High-dimensional dynamical systems: co-existence of attractors, phase transitions, maximal Lyapunov exponent and response to periodic drive

TL;DR

The paper studies high-dimensional, non-gradient dynamical systems that exhibit coexisting chaotic and fixed-point attractors and analyzes their phase structure using Dynamical Mean Field Theory (DMFT). By mapping the N-dimensional dynamics to a self-consistent stochastic process, it derives exact DMFT equations, analyzes steady-state behavior (fixed points and chaos), and computes the maximal Lyapunov exponent (MLE) via a Schrödinger-type reduction, including explicit results for the simplest model. It also investigates the system's response to periodic drives, showing frequency-selective synchronization (filters) and rich driven-phase diagrams, including cases with multiple chaotic attractors and hysteresis under external fields. The work provides a tractable analytical framework that reproduces known phenomenology of random recurrent networks while enabling precise treatment of training-like adaptations and external perturbations, with clear pathways to extensions beyond fully random models.

Abstract

We study the dynamical properties of a broad class of high-dimensional random dynamical systems exhibiting chaotic as well as fixed point and periodic attractors. We consider cases in which attractors can co-exists in some regions of the phase diagrams and we characterize their nature by computing the maximal Lyapunov exponent. For a specific choice of the dynamical system we show that this quantity can be computed explicitly in the whole chaotic phase due to an underlying integrability of a properly defined Schrödinger problem. Furthermore, we consider the response of this dynamical systems to periodic perturbations. We show that these dynamical systems act as filters in the frequency-amplitude spectrum of the periodic forcing: only in some regions of the frequency-amplitude plane the periodic forcing leads to a synchronization of the dynamics. All in all, the results that we present mirror closely the ones observed in the past forty years in the study of standard models of random recurrent neural networks. However, the dynamical systems that we consider are easier to study and we believe that this may be an advantage if one wants to go beyond random dynamical systems and consider specific training strategies.

Figures (19)

• Figure 1: Phenomenology of fixed point attractors. The results of the numerical integration of the DMFT equations for $g=0.5$, $g_1=0$, $g_2=1$, $J_0=2$, $h=0.1$. The initial condition for the dynamics is $\mathtt{C}=1$ and $\mathtt{m}=0$. We used an integration timestep ${\rm d} t=0.1$ and integrate the corresponding discrete time DMFT equations. Left panel: the dynamical correlation function as a function of $t-t'$ for increasing values of $t'$. For large enough $t'$, $C$ becomes a constant. Right panel: the behavior of $C(t,t)$ and $m(t)$. Both quantities reach a constant asymptotic value.
• Figure 2: Phenomenology of chaotic attractors. The results of the numerical integration of the DMFT equations for $g=2$, $g_1=0$, $g_2=1$, $J_0=2$, $h=0.1$. The initial condition for the dynamics is $\mathtt{C}=1$ and $\mathtt{m}=0$. We used an integration timestep ${\rm d} t=0.1$ and integrate the corresponding discrete time DMFT equations. Left panel: the dynamical correlation function as a function of $t-t'$ for increasing values of $t'$. For large enough $t'$, $C(t,t')$ becomes a function of $t-t'$ only. Right panel: the behavior of $C(t,t)$ and $m(t)$. Both quantities reach a constant asymptotic value.
• Figure 3: Correlation function $C(t,t')$ for increasing values of $t'$ as a function of $t-t'$. The curves converge to a master curve which coincides with the one predicted by Eq. \ref{['solution_C_tau']}. Left panel: Model with $g=g_2=1$, $g_1=1.5$, $J_0=h=0$, starting from $\mathtt{C}=1$ and $\mathtt{m}=0$ and integrated with ${\rm d} t=0.02$. The confining term is of the form $\hat{\mu}(z)=1+z$. Right panel: Same plot for $g=1$ and $g_1=g_2=1.5$, $J_0=h=0$ and ${\rm d} t=0.0025$ and same starting point as in the left panel. The confining potential has been chosen to be $\hat{\mu}(z)=z$.
• Figure 4: The behavior of the order parameters in the ferromagnetic chaotic phase. The dynamical system is characterized by $\hat{\mu}(z)=z$ and $1=J_0=g_1=g_2$, $g=0.5$, ${\rm d} t=0.01$, $\mathtt{C}=1$, $\mathtt{m}=0.01$ and $h=0$. Main figure: the dynamical correlation function $C(t,t')$ for increasing values of $t'$. The curves collapse on the master curve $c(\tau)$ predicted by Eq. \ref{['solution_C_tau']}. Inset: the behavior of $C(t,t)$ and $m(t)$. At long times these curves converge to the asymptotic values $C_0$ and $m_\infty$.
• Figure 5: Phase diagrams and order parameters for $\hat{\mu}(z)=z,\,g_1=1,\,g_2=1$ (Upper panel) and $\hat{\mu}(z)=1+z,\, g_1=2,\, g_2=1$ (Lower panel). (Left) Phase diagrams with the paramagnetic chaotic phase in red, ferromagnetic chaotic phase in orange, ferromagnetic fixed point phase in light blue and paramagnetic fixed point phase in dark blue. (Middle and Right) Steady-state order parameters as a function of $J_0$ for fixed $g=0.9$ (the path in the phase diagrams along which the order parameters are plotted is represented by a black dashed arrow). The gray dotted lines denote phase transitions. Note that similar phase diagrams have been found in fournier2025non.