# Feedback Stabilization Applied to Heart Rhythm Dynamics Using an Integro-Differential Method

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## Abstract

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## 1. Introduction

## 2. Formulation of a Stabilization Problem

**Example 1.**

- As noted above, small values of the control signal are needed (compared to the stabilized functions) when stabilization is achieved.
- It differs from other stabilization schemes by the absence of adjustable parameters and rough approximations in determining the control function. Control parameters $\alpha $ and $\beta $ are determined by the properties of the unstable system itself.

## 3. Mathematical Model of Heart Rythm Dynamics Based on a Modified Van der Pol Equation

#### 3.1. Problem Statement

#### 3.2. Proposed Method

## 4. Numerical Results of Numerical Stabilization of a Modified Van der Pol Equation

## 5. Stochastic Perspective

#### 5.1. Introductory Remarks

#### 5.2. Additional Examples

**Example 2.**

**Example 3.**

**Example 4.**

**Example 5.**

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Chaotic pacemaker activity. Time dependence for $0\u2a7dt\u2a7d500$. (

**b**) Chaotic pacemaker activity. Phase space. Time interval $0\u2a7dt\u2a7d1000$.

**Figure 2.**(

**a**) Numerical results of the stabilization of a modified Van der Pol Equation (16) for $\alpha =6.5,\beta =6.3$. (

**b**) Numerical results of the stabilization of a modified Van der Pol Equation (16) for $\alpha =10.5,\beta =10.3$. (

**c**) Numerical results of the stabilization of a modified Van der Pol Equation (16) for $\alpha =20.5,\beta =20.3$.

**Figure 3.**The phase trajectory of the solution of System (16) in the $(x,y)$ plane; time interval is taken to be $1000\u2a7dt\u2a7d1500$.

**Figure 4.**Control function $u\left(t\right)$ compared to the stabilized process $x\left(t\right),y\left(t\right)$.

**Figure 9.**Numerical results of the stabilization of a modified stochastic Van der Pol Equation (25).

**Figure 11.**Numerical results of the stabilization of a modified stochastic Van der Pol Equation (25).

**Figure 12.**A sample path of random process $\theta \left(t\right)$ related to the Ornstein–Uhlenbeck process.

**Figure 13.**Numerical results of the stabilization of a modified stochastic Van der Pol Equation (25).

**Figure 15.**Numerical results of the stabilization of a modified stochastic Van der Pol Equation (25).

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**MDPI and ACS Style**

Yahalom, A.; Puzanov, N.
Feedback Stabilization Applied to Heart Rhythm Dynamics Using an Integro-Differential Method. *Mathematics* **2024**, *12*, 158.
https://doi.org/10.3390/math12010158

**AMA Style**

Yahalom A, Puzanov N.
Feedback Stabilization Applied to Heart Rhythm Dynamics Using an Integro-Differential Method. *Mathematics*. 2024; 12(1):158.
https://doi.org/10.3390/math12010158

**Chicago/Turabian Style**

Yahalom, Asher, and Natalia Puzanov.
2024. "Feedback Stabilization Applied to Heart Rhythm Dynamics Using an Integro-Differential Method" *Mathematics* 12, no. 1: 158.
https://doi.org/10.3390/math12010158