延迟的Duffing振荡器的短时间延迟极限
Thomas Erneux1, Anton V Kovalev2, Evgeny A Viktorov2
1Université Libre de Bruxelles, Optique Nonlinéaire Théorique, Campus Plaine C.P. 231, 1050 Bruxelles, Belgium.
Physical review. E
|January 20, 2024
概括
研究人员使用非对称理论在延迟的达芬方程中解决了一个奇异的霍普夫分叉. 将延迟期限扩展到第三顺序是意想不到的必要,简化了复杂的激光稳定性问题.
科学领域:
- 非线性动力学是一种非线性动力学.
- 延迟微分方程 延迟微分方程
- 双分支线理论 双分支线理论
背景情况:
- 延迟的Duffing方程表现出一个Hopf分叉,在特定条件下变得单数 (ε→0, τ=O(ε)→0).
- 这种奇点在分析像激光稳定性这样的系统时带来了挑战,这些系统具有类似的数学结构.
- 现有的方法很难准确地捕捉到这个单一的极限中的分叉行为.
研究的目的:
- 在延迟的Duffing方程中开发一种解脱单一的Hopf分叉的非对称理论.
- 导出一个简化的普通微分方程 (ODE) 系统,准确地表示分叉.
- 通过将理论与任意延迟的现有非对称解决方案进行比较来验证该理论.
主要方法:
- 延迟项的泰勒扩展x(t-τ) 乘以t的权.
- 从原始延迟微分方程中导出一个最小的ODE系统.
- 对非对称行为进行分析,并与固定延迟的解决方案进行匹配.
主要成果:
- 一个新的非对称理论成功地解决了霍夫分叉的奇点.
- 与最初的预期相反,延迟期限延长至第三次订单被认为是必不可少的.
- 衍生的ODE系统准确地捕获了Hopf分叉分支,通过与其他非对称解决方案的重叠来验证.
结论:
- 开发的非对称理论提供了一种有效的方法来分析延迟系统中单一的霍夫分叉.
- 第三阶段扩张的必要性凸显了延迟动态的一个微妙方面.
- 这项工作为理解激光稳定性和其他领域的复杂现象提供了一个简化模型.
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