# Dynamic Behaviors Analysis of Asymmetric Stochastic Delay Differential Equations with Noise and Application to Weak Signal Detection

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis

#### 2.1. Stability Analysis

**Theorem**

**1**

**Proof.**

#### 2.2. Stochastic $It\widehat{o}$ Equation

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

## 3. Weak Signal Detection

#### 3.1. Chaos in the Asymmetric Stochastic Delay Differential Equation

#### 3.2. Time-Delayed Feedback and System Detection Capability

## 4. Numerical Simulation

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The time delay $\tau $ enhances the noise immunity of chaotic state: (

**a**) $\tau =0.01$; (

**b**) $\tau =1$; (

**c**) $\tau =2$; (

**d**) $\tau =3$.

**Figure 2.**Bifurcation diagrams of the asymmetric system (1) with respect to frequency $\omega $: (

**a**) $r=0.001$; (

**b**) $r=0.02$; and (

**c**) $r=0.05$.

**Figure 5.**Phase diagrams of the asymmetric system (1): (

**a**) $f=0.3541$; (

**b**) $f=0.3542$; and (

**c**) $f=0.39$.

**Figure 6.**Bifurcation diagram of the asymmetric system (17).

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**MDPI and ACS Style**

Wang, Q.; Zhang, X.; Yang, Y.
Dynamic Behaviors Analysis of Asymmetric Stochastic Delay Differential Equations with Noise and Application to Weak Signal Detection. *Symmetry* **2019**, *11*, 1428.
https://doi.org/10.3390/sym11111428

**AMA Style**

Wang Q, Zhang X, Yang Y.
Dynamic Behaviors Analysis of Asymmetric Stochastic Delay Differential Equations with Noise and Application to Weak Signal Detection. *Symmetry*. 2019; 11(11):1428.
https://doi.org/10.3390/sym11111428

**Chicago/Turabian Style**

Wang, Qiubao, Xing Zhang, and Yuejuan Yang.
2019. "Dynamic Behaviors Analysis of Asymmetric Stochastic Delay Differential Equations with Noise and Application to Weak Signal Detection" *Symmetry* 11, no. 11: 1428.
https://doi.org/10.3390/sym11111428