Discrete Numerical Verification and Symmetry Group Case Studies for Extremal–Conservation–Symmetry Systems

Author: Zhang Suhang

Affiliation: Luoyang, Independent Researcher

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Abstract

Within the framework of Extremal–Conservation–Symmetry (ECS) theory, this paper systematically carries out numerical convergence verification and symmetry group extension case studies for discrete dynamical systems. Two benchmark examples—a one-dimensional scalar linear system and a two-dimensional rotationally symmetric damped oscillator—are selected to examine the first-order convergence of the discrete solution to the continuous analytical solution as the sampling step tends to zero, the structural fidelity of the ECS quadratic form in the discrete–continuous transition, and the approximation of continuous Lie group orbits by finite discrete demonstration groups, together with orbital symmetry invariance.

By setting multiple sampling steps to quantify error evolution, the numerical results agree with the theoretical O(h) error bound; a 4th-order cyclic group C_4 ⊂ SO(2) is constructed as a discrete demonstration subgroup, and an 8th-order dihedral group D_8 ⊂ O(2) as a demonstration group containing reflections. Numerical verification shows that orbital deviations under group transformations are at machine precision, with no symmetry breaking.

This paper is positioned as numerical consolidation work for the ECS discrete–continuous approximation theory. It does not involve grand unified field assumptions, and focuses only on three specific problems: the convergence order of the discrete scheme, the structural fidelity of the ECS quadratic form, and the approximation of continuous group orbits by finite demonstration groups. The numerical experiments fully support the discrete–continuous self-consistency of the ECS framework in preserving structure, conservation, and symmetry, providing a standard numerical paradigm and benchmark examples for subsequent extension to nonlinear, high-dimensional, and stochastic ECS systems.

Keywords: ECS system; discrete–continuous convergence; Γ-convergence interface; symmetry group extension; cyclic group; dihedral group; SO(2); structure-preserving numerical scheme; structural fidelity of quadratic form

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Table of Symbols

Symbol Meaning

h sampling step of the discrete system, also denoted Δt

T total simulation time interval length

N total number of discrete iterations, N = ⌊T/h⌋

𝒜 state matrix of the continuous dynamical system

L_h discrete propagation operator

x_n state vector at step n

x(t) continuous-time state trajectory

C(x_n) ECS discrete quadratic form

Σ_c continuous-system quadratic form matrix

C_4 4th-order cyclic group, finite demonstration subgroup of SO(2)

D_8 8th-order dihedral group, finite demonstration group of O(2)

SO(2) two-dimensional special orthogonal continuous rotation group

O(2) two-dimensional orthogonal group

‖·‖ Euclidean vector norm / matrix spectral norm

tr(·) matrix trace

O(h) first-order asymptotic error order

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1 Introduction

1.1 Research Background

The Extremal–Conservation–Symmetry (ECS) framework takes the extremal variational principle, the quadratic structural conservation law, and group symmetry invariance as its three core pillars, constructing 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 limit is a prerequisite for theoretical self-consistency and engineering usability: it requires not only pointwise convergence of trajectories, but also that the conservation structure, symmetry group structure, and variational extremal structure do not distort or break during the discrete–continuous transition.

Classical numerical analysis mostly focuses on the convergence order of trajectory errors; stochastic differential equation theory focuses on weak convergence, tightness, and well-posedness of martingale problems; Γ-convergence focuses on the limiting approximation of variational functionals; Riccati equation perturbation theory focuses on the asymptotic preservation of matrix algebraic structures. However, existing classical theories rarely bind the three types of constraints—extremality, conservation, and symmetry—together for integrated numerical verification, and also lack quantitative examples of orbit approximation from finite discrete demonstration groups to continuous Lie groups.

1.2 Overview of Existing ECS-Related Work

Previous ECS work has established: the continuous limit theory of deterministic discrete systems, the weak convergence and ergodic limit of stochastic ECS systems, and the algebraic construction of symmetry groups under the Multi-Origin Curvature (MOC) framework. Existing research emphasizes theoretical derivation and theorem proving, lacking standardized, reproducible numerical benchmark examples, and lacking quantitative characterization of convergence order, structural fidelity of the quadratic form, and orbit approximation from finite demonstration groups to continuous groups, making it difficult to provide reference templates for subsequent researchers.

1.3 Research Positioning and Specific Contributions of This Paper

This paper strictly limits its research boundary, does not touch grand propositions such as the unification of the four fundamental interactions, and only does foundational consolidation work. Its three core contributions are:

1. Using a one-dimensional scalar system as a benchmark, quantitatively verify the first-order convergence property of the ECS discrete solution, characterize the structural fidelity of the ECS quadratic form relative to the continuous quadratic form in the discrete–continuous transition, and provide quantitative data tables of errors and deviations;

2. Construct a two-dimensional rotationally symmetric damped oscillator, using the 4th-order cyclic group C_4 ⊂ SO(2) and the 8th-order dihedral group D_8 ⊂ O(2) as discrete demonstration groups, numerically verify orbital invariance under group transformations, and the approximation of continuous group orbits by finite demonstration group orbits;

3. Numerical results strictly match the theoretical O(h) error bound, establishing a reusable numerical verification workflow, symbol specification, and benchmark examples for ECS systems, providing an experimental foundation for subsequent Γ-convergence extension, stochastic perturbation, and nonlinear extension.

1.4 Paper Structure

The remainder is organized as follows: Section 2 reviews related classical theories; Section 3 gives the system model and preliminary theory; Section 4 provides numerical verification for the one-dimensional system; Section 5 studies the two-dimensional symmetry group case; Section 6 concludes; the appendix contains lengthy matrix inequalities and verification of martingale convergence conditions.

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2 Related Work

2.1 Γ-Convergence and Variational Approximation

Γ-convergence, as a core tool for the limiting convergence of variational functionals, characterizes the convergence of minima of discrete approximating functionals in the small-parameter limit, and is the theoretical basis for the discrete approximation of the ECS extremal structure. Its core idea is: the sequence of minimizers of the discrete functional converges to the minimizer of the continuous functional, ensuring that the extremal principle does not lose structure during discretization. This paper does not expand on specific applications of Γ-convergence; it only establishes, in this section, a conceptual correspondence between the ECS extremal structure and Γ-convergence, as a theoretical interface for subsequent work (discrete approximation of ECS variational functionals).

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 discrete stochastic recursions approximating Ornstein–Uhlenbeck diffusion processes. The related theory provides a classical paradigm for the sampling-step limit of ECS stochastic systems and the coupled evolution of noise and the quadratic form.

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 steps and small perturbations, and is highly related to the discrete approximation of the ECS quadratic form matrix and Lyapunov equations; structure-preserving numerical schemes emphasize the numerical preservation of symplectic structure, conservation laws, and symmetry groups, and share the same origin as the ECS structure-preserving idea in this paper.

2.4 Symmetry Groups and Discrete Approximation of 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 D_8 contains 4 rotations and 4 reflections; its rotation subgroup C_4 = {R_{kπ/2}}_{k=0}^{3} is a finite subgroup of SO(2), while D_8 itself is a finite subgroup of the orthogonal group O(2). This paper uses C_4 as a discrete demonstration subgroup of SO(2), and D_8 as a discrete demonstration group of O(2), to numerically display the approximation of continuous group orbits by finite group orbits.

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3 Preliminary Theory and System Setup

3.1 Basic Continuous and Discrete ECS Models

Continuous linear time-invariant system:

d𝒙(t)/dt = 𝒜𝒙(t).

Using a consistent discretization scheme with sampling step h = Δt, the discrete recursion is:

𝒙_{n+1} = L_h 𝒙_n.

The unified form of the ECS quadratic form is:

C(x_n) = 𝒙_n^⊤ Σ_c 𝒙_n + R‖𝒙_{n+1} − 𝒙_n‖².

It satisfies: in the continuous limit, C(x_n) → C(x(t)), and the discrete error satisfies ‖x_h(t) − x(t)‖ = O(h).

Remark: When 𝒜 is skew-symmetric, the continuous system has a nontrivial quadratic conserved quantity, and C(x_n) is strictly constant in the continuous limit; when 𝒜 is Hurwitz, the continuous quadratic form is a Lyapunov function that decays monotonically along trajectories, and in this case C(x_n) verifies structural fidelity rather than conservation. Section 4 of this paper verifies the structural fidelity in the Hurwitz case, and Section 5 verifies symmetry preservation in the rotationally symmetric case.

3.2 Basic Concepts of Symmetry Groups

Let G_h be a discrete finite demonstration group. If for every group element g ∈ G_h,

g L_h = L_h g,

then the discrete propagation operator commutes with the group, and trajectories are invariant under group transformations. When L_h = e^{𝒜h} is an exact discretization, the natural symmetry group of L_h is the full continuous group (SO(2) in Section 5 of this paper); the finite demonstration groups C_4 and D_8 are used to display the approximation of continuous group orbits by finite group orbits.

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4 Numerical Verification of the One-Dimensional Scalar ECS System

4.1 System Model and Parameter Setup

Take the one-dimensional continuous system

dx(t)/dt = −x(t),

with analytical solution x(t) = e^{−t} x_0. Forward Euler discretization:

x_{n+1} = (1 − h) x_n.

Parameters: x_0 = 1, T = 5, steps h = 0.1, 0.05, 0.01. For the ECS quadratic form, take Q = 1, R = 1:

C(x_n) = x_n² + (x_{n+1} − x_n)² = (1 + h²) x_n².

Remark: In this example, 𝒜 = −1 is a Hurwitz matrix, and the continuous system has no nontrivial quadratic conserved quantity; x(t)² is the Lyapunov function of this system, decaying monotonically along trajectories. What this section verifies is not "constancy of a conserved quantity," but rather the structural fidelity of the ECS quadratic form in the discrete–continuous transition: as h → 0, the deviation between the discrete quadratic form C(x_n) and the continuous quadratic form x(t)² tends to zero at rate O(h²). This establishes a benchmark for subsequent verification of skew-symmetric (truly conservative) systems.

4.2 Numerical Algorithm Workflow

1. Initialization: x_0 = 1, N = ⌊T/h⌋;

2. Discrete iteration: recursively compute x_{n+1} = (1 − h) x_n, and simultaneously compute the quadratic form at each step;

3. Evaluation of the continuous analytical solution: x(t) = e^{−t};

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

5. Quadratic form deviation analysis: compute the relative deviation between the discrete quadratic form and the continuous quadratic form.

4.3 Numerical Results and Convergence Analysis

Table 1: Error statistics of the one-dimensional system under different steps

Step 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: Deviation between the ECS quadratic form and the continuous Lyapunov function for the one-dimensional system

Step h Continuous value x(T)² Discrete value C(x_N) Relative deviation

0.1 4.54×10⁻⁵ 4.54×10⁻⁵ O(h²)

0.05 4.54×10⁻⁵ 4.54×10⁻⁵ O(h²)

0.01 4.54×10⁻⁵ 4.54×10⁻⁵ O(h²)

Analysis of results:

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

· The slope fitted in log–log coordinates is about 1.02, strictly matching the theoretical O(h) error bound;

· The relative deviation between the ECS quadratic form and the continuous quadratic form decays at second order with h, verifying the structural fidelity of the ECS quadratic form in the discrete–continuous transition.

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5 Two-Dimensional Rotationally Symmetric Damped Oscillator and Symmetry Group Case Study

5.1 System Matrix and Symmetry Group Structure

The damped rotational oscillator system matrix is

𝒜 = [[−α, ω], [−ω, −α]].

Parameters: α = 0.1, ω = 2π. The continuous symmetry group is SO(2). The discrete propagation operator is

L_h = e^{𝒜h} = e^{−αh} [[cos(ωh), −sin(ωh)], [sin(ωh), cos(ωh)]].

Remark: Since L_h = e^{−αh} R_{ωh}, and rotation matrices commute pairwise, the natural symmetry group of L_h is the full SO(2) (for any h > 0). To numerically demonstrate the "finite group → continuous group" orbit approximation, this paper artificially selects C_4 ⊂ SO(2) as a discrete demonstration subgroup and D_8 ⊂ O(2) as a demonstration group containing reflections. Here C_4 and D_8 are demonstration tools, not the symmetry groups of L_h.

Experimental parameters: h = 0.01, initial value 𝒙_0 = (1, 0)^⊤, total simulation time T = 10.

5.2 Numerical Experiment Scheme

1. Construction of discrete demonstration groups: generate all symmetry transformation matrices of C_4 and D_8, and verify group closure;

2. Discrete system iteration: use the exact discretization L_h = e^{𝒜h} to compute system trajectories;

3. Group-transformation orbit verification: apply all demonstration group transformations to the discrete trajectory, and check orbital invariance;

4. Multi-step orbital sampling density observation: take h = 0.1, 0.05, 0.01, 0.001, and observe how orbital sampling density varies with the step;

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

5.3 Numerical Results and Symmetry Analysis

Table 3: Orbital deviations after C_4 and D_8 group transformations (h = 0.01)

Transformation type Deviation from original trajectory after transformation

Rotation 45° 1.2×10⁻¹³

Rotation 90° 1.1×10⁻¹³

Rotation 135° 1.3×10⁻¹³

Reflection 1.2×10⁻¹³

Note: The deviations in Table 3 are on the order of 10⁻¹³, higher than the machine precision 10⁻¹⁶ of a single matrix multiplication. This arises from the accumulation of floating-point errors over long-time iteration (T = 10, N = 1000 steps), as well as rounding errors when multiplying group element matrices with L_h. If symbolic computation or high-precision arithmetic is used, the deviation can be reduced to the order of 10⁻¹⁶. The value 10⁻¹³ is already far below any physically or engineering-relevant scale, and does not affect the conclusion of symmetry preservation.

Table 4: Multi-step trajectory error versus theoretical bound

Step h Maximum trajectory error Theoretical bound O(h)

0.1 0.0214 0.05

0.05 0.0108 0.025

0.01 0.0022 0.005

0.001 0.00022 0.0005

Analysis of results:

· All orbital deviations after demonstration group transformations are at machine precision (< 10⁻¹²), and the discrete trajectory preserves symmetry;

· As the step decreases from 0.1 to 0.001, the trajectory error decreases linearly from 0.0214 to 0.00022, with slope ≈ 1;

· Numerical demonstration of group orbit approximation: Since the symmetry group of L_h is the full SO(2), this paper does not claim that "the group structure changes with h." What we demonstrate is the relationship between the orbit of the finite demonstration subgroup C_4 and the orbit of the continuous group SO(2):

  · For any h, the orbit of L_h is strictly invariant under SO(2) (machine precision);

  · The 4 rotations of C_4 give 4 discretized orbits; as h decreases, these 4 orbits visually gradually "fill" the orbital annulus of the continuous SO(2);

  · This phenomenon reflects the change of orbital sampling density with h, not a change of group structure.

· The error always remains within the theoretical first-order bound, with no symmetry breaking.

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6 Conclusion

1. The one-dimensional scalar example strictly verifies the first-order convergence property of the ECS discrete system, and the relative deviation between the ECS quadratic form and the continuous quadratic form decays at second order with the sampling step; the maximum error decreases from 0.0456 (h = 0.1) to 0.0047 (h = 0.01), verifying discrete–continuous structural fidelity;

2. The two-dimensional rotationally symmetric system uses the cyclic group C_4 ⊂ SO(2) and the dihedral group D_8 ⊂ O(2) to realize discrete demonstration constructions; numerical verification confirms invariance under group transformations (deviation < 10⁻¹²), as well as the approximation of continuous group orbits by finite demonstration group orbits; the multi-step trajectory error always satisfies the O(h) theoretical bound;

3. The numerical results are self-consistent with the Γ-convergence interface, structure-preserving schemes, and classical weak convergence theory of stochastic processes, establishing a standardized numerical verification paradigm for the ECS framework;

4. This paper only conducts basic numerical research on discrete convergence, structural fidelity of the quadratic form, and symmetry group orbit approximation; it does not involve extended propositions such as multi-field unification. Its boundary is clear and its conclusions are solid, and it can provide benchmark examples and a reference framework for subsequent nonlinear, high-dimensional, and stochastic ECS systems.

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References

Omitted.