $(-1,-0.5)$ for all the figures. Trajectories are generated for (a) $J_A=0.5$, (b) $J_A=-0.5$, and (c) $J_A=-3$. The square in the top corner of (a) shows a magnification of the circled trajectories, and the ones in (d) show correlations in $x-y$ coordinates depending on the values of $J_A$ (see movies S11 and S12 in the Supplemental Material [53]). The green dashed lines represent the change from static phase wave to chaotic states (left) and the breaking of $x-y$ coordinates correlation (right).

Figure 5

Top row: Trajectories (blue) followed by single particles in the $x-\theta$ torus. The red circles are positioned at the start of each trajectory. Bottom row: Evolution of $S_{(\theta \pm x)}^{\max }$ and $S_{(x\pm y)}^{\max }$ for the original initial conditions (purple and gold, respectively) and the perturbed initial conditions (orange and green, respectively). Parameters $(K,J_A,J_R)$ are set as [(a) and (e)] $(-1,0,-0.5)$, [(b) and (f)] $(-1,-0.5,-0.5)$, [(c) and (g)] $(-1,-1,-0.5)$, and [(d) and (h)] $(-1,-3,-0.5)$. The small rectangle in (a) shows a magnification of the circled lines. See movies S13, S14, S15, and S16 in the Supplemental Material [53] (for trajectories of two particles see movies S17, S18, S19, and S20).

Figure 6

Evolution of several Lyapunov exponents of the system in function of $J_A$. The largest exponent is given by ${\lambda }_1$ and the subsequent ones, in terms of amplitude, follow the respective numeration. Panels (a), (b), and (c) display sets of five consecutive exponents within each panel but nonconsecutive across panels. The dashed line is positioned close to where at least one exponent becomes significantly positive.

Figure 7

Evolution of the five largest Lyapunov exponents, estimated considering only position degrees of freedom, in function of $J_A$. The dashed line is positioned close to where at least one exponent becomes significantly positive.

Figure 8

Regions where states emerge on the $J_A-J_R$ plane for (a) $K<0$, (b) $K=0$, and (c) $K>0$. Arrows and gradients in (a) represent state variations in the chaotic regime, transitioning from swirling to bouncing states. Colors labeling different states are an indicator of multistability.