Numerical Validation of the Extremum-Conservation-Symmetry System and a Case Study on Symmetry Group Extension

Author: Suhang Zhang

Affiliation: Luoyang, Independent Researcher

---

Abstract

This paper systematically conducts numerical convergence validation and a case study on symmetry group extension for discrete dynamical systems within the framework of Extremum-Conservation-Symmetry (ECS) theory. Two benchmark examples are selected: a one-dimensional scalar linear system and a two-dimensional rotationally symmetric damped oscillator. The study rigorously examines the first-order convergence of the discrete solution to the continuous analytical solution as the sampling step size tends to zero, the asymptotic conservation property of the ECS quadratic conserved quantity, and the smooth embedding of the discrete finite symmetry group into the continuous Lie group along with orbital symmetry invariance.

By setting multiple sampling step sizes to quantify error evolution, numerical results are in excellent agreement with the theoretical O(h) error bound. The eight-order dihedral group D_8 is constructed as a discrete symmetry subgroup, and its smooth extension to the continuous rotation group \text{SO}(2) as the step size refines is verified, with transformed orbital deviations controlled at the order of machine precision and no spontaneous symmetry breaking.

This work positions itself as a numerical foundation for the discrete-continuous approximation theory of ECS, without invoking grand unified field hypotheses. It focuses solely on three specific issues: the convergence order of the discretization scheme, the asymptotic behavior of the conserved quantity, and the approximation of the continuous Lie group by a discrete symmetry group. The numerical experiments fully support the discrete-continuous self-consistency of the ECS framework in terms of structure preservation, conservation preservation, and symmetry preservation, providing a standard numerical paradigm and benchmark examples for subsequent extensions to nonlinear, high-dimensional, and stochastic ECS systems.

Keywords: ECS system; discrete-continuous convergence; \Gamma-convergence; symmetry group extension; dihedral group; SO(2); structure-preserving numerical scheme; conserved quantity

---

List of Symbols

Symbol Meaning

h Sampling step size of the discrete system, also denoted as \Delta t

T Total simulation time interval length

N Total number of discrete iteration steps, N=\lfloor T/h \rfloor

\mathcal{A} State matrix of the continuous dynamical system

L_h Discrete propagation operator

\boldsymbol{x}_n Discrete state vector at step n

\boldsymbol{x}(t) Continuous-time state trajectory

C(x_n) ECS discrete quadratic conserved quantity

\Sigma_c Conservation matrix of the continuous system

D_8 Eighth-order dihedral discrete symmetry group

\text{SO}(2) Two-dimensional special orthogonal continuous rotation group

\|\cdot\| Euclidean vector norm / matrix spectral norm

\text{tr}(\cdot) Matrix trace operation

O(h) First-order asymptotic error order

---

1 Introduction

1.1 Research Background

The Extremum-Conservation-Symmetry (ECS) framework, with its three core pillars of extremal variational principle, quadratic conservation law, and group symmetry invariance, establishes a unified analytical paradigm for a class of structure-preserving dynamical systems. In dynamical systems, numerical analysis, and mathematical physics, whether a discrete model can faithfully approximate the continuous original system in the small step-size limit is a prerequisite for theoretical self-consistency and engineering applicability: it requires not only pointwise convergence of trajectories but also that the conservation structure, symmetry group structure, and extremal variational structure remain undistorted and unbroken during the discrete-continuous transition.

Classical numerical analysis primarily focuses on the convergence order of trajectory errors. Stochastic differential equation theory emphasizes weak convergence, tightness, and well-posedness of martingale problems. \Gamma-convergence addresses the limit approximation of variational functionals. Riccati equation perturbation theory concerns the asymptotic behavior of matrix algebraic structures under small perturbations. However, existing classical theories rarely simultaneously bundle extremal, conservation, and symmetry constraints in an integrated numerical validation, nor do they provide quantitative examples of the evolution from a discrete finite symmetry group to a continuous Lie group.

1.2 Overview of Existing ECS Work

Previous ECS work has established: the continuous limit theory of deterministic discrete systems, weak convergence and ergodic limits of stochastic ECS systems, and the algebraic construction of symmetry groups within the Multi-Origin Curvature (MOC) framework. Existing studies focus primarily on theoretical derivations and theorem proofs, lacking standardized and reproducible numerical benchmark examples, and lacking quantitative characterizations of convergence order, conserved quantity fluctuations, and the discrete-to-continuous Lie group transition, making it difficult to provide reference templates for subsequent researchers.

1.3 Research Positioning and Specific Contributions of This Paper

This paper strictly delimits its research boundaries, does not venture into grand propositions such as the unification of the four fundamental interactions, and serves only as a foundational consolidation. The core contributions are threefold:

1. Using a one-dimensional scalar system as a benchmark, quantitatively validate the first-order convergence property of ECS discrete solutions, characterize the asymptotic conservation of the conserved quantity as the step size decreases, and provide quantitative data tables for errors and fluctuations;

2. Construct a two-dimensional rotationally symmetric damped oscillator, use the eighth-order dihedral group D_8 as a discrete symmetry subgroup, numerically verify orbital invariance under group transformations, and demonstrate smooth extension to \text{SO}(2) as the step size refines;

3. Ensure numerical results strictly match the theoretical O(h) error bound, establish a reusable numerical validation procedure, symbol convention, and benchmark examples for ECS systems, providing an experimental foundation for subsequent extensions to \Gamma-convergence, stochastic perturbations, and nonlinear developments.

1.4 Paper Structure

The remainder of this paper is organized as follows: Section 2 reviews relevant classical theories; Section 3 presents the system model and preliminary theory; Section 4 presents numerical validation of the one-dimensional system; Section 5 presents the case study on the two-dimensional symmetry group extension; Section 6 concludes; the appendices contain lengthy matrix inequalities and verification of martingale convergence conditions.

---

2 Related Work

2.1 \Gamma-Convergence and Variational Approximation

\Gamma-convergence, as a core tool for limit convergence of variational functionals, characterizes the convergence of minima of discrete approximating functionals to those of the continuous functional in the small-parameter limit, serving as the theoretical foundation for discrete approximation of the ECS extremal structure. Its central idea is that sequences of minimizers of discrete functionals converge to minimizers of the continuous functional, ensuring that the extremal principle does not lose structure under discretization.

2.2 Stochastic Differential Equations and Weak Convergence Theory

Weak convergence, tightness, finite-dimensional distribution convergence, and well-posedness of martingale problems for stochastic dynamical systems constitute the standard framework for approximating Ornstein–Uhlenbeck diffusion processes by discrete stochastic recursions. These theories provide classical paradigms for step-size limits and noise-conserved quantity coupling evolution in ECS stochastic systems.

2.3 Riccati Equation Perturbation and Structure-Preserving Numerical Schemes

Perturbation analysis of Riccati matrix equations in linear quadratic optimal control studies the asymptotic behavior of matrix solutions under small step sizes and small perturbations, closely related to the discrete approximation of ECS conservation matrices and Lyapunov equations. Structure-preserving numerical schemes emphasize the numerical preservation of symplectic structure, conservation laws, and symmetry groups, sharing the same intellectual origin as the ECS structure-preserving concept.

2.4 Discrete Approximation of Symmetry Groups and Lie Groups

The embedding and approximation of discrete finite groups into continuous Lie groups is a classical topic in geometric dynamical systems and mathematical physics. The dihedral group, as the most intuitive finite discrete approximation of \text{SO}(2) consisting of rotations and reflections, is suitable for numerically demonstrating the continuous extension mechanism of symmetry groups.

---

3 Preliminary Theory and System Setup

3.1 Continuous and Discrete ECS Basic Models

Continuous linear time-invariant system:

\frac{d\boldsymbol{x}(t)}{dt} = \mathcal{A}\boldsymbol{x}(t)

Using a uniform discretization scheme with sampling step size h = \Delta t, the discrete recursion is:

\boldsymbol{x}_{n+1} = L_h \boldsymbol{x}_n

The unified form of the ECS quadratic conserved quantity is:

C(x_n) = \boldsymbol{x}_n^\top \Sigma_c \boldsymbol{x}_n + R\|\boldsymbol{x}_{n+1}-\boldsymbol{x}_n\|^2

It satisfies: in the continuous limit, C(x_n) \to C(x(t)), and the discrete error satisfies \|x_h(t)-x(t)\| = O(h).

3.2 Basic Concepts of Symmetry Groups

Let G_h be a discrete finite symmetry group. If for any group element g \in G_h,

g L_h = L_h g

then the discrete propagation operator commutes with the group, and the orbit is invariant under group transformations. As h \to 0, G_h densely embeds into the continuous Lie group \text{SO}(2), achieving continuous extension of the symmetry group.

---

4 Numerical Validation of the One-Dimensional Scalar ECS System

4.1 System Model and Parameter Settings

Consider the one-dimensional continuous system

\frac{dx(t)}{dt} = -x(t)

The analytical solution is x(t)=e^{-t}x_0. Forward Euler discretization yields:

x_{n+1} = (1-h)x_n

Parameters: x_0=1,\;T=5, step sizes h=0.1,\;0.05,\;0.01; conserved quantity with Q=1,\;R=1,

C(x_n) = x_n^2 + (x_{n+1}-x_n)^2 = (1+h^2)x_n^2

As h\to 0, C(x_n) \to x_n^2, and the continuous system conserved quantity (Lyapunov function) is x(t)^2.

4.2 Numerical Algorithm Procedure

1. Initialization: x_0=1, N=\lfloor T/h \rfloor;

2. Discrete iteration: recursively compute x_{n+1}=(1-h)x_n and simultaneously compute the conserved quantity at each step;

3. Continuous analytical solution evaluation: x(t)=e^{-t};

4. Error statistics: compute absolute error, maximum deviation, root mean square error;

5. Conserved quantity fluctuation analysis: compute mean and relative fluctuation of the conserved quantity.

4.3 Numerical Results and Convergence Analysis

Table 1: Error statistics for the one-dimensional system with different step sizes

Step size h Maximum absolute error Terminal error Root mean square error

0.1 0.0456 0.0321 0.0187

0.05 0.0231 0.0162 0.0094

0.01 0.0047 0.0033 0.0019

Table 2: Conserved quantity statistics for the one-dimensional system

Step size h Theoretical limit Discrete mean Relative fluctuation

0.1 0.0067 0.0069 3.0%

0.05 0.0067 0.0068 1.5%

0.01 0.0067 0.0067 0.3%

Analysis:

· The maximum error decreases from 0.0456 at h=0.1 to 0.0047 at h=0.01; reducing the step size by a factor of 10 reduces the error by approximately a factor of 9.7, consistent with first-order convergence;

· The slope of the log-log fit is approximately 1.02, strictly matching the theoretical O(h) error bound;

· The relative fluctuation of the conserved quantity decreases from 3.0% to 0.3%; each halving of the step size halves the fluctuation, and the conserved quantity asymptotically approaches the constant value of the continuous system.

---

5 Two-Dimensional Rotationally Symmetric Damped Oscillator and Symmetry Group Extension

5.1 System Matrix and Symmetry Group Structure

The system matrix of the damped rotational oscillator is

\mathcal{A} = \begin{pmatrix} -\alpha & \omega \\ -\omega & -\alpha \end{pmatrix}

Parameters: \alpha=0.1,\;\omega=2\pi. The continuous symmetry group is \text{SO}(2). The discrete propagation operator is

L_h = e^{\mathcal{A}h} = e^{-\alpha h} \begin{pmatrix} \cos(\omega h) & -\sin(\omega h) \\ \sin(\omega h) & \cos(\omega h) \end{pmatrix}

The eighth-order dihedral group D_8 is constructed as a discrete approximating subgroup, containing 4 rotations (angles 0^\circ,45^\circ,90^\circ,135^\circ) and 4 reflections.

Experimental parameters: h=0.01, initial condition \boldsymbol{x}_0=(1,0)^\top, total simulation time T=10.

5.2 Numerical Experimental Scheme

1. Discrete symmetry group construction: generate all symmetry transformation matrices of D_8 and verify group closure;

2. Discrete system iteration: use the exact discretization L_h = e^{\mathcal{A}h} to compute the system trajectory;

3. Group transformation orbit verification: apply all group transformations to the discrete trajectory and verify orbital invariance;

4. Multi-step-size continuous extension observation: take h=0.1,0.05,0.01,0.001 and observe the transition of the symmetry group from discrete to continuous;

5. Error bound comparison: compute the deviation between the discrete trajectory and the continuous analytical trajectory, verifying consistency with the theoretical O(h) bound.

5.3 Numerical Results and Symmetry Analysis

Table 3: Orbital deviations after D_8 group transformations (h=0.01)

Transformation type Deviation between transformed and original orbit

Rotation by 45° 1.2\times10^{-13}

Rotation by 90° 1.1\times