# Resonance Enhancement by Suitably Chosen Frequency Detuning

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## Abstract

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**2016**, 133(25), 6862–6867). Moreover, similar study can be conducted for a generic three-wave system taken in the Hamiltonian form which makes our results applicable for an arbitrary Hamiltonian three-wave system met in climate prediction theory, geophysical fluid dynamics, plasma physics, etc.

## 1. Introduction

In order to better understand these issues, we believe that it is important to move beyond the kinematic picture of resonance broadening and attempt to devise methods of studying these effects dynamically.

## 2. Barotropic Vorticity Equation

## 3. Model Equations

## 4. Amplitudes

`Matlab`

^{TM}software along with its standard ODE Suite [21]. In particular, the standard

`ode45`solver was employed with stringent error tolerance settings. In Figure 1 we show the energy evolution in the resonant triad given in Table 1 for several values of the frequency detuning $\Delta \omega =\tilde{\Delta \omega}/\epsilon \in [-\frac{1}{2},\frac{1}{2}]$; e.g., in geophysical applications $\epsilon \sim \mathcal{O}({10}^{-2})$. From these graphs it can be seen that the period $\tau $ and the range of the energy variation, defined as

- (I)
- $\Delta \omega \in \left(-\infty ,\Delta {\omega}_{max}^{(1)}\right]$;
- (II)
- $\Delta \omega \in \left(\Delta {\omega}_{max}^{(1)},min\{0,\Delta {\omega}_{\mathrm{st}}\}\right]$;
- (III)
- $\Delta \omega \in \left(min\{0,\Delta {\omega}_{\mathrm{st}}\},max\{0,\Delta {\omega}_{\mathrm{st}}\}\right]$;
- (IV)
- $\Delta \omega \in \left(\Delta {\omega}_{\mathrm{st}},\Delta {\omega}_{max}^{(2)}\right]$;
- (V)
- $\Delta \omega \in \left(\Delta {\omega}_{max}^{(2)},+\infty \right)$.

## 5. Phase Space Analysis

`Maple`

^{TM}software. On Figure 4, Figure 5 and Figure 6 we depict the typical phase portraits of the dynamical system of Equations (24)–(28) in phase-amplitude variables. For illustration we choose the triad given in Table 1 with the initial energy distribution (a). In these pictures we represent the high-frequency mode ${C}_{3}$ on the horizontal axis, while the dynamical phase $\psi $ is on the vertical.

## 6. Conclusions

- The amplitude of energy variation Equation (29) in a triad with suitably chosen detuning ($\Delta \omega \ne 0$) can be significantly higher than in the case of exact resonance, i.e., $\Delta \omega \equiv 0$. The maximal amplification as compared to exact resonance is attained when $\Delta {\omega}_{\mathrm{st}}$ coincides with the point of exact resonance. In this case one of the zones (III) or (IV) disappears.
- The phase portraits (see Figure 4, Figure 5 and Figure 6) along with the shape and size of the periodic cycles are substantially different for $\Delta \omega >0$ and $\Delta \omega >0$ (c.f. Figure 6). This means that any complete analysis of detuned resonance must include both positive and negative values of the detuning parameter $\Delta \omega $.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Energy evolution in the triad given in Table 1, for different values of the detuning $\Delta \omega $.

**Figure 2.**Typical dependency of the energy variation range $\Delta \mathcal{E}$ on the frequency detuning $\Delta \omega $ for the case when the high frequency mode ${\omega}_{3}$ has the maximal energy (initial condition from Table 1). The vertical red dashed line shows the location of $\Delta {\omega}_{max}^{(1)}$, while the vertical black solid line shows the location of $\Delta {\omega}_{max}^{(2)}$. Finally, the blue dash-dotted line shows the amplitude obtained the exact resonance.

**Figure 3.**Typical dependence of the energy variation range $\Delta \mathcal{E}$ on the frequency detuning $\Delta \omega $ for the case when the high frequency mode ${\omega}_{3}$ has the lowest energy (initial condition from Table 1).

Parameter | Value |
---|---|

Resonant wave vectors, $[{m}_{j},{n}_{j}]$ | $[4,12]$, $[5,14]$, $[9,13]$ |

Resonant frequencies, $2{m}_{j}/{n}_{j}({n}_{j}+1)$ | $0.0513$, $0.0476$, $0.0989$ |

Resonant triad parameters, ${N}_{j}$ | 156, 210, 182 |

Interaction coefficient, Z | $7.82$ |

Initial energy distribution (a), % | 20%, 30%, 50% |

Initial energy distribution (b), % | 40%, 40%, 20% |

Initial dynamical phase, $\psi $ | 0.0 |

**Table 2.**Behaviour of physical parameters $\Delta \mathcal{E}$, $\tau $ and $\psi $ in different regions.

Region/Range | ⟶ | ⟵ |
---|---|---|

(I) | $\Delta \mathcal{E}+$, $\tau +$, $\Delta \psi -$ | $\Delta \mathcal{E}-$, $\tau -$, $\Delta \psi +$ |

(II) | $\Delta \mathcal{E}-$, $\tau +$, $\Delta \psi -$ | $\Delta \mathcal{E}+$, $\tau -$, $\Delta \psi +$ |

(III) | $\Delta \mathcal{E}-$, $\tau +$, $\Delta \psi -$ | $\Delta \mathcal{E}+$, $\tau -$, $\Delta \psi +$ |

(IV) | $\Delta \mathcal{E}+$, $\tau -$, $\Delta \psi +$ | $\Delta \mathcal{E}-$, $\tau +$, $\Delta \psi -$ |

(V) | $\Delta \mathcal{E}-$, $\tau -$, $\Delta \psi +$ | $\Delta \mathcal{E}+$, $\tau +$, $\Delta \psi -$ |

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**MDPI and ACS Style**

Dutykh, D.; Tobisch, E.
Resonance Enhancement by Suitably Chosen Frequency Detuning. *Mathematics* **2020**, *8*, 450.
https://doi.org/10.3390/math8030450

**AMA Style**

Dutykh D, Tobisch E.
Resonance Enhancement by Suitably Chosen Frequency Detuning. *Mathematics*. 2020; 8(3):450.
https://doi.org/10.3390/math8030450

**Chicago/Turabian Style**

Dutykh, Denys, and Elena Tobisch.
2020. "Resonance Enhancement by Suitably Chosen Frequency Detuning" *Mathematics* 8, no. 3: 450.
https://doi.org/10.3390/math8030450