[Paper Review] Exceptional points in nonlinear and stochastic dynamics

Cheyne Weis, Michel Fruchart+4 more|arXiv (Cornell University)|Jul 24, 2022

stochastic dynamics and bifurcation4 citations

TL;DR

This paper introduces generalized exceptional points (EPs) in nonlinear and stochastic dynamical systems, where extended attractors like limit cycles coalesce during bifurcations. It shows that this coalescence is marked by the tangency of covariant Lyapunov vectors (CLVs) with vanishing Lyapunov exponents, revealing non-reciprocal dynamics, enhanced noise sensitivity, and broken isochrons, with applications to ecological, neural, and physical systems.

ABSTRACT

We study a class of bifurcations generically occurring in dynamical systems with non-mutual couplings ranging from models of coupled neurons to predator-prey systems and non-linear oscillators. In these bifurcations, extended attractors such as limit cycles, limit tori, and strange attractors merge and split in a similar way as fixed points in a pitchfork bifurcation. We show that this merging and splitting coincides with the coalescence of covariant Lyapunov vectors with vanishing Lyapunov exponents, generalizing the notion of exceptional points to non-linear dynamical systems. We distinguish two classes of bifurcations, corresponding respectively to continuous and discontinuous behaviors of the covariant Lyapunov vectors at the transition. We outline some physical consequences of generalized exceptional points on the dynamics of the system, including non-reciprocal responses, the destruction of isochrons, and enhanced sensitivity to noise. We illustrate our results with concrete examples from neuroscience, ecology, and physics. When applied to interpret existing experimental observations, our analysis suggests a simple explanation for the non-trivial phase delays observed in the population dynamics of plankton communities and the recently measured statistics of rotation reversals for a solid body immersed in a Rayleigh-Bénard convection cell.

Motivation & Objective

• To identify and characterize bifurcations in non-reciprocal nonlinear dynamical systems where extended attractors—such as limit cycles, tori, or strange attractors—merge and split.
• To establish a connection between the coalescence of such attractors and the tangency of covariant Lyapunov vectors (CLVs) with zero Lyapunov exponents, generalizing the concept of exceptional points beyond linear systems.
• To distinguish between continuous and discontinuous transitions in CLV behavior at bifurcation points, revealing distinct dynamical regimes.
• To explore physical consequences such as non-reciprocal responses, destruction of isochrons, and enhanced sensitivity to noise arising from generalized EPs.
• To apply the framework to real-world systems, including predator-prey models, neural oscillators, and Rayleigh-Bénard convection, to explain experimental observations like phase delays and rotation reversal statistics.

Proposed method

• Model nonlinear dynamical systems with non-mutual (non-reciprocal) couplings using coupled differential equations, such as a four-dimensional predator-prey system with asymmetric predation coupling.
• Compute Lyapunov exponents (LEs) and covariant Lyapunov vectors (CLVs) via time-averaged numerical integration of the variational equations along periodic orbits.
• Use the Floquet theory to compute the monodromy matrix over one period of the limit cycle, then diagonalize it to extract LEs and eigenvectors, which are mapped to CLVs at initial time.
• Track the time-dependent evolution of CLVs using the flow map, and compute the angle between CLVs via the dot product to quantify their alignment.
• Define a time-averaged measure of CLV alignment using ⟨sin²θ₁₂(t)⟩ over one period to detect generalized EPs at bifurcation points.
• Vary the bifurcation parameter β (predation coupling strength) and analyze transitions in LEs, CLV angles, and phase dephasing to identify critical behavior.

Experimental results

Research questions

• RQ1 How do attractor coalescence events in nonlinear dynamical systems relate to the behavior of covariant Lyapunov vectors and Lyapunov exponents?
• RQ2 What distinguishes continuous from discontinuous transitions in the alignment of CLVs at bifurcation points in non-reciprocal systems?
• RQ3 How do generalized exceptional points—defined by coalescing CLVs with vanishing LEs—affect the system's response to noise and its dynamical stability?
• RQ4 Can generalized EPs explain non-trivial phase delays in ecological population dynamics, such as in plankton communities?
• RQ5 Do generalized EPs account for the observed statistics of rotation reversals in Rayleigh-Bénard convection cells?

Key findings

• A bifurcation occurs at β_c ≈ 0.02275 in a four-dimensional predator-prey model, where a single limit cycle splits into two stable limit cycles with non-zero dephasing.
• Lyapunov exponents λ₁ and λ₂ both vanish at the bifurcation point, confirming the presence of a generalized exceptional point.
• The angle θ₁₂ between the two CLVs corresponding to the zero LEs vanishes continuously below β_c and discontinuously at β_c, indicating a discontinuous transition in CLV behavior.
• The average alignment measure ⟨sin²θ₁₂⟩ becomes arbitrarily small as β approaches β_c from above, signaling near-coalescence of CLVs along the entire limit cycle.
• The instantaneous angle θ₁₂(t) between CLVs depends on position along the periodic orbit, with minimal values occurring at specific phases, even away from the bifurcation.
• The two resulting limit cycles exhibit asymmetric population dynamics: the roles of predators and prey are exchanged between the two cycles, indicating broken symmetry and non-reciprocal behavior.