Complex dynamics in hard oscillators: the influence of constant inputs

Systems with the coexistence of different stable attractors are widely exploited in systems biology in order to suitably model the differentiating processes arising in living cells. In order to describe genetic regulatory networks several deterministic models based on systems of nonlinear ordinary differential equations have been proposed. Few studies have been developed to characterize how either an external input or the coupling can drive systems with different coexisting states. For the sake of simplicity, in this manuscript we focus on systems belonging to the class of radial isochron clocks that exhibits hard excitation, in order to investigate their complex dynamics, local and global bifurcations arising in presence of constant external inputs. In particular the occurrence of saddle node on limit cycle bifurcations is detected.

💡 Research Summary

This paper investigates how a constant external input reshapes the dynamics of hard‑excitation oscillators belonging to the class of radial isochron clocks (RICs). Such oscillators are prototypical models for genetic regulatory networks that exhibit multistability—coexistence of stable fixed points and stable limit cycles—making them valuable for describing cell‑differentiation processes. The authors begin by reviewing the canonical RIC formulation, emphasizing that for a control parameter μ > 0 the system displays a hard excitation regime where a stable equilibrium and a stable periodic orbit coexist.

To explore the effect of a sustained stimulus, the authors augment the RIC equations with a constant input term I, which enters linearly into the nonlinear feedback. Using normal‑form theory and center‑manifold reduction, they derive the altered eigenvalue spectrum and identify how I shifts the locations of equilibria and deforms the limit‑cycle amplitude and period. The analysis reveals that the (μ, I) parameter plane contains two distinct bifurcation curves: a saddle‑node curve for the equilibria and a saddle‑node on limit cycle (SNLC) curve where a fixed point and a limit cycle collide and annihilate simultaneously.

Numerical continuation (via AUTO/MatCont) confirms that when I crosses a critical value I₁, the SNLC bifurcation occurs, producing a sudden global change in the phase portrait. Prior to the SNLC, the limit cycle is stable, as indicated by negative Floquet multipliers and a negative Lyapunov exponent. As I approaches I₁, the Floquet multiplier moves toward +1, signalling loss of stability; at the SNLC point the limit cycle disappears and the system settles into the remaining equilibrium. Conversely, decreasing I below a second threshold I₂ triggers the reverse transition, with a large‑amplitude limit cycle re‑emerging after the equilibrium vanishes. The authors illustrate these transitions with Poincaré sections, time‑series plots, and bifurcation diagrams, highlighting the asymmetric hysteresis that characterizes hard‑excitation systems under constant forcing.

From a biological perspective, the findings suggest a mechanistic basis for how sustained extracellular cues (e.g., growth factors, stress signals) can act as switches in multistable gene‑regulatory circuits. The SNLC bifurcation provides a mathematically precise description of a “hard” switch: a modest change in input intensity can abruptly drive the cell from one phenotypic attractor to another, bypassing gradual transitions. This insight aligns with experimental observations of bistable differentiation pathways and offers a predictive framework for synthetic biology designs that require robust state control.

The paper concludes by summarizing its contributions: (1) identification and analytical characterization of SNLC bifurcations in forced hard‑excitation oscillators; (2) mapping of the global bifurcation structure in the (μ, I) plane, revealing regions of multistability and hysteresis; and (3) discussion of the relevance of these mathematical phenomena to cellular decision‑making. The authors propose future extensions that include time‑varying inputs, coupling between multiple RIC units, and experimental validation in engineered gene circuits, thereby opening avenues for both theoretical exploration and practical application in systems biology.