Singular basins in multiscale systems: tunneling between stable states

TL;DR

The paper reveals that singular funnels, narrow tunnels in basins of attraction, emerge universally in slow–fast (multiscale) systems and can connect distant regions of phase space, preventing standard reductions like adiabatic elimination and averaging from fully capturing system resilience. Through canonical slow–fast models (a pitchfork normal form with adaptive parameter and an adaptive phase rotator) and mean-field networks of adaptive rotators, it shows that the full system admits transitions between coexisting attractors via SFs, with SF volumes shrinking as $V(\varepsilon)\sim \exp(-C/\varepsilon)$ when the timescale ratio is large. Numerical Monte Carlo methods quantify SF volumes, while reduced slow dynamics obtained by adiabatic elimination or averaging fail to reproduce certain cross-boundary transitions, underscoring the need for caution in using dimension-reduced models for resilience analysis. The results generalize to higher dimensions, though SF scaling can deviate, highlighting the potential for hidden fast dynamics to trigger unexpected critical transitions in real-world multistable systems.

Abstract

Real-world systems often evolve on different timescales and possess multiple coexisting stable states. Whether or not a system returns to a given stable state after being perturbed away from it depends on the shape and extent of its basin of attraction. We show that basins of attraction in multiscale systems can exhibit special geometric properties in the form of singular funnels. Although singular funnels are narrow, they can extend to different regions of the phase space and, unexpectedly, impact the system's resilience to perturbations. Consequently, singular funnels may prevent common dimensionality reductions in the limit of large timescale separation, such as the quasi-static approximation, adiabatic elimination and time-averaging of the fast variables. We refer to basins of attraction with singular funnels as singular basins. We show that singular basins are universal and occur robustly in a range of multiscale systems: the normal form of a pitchfork bifurcation with a slowly adapting parameter, an adaptive active rotator, and an adaptive network of phase rotators.

Figures (6)

• Figure 1: (a) Singular basin with a singular funnel (SF) in a bistable pitchfork normal form (\ref{['eq:normal-form-a']}--\ref{['eq:normal-form-b']}) with slowly adapting parameter $\mu$. The red (white) region is the basin of attraction of the stable equilibrium $e_1$ ($e_3$). The boundary of the two basins is given by the stable manifold of the saddle equilibrium $e_2$ (blue curves). The SF near $x=0$ shrinks exponentially with $\mu$. (b) The bistable potential of the corresponding slow subsystem after adiabatic elimination of the fast variable $x$ (equivalently, reduction to the stable critical manifold). Parameters: $a=3$ , $b=2$, and $\varepsilon=0.1$. (c)-(d) The same as (a)-(b) but for system \ref{['eq:new_normal-form-mu']} with $a=5$ , $b=10$, and $\varepsilon=0.1$.
• Figure 2: (a,c) Singular basin with a singular funnel (SF) in an adaptive phase-oscillator (\ref{['eq:sing_osc-a']}--\ref{['eq:sing_osc-b']}). The basin of attraction of the periodic rotation $\gamma_c$ is plotted in red and the basin of attraction of the stable equilibrium $e_1$ is plotted in white. The (blue) boundary of the two basins is given by the stable manifold of the saddle equilibrium $e_2$. (b) The bistable potential of the corresponding slow subsystem after adiabatic elimination and averaging of the fast variable $\varphi$. (d) The volume of the SF limited to $-10 \le \mu\le 3$ as a function of $\varepsilon$ obtained using equation \ref{['eq:scaling']} (black curve) and Monte Carlo simulation (red open circles) see the supplemental material for details. Other parameters: $\eta = 10$, $\omega = -4$; (a): $\alpha = \pi/2$, $\varepsilon = 0.1$; (c): $\alpha = \pi/2$, $\varepsilon = 0.01$.
• Figure 3: (a) Bistable potential of the slow subsystem of two adaptive phase oscillators (\ref{['eq:N_osc']}-\ref{['eq:mudot']}), for $N = 2$, after adiabatic elimination and averaging of the fast variables $\varphi_{1,2}$. (b, d) Two cross-sections of the singular basin with a singular funnel (SF) of the same system: (b) $\varphi_1 = 1.2461$ and (d) $\mu = 2.86$. The basin of attraction of the periodic rotation $\gamma_c$ is plotted in red and the basin of attraction of the stable equilibrium $e_1$ is plotted in white. The (blue) boundary of the two basins is given by the stable manifold of the saddle equilibrium $e_2$. (c) The volume of the SF limited to $-10 \le \mu\le 0$ as a function of $\varepsilon$ obtained using Monte Carlo simulation for different values of the parameter $\Delta_\omega=\omega_1-\omega_2$, showing the robustness of the SF. Other parameters: $\omega_1 = -4$, $\kappa = 1$, $\eta = 10$; (b-d) $\varepsilon = 0.1$.
• Figure 4: (a) The volume of the basin of attraction of ten adaptive phase oscillators (\ref{['eq:N_osc']}-\ref{['eq:mudot']}) for $N = 10$, limited to $-10 \le \mu\le 0$, as a function of $\varepsilon$ obtained using Monte Carlo simulation see the supplemental material fro more details. Its decrease with $\varepsilon$ indicates the presence of a singular funnel (SF). (b,c) Two trajectories starting from close initial conditions: $\varphi_i(0) = 6$ for $i = 1,\dots,10$, with $\mu(0) = -5$ (red) and $\mu(0) = -5.1$ (blue). The red trajectory starts from SF and therefore it is attracted to a stable rotating state. Other parameters: $\omega_i = -4 + (i-1)/9$, $i = 1,\dots,10$, $\eta = 10$, $\alpha = \pi/2$, and $\kappa = 1$.
• Figure 5: Phase portrait of the pitchfork normal form, system \ref{['eq:normal-form-x']}--\ref{['eq:normal-form-mu']}. (a) $\varepsilon = 0.1$, the red region (white) region is the basin of attraction of $e_0 (e_2)$, the stable $W^s(e_1)$ (unstable $W^u(e_1)$) manifolds of the saddle point $e_1$ in blue. (b) $\varepsilon \to 0$, critical manifolds $S_{0,1}$ in thick black, the attracting parts in solid and the repelling part of $S_0$ is dashed. The double arrowed vertical black lines indicate the direction of the fast layer system. Parameters: $a = 3, \, b = 2$.