A Radial and Tangential Framework for Studying Transient Reactivity

TL;DR

This work introduces a radial–tangential decomposition for two-dimensional linear ODEs to study transient reactivity, defining R(θ) and T(θ) as sinusoidal components that govern radial and angular motion on the unit circle. It connects transient reactivity to the eigenstructure through zeros of T and the newly defined orthovectors/orthovalues, and it offers four rotation-based standard matrix forms that preserve both transient and asymptotic behavior for clearer analysis. The authors establish precise bounds and exact formulas for maximal amplification ρ_max and demonstrate that reactivity can be unbounded even as the origin remains attracting, while nonautonomous systems can accumulate reactivity under suitable rotation rates. The framework yields geometric insight into reactive regions and provides practical tools for analyzing stability and transient dynamics in 2D linear systems, with extensions to nonautonomous settings. Overall, the radial–tangential approach unifies transient and long-term behavior, enabling explicit computation of key dynamical quantities such as δ_R, δ_T, and ρ_max.

Abstract

In modeling biological systems and other applications, an important recurring question is whether those systems maintain healthy regimes not just at dynamic attractors but also during transient excursions away from those attractors. For ODE models, these excursions are not due only to nonlinearities. Even in a linear, autonomous system with a global attractor at the origin, some trajectories will move transiently away from the origin before eventually being attracted asymptotically. Reactivity, defined by Neubert and Caswell in 1997, captures this idea of transient amplification of perturbations by measuring the maximum instantaneous rate of radial growth. In this paper, we introduce a novel framework for analyzing reactivity and transient dynamics in two-dimensional linear ODEs using a radial and tangential decomposition of the vector field. We establish how to view the eigen-structure of the system through this lens and introduce a matching structure of orthovectors and orthovalues that characterize where the radial velocity becomes positive. From this perspective, we gain geometric insight about the regions of state space where positive radial growth occurs and how solutions trajectories traverse through these regions. Additionally, we propose four standard matrix forms that characterize both transient and asymptotic behavior and which highlight reactivity features more directly. Finally, we apply our framework to explore the limits of reactivity and maximal amplification and to characterize how transient reactivity can accumulate in nonautonomous linear systems to result in asymptotically unstable behavior.

Figures (12)

• Figure 1: (a) The phase portrait of linear ODE with coefficient matrix $A=\left(-1-80-3\right)$ is plotted with concentric circles in grey indicating contours of equal distance from the origin. Many trajectories exhibit reactive transient growth (highlighted in red) away from the origin before being asymptotically attracted. The solution trajectory with initial condition $(x,y)\approx (-0.4,0.9)$ on the unit circle is selected to illustrate maximum amplification. (b) The distance of this selected trajectory from the origin is plotted versus time with transient amplification highlighted in red. The trajectory is amplified from $t=0$ to $t=t_{\max}$ where it attains maximum amplification $\rho_{\max} \approx 1.67$.
• Figure 1: (a) The vector field defined by $AX$ is decomposed in to components in the direction radial to and tangential to the vector $X$. The radial component, in red, and the tangential component, in dark blue, are both given as functions of the angle $\theta$. (b) Graphs of $\mathcal{R}$ and $\mathcal{T}$ as functions of $\theta$. Each is sinusoidal with period $\pi$ and amplitude $p$. $\mathcal{R}$, in red, has maximum and minimum values $\rho_1$ and $\rho_2$, respectively, midline $m_R$ and maximum at $\theta_R$. $\mathcal{T}$, in dark blue, has maximum and minimum values $\tau_1$ and $\tau_2$, respectively, midline $m_T$ and maximum at $\theta_T=\theta_R-\pi/4$. See Corollary \ref{['cor:sinusoidal']}.
• Figure 1: (a) The phase portrait for the system with coefficient matrix $A=\left(-1-80-3\right)$ is shown with the reactive region $\mathcal{U}_R$ highlighted in pink. (b) The corresponding radial and tangential components, $\mathcal{R}$ and $\mathcal{T}$, are plotted in red and dark blue, respectively. The reactive set $\mathcal{S}_R$, where $\mathcal{R}(\theta)>0$ is also shown. Notice the symmetry of the zeros of $\mathcal{R}$ around $\theta_R$ with radius $\delta_R$ so that $\mathcal{S}_R = (\theta_R -\delta_R, \theta_R + \delta_R)\in \mathbb{R}\ (\mathrm{mod}\ \pi)$. Similarly, the zeros of $\mathcal{T}$ are symmetric around $\theta_T$ where $\delta_T$ is the angular distance between the maximum of $\mathcal{T}$ and the zeros of $\mathcal{T}$.
• Figure 1: (a) The phase portrait for the system with coefficient matrix $A=\left(-2121\right)$ is shown. The origin is a saddle whose eigenvectors are highlighted in dark purple (repelling direction) and in light green (attracting direction). The associated eigenvalues are $\lambda_1=m_R+p_R>0$ and $\lambda_2=m_R-p_R<0$ as given in Corollary \ref{['cor:pR']}. (b) The radial and tangential components, $\mathcal{R}(\theta)$ and $\mathcal{T}(\theta)$, are plotted in red and dark blue, respectively. Notice the eigenvectors in (a) correspond to the angles $\theta_1$ and $\theta_2$ in (b) and the eigenvalues $\lambda_1$ and $\lambda_2$ equal $\mathcal{R}(\theta_1)$ and $\mathcal{R}(\theta_2)$, respectively. Below the axes, we show the phase line of angular dynamics (Equation \ref{['eqn:dtheta']}) which highlights how trajectories move towards the repelling eigendirection (dark purple) in their long-term behavior (Corollary \ref{['cor:theta-dynamics']}).
• Figure 1: The radial and tangential decompositions of the vector fields $AX$ and $JAX$ are depicted in (a) and (b), respectively. When the vector field given by $AX$ is rotated counter-clockwise by the matrix $J$, the components $\mathcal{R}_A(\theta)X$ and $\mathcal{T}_A(\theta)X^\perp$ are also rotated counter-clockwise. The radial component $\mathcal{R}_{JA}(\theta)X$ of the new rotated vector field given by $JAX$ is equal to the rotation of what was tangential component of $AX$: $\mathcal{R}_{JA}(\theta)X = J(\mathcal{T}_A(\theta)X^\perp)$. Similarly, the tangential component $\mathcal{T}_{JA}(\theta)X^\perp$ of the new rotated vector field is equal to the rotation of of what was the radial component of $AX$: $\mathcal{T}_{JA}(\theta)X^\perp = J(\mathcal{R}_A(\theta)X)$. This geometric intuition is at the heart of Lemma \ref{['lem:duality']}.

Theorems & Definitions (71)

• Definition 1
• Theorem 2.1
• Proof 1: Proof of Theorem \ref{['theo:RadTang']}
• Corollary 2.2
• Proof 2
• Corollary 2.3
• Proof 3
• Corollary 2.4
• Corollary 3.1
• Proof 4