REGREACT Framework Analysis

• REGREACT framework is a canonical, structure-driven method for analyzing transient reactivity in 2D linear systems by decomposing vector fields into radial and tangential components.
• It computes maximum instantaneous radial growth and distinguishes between growth and rotational dynamics, enabling precise identification of reactive sectors.
• Its use of canonical matrix forms and geometric decomposition delivers actionable insights into both autonomous and nonautonomous system behaviors.

The REGREACT framework provides a canonical, structure-driven approach for analyzing transient reactivity in two-dimensional linear dynamical systems. By decomposing the system’s vector field into radial and tangential components, REGREACT establishes a unified language to quantify and visualize how perturbations amplify or decay and how the geometric arrangement of eigendirections, orthovectors, and “reactive regions” governs dynamical transients. The framework enables explicit computation of the directions and rates of maximal radial growth, a precise separation of growth and rotational influences, and provides canonical matrix forms central to the study of both autonomous and nonautonomous ODEs (Broda et al., 30 Oct 2025).

1. Foundational Definitions and Instantaneous Reactivity

Consider the autonomous linear ODE: $$\dot X = A\,X,\qquad X\in\mathbb R^2,\,\,A\in\mathbb R^{2\times2}$$ For initial data $X(0)=X_0$, let $r(t)=\|X(t)\|$. The instantaneous radial growth rate at $X_0$ is: $$\left.\frac{d}{dt}\|X(t)\|\right|_{t=0} = \frac{X_0^T\,A\,X_0}{\|X_0\|}$$ The system’s reactivity $\rho$ is the maximal instantaneous rate over all directions: $$\rho = \max_{\|v\|=1} v^T A v = \lambda_{\max}\left(\tfrac12(A+A^T)\right)$$ where $\lambda_{\max}$ denotes the principal eigenvalue of the symmetric part of $A$. Thus, REGREACT formalizes how certain initial perturbations may experience transient amplification, even in globally attractive systems, with the growth direction and rate determined by the symmetrized dynamics.

2. Radial and Tangential Decomposition

Let $X = r(\cos\theta, \sin\theta)$. With the 90° rotation $X(0)=X_0$0 and $X(0)=X_0$1, the action of $X(0)=X_0$2 decomposes as: $X(0)=X_0$3 Here, $X(0)=X_0$4 is the radial field (controls $X(0)=X_0$5, i.e., growth/decay) and $X(0)=X_0$6 is the tangential field (governs instantaneous angular velocity). Explicitly,

$X(0)=X_0$7

$X(0)=X_0$8

where $X(0)=X_0$9. The radial component identifies the directions in which transients amplify, while the angular component parameterizes the local rotation.

3. Orthovectors, Orthovalues, and Their Geometric Interpretation

An orthovector $r(t)=\|X(t)\|$0 with orthovalue $r(t)=\|X(t)\|$1 satisfies

$r(t)=\|X(t)\|$2

This corresponds to directions where the radial growth rate vanishes ($r(t)=\|X(t)\|$3), and the flow is purely rotational ($r(t)=\|X(t)\|$4). The equation $r(t)=\|X(t)\|$5 yields the orthovalues: $r(t)=\|X(t)\|$6 Their corresponding orthovector angles are

$r(t)=\|X(t)\|$7

where

$r(t)=\|X(t)\|$8

This structure reveals a duality between eigenvectors (solving $r(t)=\|X(t)\|$9) and orthovectors (solving $X_0$0), central to identifying “reactive sectors” of transient growth.

4. Canonical Matrix Forms and Their Structural Roles

REGREACT defines four canonical forms, facilitating direct geometric and algebraic interpretations of system behavior:

Form	Alignment	Canonical Matrix $X_0$1
R-centered	Max $X_0$2 on $X_0$3-axis	$X_0$4
T-centered	Max $X_0$5 on $X_0$6-axis	$X_0$7
R-zeroed	Orthovector on $X_0$8-axis	$X_0$9
T-zeroed	Eigenvector on $$\left.\frac{d}{dt}\|X(t)\|\right|_{t=0} = \frac{X_0^T\,A\,X_0}{\|X_0\|}$$0-axis	$$\left.\frac{d}{dt}\|X(t)\|\right|_{t=0} = \frac{X_0^T\,A\,X_0}{\|X_0\|}$$1

A careful choice of coordinate axes—via similarity transformations—places reactive or rotational directions into standard orientation. This structural clarity enables explicit determination of reactive sectors and the geometric relation between transient growth and eigenstructure.

5. Bounds and Limits of Reactivity and Amplification

REGREACT characterizes both the unboundedness and the limitations of transient amplification. Unbounded reactivity arises even with attracting eigenvalues or fixed nonorthogonal eigendirections, as the radial maximum ($$\left.\frac{d}{dt}\|X(t)\|\right|_{t=0} = \frac{X_0^T\,A\,X_0}{\|X_0\|}$$2) diverges. However, maximal amplification over time, when trajectories transit the reactive sector exactly once (for $$\left.\frac{d}{dt}\|X(t)\|\right|_{t=0} = \frac{X_0^T\,A\,X_0}{\|X_0\|}$$3), is bounded above: $$\left.\frac{d}{dt}\|X(t)\|\right|_{t=0} = \frac{X_0^T\,A\,X_0}{\|X_0\|}$$4 In the real‐eigenvalue case, a weaker bound is

$$\left.\frac{d}{dt}\|X(t)\|\right|_{t=0} = \frac{X_0^T\,A\,X_0}{\|X_0\|}$$5

Exact bounds for general systems are provided in closed form in the cited work. This analysis demarcates the extent to which transient excursions can amplify perturbations in a globally stable regime (Broda et al., 30 Oct 2025).

6. Nonautonomous Extensions and Dynamical Consequences

Under nonautonomous forcing, such as a time-varying co-rotation

$$\left.\frac{d}{dt}\|X(t)\|\right|_{t=0} = \frac{X_0^T\,A\,X_0}{\|X_0\|}$$6

the reactive and tangential fields transform as $$\left.\frac{d}{dt}\|X(t)\|\right|_{t=0} = \frac{X_0^T\,A\,X_0}{\|X_0\|}$$7, $$\left.\frac{d}{dt}\|X(t)\|\right|_{t=0} = \frac{X_0^T\,A\,X_0}{\|X_0\|}$$8, rigidly shifting the angular rate without altering the radial sector. Instability arises if the rotation “locks” trajectories inside the reactive region, formalized as

$$\left.\frac{d}{dt}\|X(t)\|\right|_{t=0} = \frac{X_0^T\,A\,X_0}{\|X_0\|}$$9

where $\rho$0 delineate the orthovalue interval. This mechanism illustrates how nonautonomous rotations can drive persistent growth from transient sectors, providing a rigorous route to instability by dynamical resonance with inherent reactive structure.

7. Analytical and Geometric Implications

The REGREACT framework delivers a direct mapping from matrix parameters to dynamical properties: identifying every direction’s propensity for radial growth or decay; localizing transient amplification precisely; and separating the influence of eigenvalues from geometric orientation. It establishes a geometric toolkit—distinct from but deeply connected to eigendecomposition—for quantifying and visualizing transient “reactivity” in linear systems, with applicability across the analysis of biological, physical, and engineering models (Broda et al., 30 Oct 2025).