Geometric decomposition of planar vector fields with a limit cycle

TL;DR

The paper addresses how to describe planar limit-cycle dynamics with a deterministic geometric framework. It proposes a decomposition of planar vector fields into a generalized Hamiltonian structure using a scalar energy-like function $H$ and a bilinear form $\kappa$, yielding $dx/dt = (p I + w R_{\perp}) \nabla H$, which unifies potential- and rotation-dominated behavior without requiring stochasticity. Analytically, it connects to the Helmholtz–Hodge perspective and to a polar-like interpretation near fixed points, and it introduces a Level-Set Integration algorithm to numerically compute $H$ and $\kappa$ for systems with a single limit cycle. The approach is demonstrated on radial-symmetric and linearized systems and applied to a van der Pol-type oscillator, showing the method can reconstruct energy contours and reveal the underlying geometry of oscillations, with potential implications for modeling biological time and oscillatory processes.

Abstract

Mathematical modelling is a cornerstone of computational biology. While mechanistic models might describe the interactions of interest of a system, they are often difficult to study. On the other hand, abstract models might capture key features but remain disconnected from experimental manipulation. Geometric methods have been useful in connecting both approaches, although they have only been established for specific type of systems. Phenomena of biological relevance, such as limit cycles, are still difficult to study using conventional methods. In this paper, I explore an alternative description of planar dynamical systems and I present an algorithm to compute numerically the geometric structure of planar systems with a limit cycle.

Figures (10)

• Figure 1: Example of a potential system. (A) Simulation of an arbitrary potential system for different initial conditions (blue points). Each trace is shown as a black line and final simulation values are shown as red points. The value of the potential $\psi$ is shown as color, with yellow and dark blue corresponding to regions of high and low potential, respectively. Note that all trajectories converge to the regions of lowest potential, the fixed points. (B) 3D representation of the potential of a system. Color and height represent the potential $\psi$ for different values of $x$ and $y$.
• Figure 2: The potential structure of a planar dynamical system. (A) The solution of Equation \ref{['symm_rad']} is shown in red for the initial condition $x_0 = y_0 = 3.4\cdot 10^{-3}$ (Parameter values: $\beta = 10$, $k = 0.05$, $\omega = -1$). Energy level contours at specific points are shown as grey lines. The system converges to the trajectory given by $x^2 + y^2 = \beta$ (the circle of radius $r = \sqrt{10}$). (B) Potential landscape for the system. Color and height represent the potential $\psi$ for different values of $x$ and $y$. Points satisfying the inequality $x + y>3$ have been removed to reveal the internal structure of the potential well. The limit cycle lies at the bottom of the potential.
• Figure 3: Decomposition of a vector field. Vector field (black arrows) by Equation \ref{['symm_rad']} (Parameter values: $\beta = 10$, $k = 0.05$, $\omega = -1$). The limit cycle is shown as a dashed red line. Some computed energy level contours for the system are shown in grey. Potential and rotational components, $\nabla \psi$ and $\nabla \phi$ respectively, are shown as blue and purple arrows. The gradient of the Hamiltonian function, $\nabla H$ is shown as warm colour arrows. Note that while $\nabla \psi$ vanishes at the limit cycle, $\nabla H$ is non-zero. The action of $\kappa$ on $\nabla H$ is shown in green. Note that at the limit cycle, it rotates $\nabla H$ by $\pi/2$.
• Figure 4: Analysis and linear approximation of an asymmetric system. (A) The solution of Equation \ref{['asymm']} is shown in red for the initial condition $x_0 = y_0 = 0.1$ (Parameter values: $\beta = 10$, $k = 0.01$, $\omega = -\pi/10$, $a = 1.5$, $\mu = 2$). Energy level contours at specific points are shown as grey lines. The system converges to the trajectory given by $H = 5$ (Equation \ref{['H_asymm']}). Dashed purple squared represents the region plotted in B. (B) Linear approximation $H_0 = \frac{1}{2} (x^2 + y^2)$ (dashed blue lines) of the generalised Hamiltonian $H$ (solid grey lines) given by Equation \ref{['H_asymm']}. Green arrows represent the result of the action of $\kappa_0$ on $\nabla H_0$. The simulation result is shown in red.
• Figure 5: Level-set integration algorithm outline. Computation of $\nabla H$ and $\nabla H^\perp$ for a point in relation to the limit cycle of the system of choice (Equation \ref{['asymm']}). Selected point of reference shown as a vertical dashed red line. Algorithm values are $p_l = 0.02$ and $\Delta H = 0.2$. (A) Distance $\Delta s$ for the point to all the points in the reference set. (B) Estimated angle of $\kappa$ (green line) and angular difference (blue line). Difference between both angles is shown as an orange line. The red line represent the value at which the absolute difference is minimal. (C) Estimated values for $w$ for every point in the reference set. Computed value for $w = -0.336$ (Given value of $w_{0} = -\pi/10 \approx -0.314$). (D) Computed vectors $\nabla \hat{H}$ and $\nabla \hat{H}^\perp$ (green and red arrows, respectively) for the selecte point (red). Reference point is shown in blue.