# Stability and Boundedness of the Solutions of Multi-Parameter Dynamical Systems with Circulatory Forces

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Corollary**

**1.**

**Corollary**

**2.**

## 3. Main Results: Circulatory System with Three Degrees of Freedom

**Example**

**1.**

## 4. Special Case: Multiple Eigenvalues of Matrix K

**Definition**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

**Proof.**

**Example**

**2.**

**Remark**

**1.**

**Example**

**3.**

## 5. Nonlinear Systems with Unstable Equilibrium

#### 5.1. Some General Remarks

#### 5.2. A Case Study: The Bounded Solutions of a 2-DoF System with Unstable Origin

**Lemma**

**1.**

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 2.**Dynamic model of the system subjected to a follower load (adopted from [30]).

**Figure 6.**Projection of the phase trajectory: (a) ${x}_{1}(0)=0.001$; (b) ${x}_{1}$ is changed to $-0.001$.

**Figure 11.**The domains where all trajectories starting in ${D}_{1}$ are limited by the threshold of $\partial {D}_{2}$.

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**MDPI and ACS Style**

Awrejcewicz, J.; Losyeva, N.; Puzyrov, V.
Stability and Boundedness of the Solutions of Multi-Parameter Dynamical Systems with Circulatory Forces. *Symmetry* **2020**, *12*, 1210.
https://doi.org/10.3390/sym12081210

**AMA Style**

Awrejcewicz J, Losyeva N, Puzyrov V.
Stability and Boundedness of the Solutions of Multi-Parameter Dynamical Systems with Circulatory Forces. *Symmetry*. 2020; 12(8):1210.
https://doi.org/10.3390/sym12081210

**Chicago/Turabian Style**

Awrejcewicz, Jan, Nataliya Losyeva, and Volodymyr Puzyrov.
2020. "Stability and Boundedness of the Solutions of Multi-Parameter Dynamical Systems with Circulatory Forces" *Symmetry* 12, no. 8: 1210.
https://doi.org/10.3390/sym12081210