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Forming invariant stochastic differential systems with a given first integral

Konstantin A. Rybakov

TL;DR

The paper addresses the inverse dynamics problem of forming invariant stochastic differential systems whose trajectories stay on a manifold defined by a first integral $M(t,x)=\text{const}$. It introduces a tangent-hyperplane basis construction to synthesize Itô or Stratonovich SDEs with coefficients that depend linearly on a minimal set of scalar functions, ensuring invariance and enabling degenerate diffusion in chosen state components. It generalizes the approach to second, fourth, and eighth order systems using basis structures tied to $G=\nabla_x M$ and orthogonal vectors, and provides explicit constructions and degeneracy-handling strategies, including $\Lambda$-type bases for stability. Numerical experiments using Milstein and Artemiev methods validate first-order convergence and demonstrate invariant dynamics on manifolds such as a catenoid and a time-dependent paraboloid, illustrating practical utility for control synthesis and numerical verification of SDE solvers.

Abstract

This article proposes a method for forming invariant stochastic differential systems, namely dynamic systems with trajectories belonging to a given smooth manifold. The Itô or Stratonovich stochastic differential equations with the Wiener component describe dynamic systems, and the manifold is implicitly defined by a differentiable function. A convenient implementation of the algorithm for forming invariant stochastic differential systems within symbolic computation environments characterizes the proposed method. It is based on determining a basis associated with a tangent hyperplane to the manifold. The article discusses the problem of basis degeneration and examines variants that allow for the simple construction of a basis that does not degenerate. Examples of invariant stochastic differential systems are given, and numerical simulations are performed for them.

Forming invariant stochastic differential systems with a given first integral

TL;DR

The paper addresses the inverse dynamics problem of forming invariant stochastic differential systems whose trajectories stay on a manifold defined by a first integral . It introduces a tangent-hyperplane basis construction to synthesize Itô or Stratonovich SDEs with coefficients that depend linearly on a minimal set of scalar functions, ensuring invariance and enabling degenerate diffusion in chosen state components. It generalizes the approach to second, fourth, and eighth order systems using basis structures tied to and orthogonal vectors, and provides explicit constructions and degeneracy-handling strategies, including -type bases for stability. Numerical experiments using Milstein and Artemiev methods validate first-order convergence and demonstrate invariant dynamics on manifolds such as a catenoid and a time-dependent paraboloid, illustrating practical utility for control synthesis and numerical verification of SDE solvers.

Abstract

This article proposes a method for forming invariant stochastic differential systems, namely dynamic systems with trajectories belonging to a given smooth manifold. The Itô or Stratonovich stochastic differential equations with the Wiener component describe dynamic systems, and the manifold is implicitly defined by a differentiable function. A convenient implementation of the algorithm for forming invariant stochastic differential systems within symbolic computation environments characterizes the proposed method. It is based on determining a basis associated with a tangent hyperplane to the manifold. The article discusses the problem of basis degeneration and examines variants that allow for the simple construction of a basis that does not degenerate. Examples of invariant stochastic differential systems are given, and numerical simulations are performed for them.
Paper Structure (6 sections, 7 theorems, 82 equations, 3 figures)

This paper contains 6 sections, 7 theorems, 82 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Invariant Stochastic Differential Systems
  3. Forming Invariant Stochastic Differential Systems
  4. Invariant Stochastic Differential Systems of the Second, Fourth, and Eighth Orders
  5. Computational Experiments
  6. Conclusion

Key Result

Proposition 1

If $g_2 \neq 0$, $g_3 \neq 0$, …, $g_{n-1} \neq 0$, then vectors $G,N_1,\dots,N_{n-1}$ are linearly independent, and the determinant of the matrix formed by these vectors is equal to $(-1)^{n-1} |G|^2 g_2 g_3 \dots g_{n-1}$.

Figures (3)

  • Figure 1: Sample phase trajectories of the numerical solution corresponding to the random process $X(t)$, $x_0 = [ \, 0 ~~ 1 ~~ 0 \, ]^{ \mathrm{T}}$.
  • Figure 2: Sample phase trajectories of the numerical solution corresponding to the random process $X(t)$, $x_0 = [ \, 1 ~~ 0 ~~ 0 \, ]^{ \mathrm{T}}$.
  • Figure 3: Sample trajectories of the numerical solution corresponding to the random process $X(t)$.

Theorems & Definitions (18)

  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Proposition 3
  • ...and 8 more