# The Verhulst-Like Equations: Integrable OΔE and ODE with Chaotic Behavior

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## Abstract

**:**

## 1. Introduction

## 2. Integrable O$\Delta $E

## 3. Continualization with Padé Approximants

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

O$\Delta $E | Ordinary Difference Equation |

ODE | Ordinary Differential Equation |

## References

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**Figure 1.**Numerical solution of Cauchy problem (Equations (17) and (19)) for R = 2.5 shows periodic motion.

**Figure 2.**With increasing of parameter R (R = 2.86 for this figure) numerical solution of Cauchy problem (Equations (17) and (19)) shows periodic motion, slightly different from that shown in Figure 1.

**Figure 4.**Numerical solution of Cauchy problem (Equations (17) and (19)) for R = 2.95 shows the appearance of subharmonics in periodic oscillations.

**Figure 6.**Numerical solution of Cauchy problem (Equations (17) and (19)) for R = 3.0. Phase trajectory and Poincaré section (

**a**) and trajectories in 3D space (

**b**). The dependence on initial conditions is small (see (

**c**)).

**Figure 9.**Numerical solution of Cauchy problem (Equations (17) and (19)) for R = 3.1. Phase trajectory and Poincaré section (

**a**) and trajectories in 3D space (

**b**). A very small change in initial conditions created a significantly different outcome (see (

**c**)).

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**MDPI and ACS Style**

Andrianov, I.; Starushenko, G.; Kvitka, S.; Khajiyeva, L.
The Verhulst-Like Equations: Integrable OΔE and ODE with Chaotic Behavior. *Symmetry* **2019**, *11*, 1446.
https://doi.org/10.3390/sym11121446

**AMA Style**

Andrianov I, Starushenko G, Kvitka S, Khajiyeva L.
The Verhulst-Like Equations: Integrable OΔE and ODE with Chaotic Behavior. *Symmetry*. 2019; 11(12):1446.
https://doi.org/10.3390/sym11121446

**Chicago/Turabian Style**

Andrianov, Igor, Galina Starushenko, Sergey Kvitka, and Lelya Khajiyeva.
2019. "The Verhulst-Like Equations: Integrable OΔE and ODE with Chaotic Behavior" *Symmetry* 11, no. 12: 1446.
https://doi.org/10.3390/sym11121446