# Finite Difference Models of Dynamical Systems with Quadratic Right-Hand Side

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preservation and Inheritance of Integrals of Motion

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1**

**Theorem**

**2**

## 3. Implicit Schemes and the Problem of Extra Roots

## 4. Reversible Schemes

## 5. Motion of a Top in the Euler–Poinsot Case

#### 5.1. Integral Curves

**Theorem**

**3.**

**Theorem**

**4.**

#### 5.2. Representation of a Difference Scheme in the Form of a Quadrature

**Theorem**

**5.**

**Proof.**

#### 5.3. Periodicity of Approximate Solution

**Definition**

**3.**

## 6. Approximate Trajectories in Projective Spaces

## 7. Closure of the Approximate Trajectory

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Change of quadrature increment (19) at $A=1,B=2,C=3$; initial data $p=0,q=r=1$; and step $\Delta t={\textstyle \frac{1}{10}}$.

**Figure 3.**Exact (red) and approximate (blue) solutions of Equation (21), satisfying the initial condition $p\left(0\right)=0$.

**Figure 4.**Approximate solution of a top whose center of gravity is shifted relative to the anchor point, found using a reversible scheme.

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**MDPI and ACS Style**

Malykh, M.; Gambaryan, M.; Kroytor, O.; Zorin, A.
Finite Difference Models of Dynamical Systems with Quadratic Right-Hand Side. *Mathematics* **2024**, *12*, 167.
https://doi.org/10.3390/math12010167

**AMA Style**

Malykh M, Gambaryan M, Kroytor O, Zorin A.
Finite Difference Models of Dynamical Systems with Quadratic Right-Hand Side. *Mathematics*. 2024; 12(1):167.
https://doi.org/10.3390/math12010167

**Chicago/Turabian Style**

Malykh, Mikhail, Mark Gambaryan, Oleg Kroytor, and Alexander Zorin.
2024. "Finite Difference Models of Dynamical Systems with Quadratic Right-Hand Side" *Mathematics* 12, no. 1: 167.
https://doi.org/10.3390/math12010167