Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, People's Republic of China.
College of Mathematics and Information Science, Shaan'xi Normal University, Xi'an 710062, People's Republic of China.
Chaos. 2019 Oct;29(10):103150. doi: 10.1063/1.5114998.
Recently, a large number of studies have concentrated on aging transition, but they have so far been restricted to coupled integer-order oscillators. Here, we report the first study of aging transition in mixed active and inactive fractional-order oscillators. It has been demonstrated that while the heterogeneity is caused by the distance parameter, both the coupling strength and the fractional-order derivative can modulate the critical ratio at which aging transition occurs. In addition, a small fractional-order derivative may ruin the ability of oscillation and, thus, reduce the critical ratio in globally coupled fractional-order Stuart-Landau oscillators. Remarkably, the larger the natural frequency is the more easily the aging transition occurs in coupled fractional-order oscillators. Further studies have shown that, being diverse from an integer-order Stuart-Landau oscillator, the natural frequency may induce a Hopf bifurcation in a fractional-order Stuart-Landau oscillator, accordingly, introducing a new heterogeneity in the coupled fractional-order Stuart-Landau oscillators. Therein, a counterintuitive phenomenon has been found that the critical ratio depends unmonotonously on the coupling strength, which implies that the coupled fractional-order Stuart-Landau oscillators possess the weakest robustness of oscillation at a certain level of coupling strength.
最近，大量研究集中在衰老转变上，但迄今为止，这些研究仅限于耦合整数阶振荡器。在这里，我们报告了第一个混合主动和非活动分数阶振荡器的衰老转变研究。研究表明，虽然异质性是由距离参数引起的，但耦合强度和分数阶导数都可以调节衰老转变发生的临界比。此外，小的分数阶导数可能会破坏振荡的能力，从而降低全局耦合分数阶 Stuart-Landau 振荡器中的临界比。值得注意的是，自然频率越大，耦合分数阶振荡器中衰老转变越容易发生。进一步的研究表明，与整数阶 Stuart-Landau 振荡器不同，自然频率可能会在分数阶 Stuart-Landau 振荡器中引起 Hopf 分岔，从而在耦合分数阶 Stuart-Landau 振荡器中引入新的异质性。在此，我们发现了一个违反直觉的现象，即临界比不单调地依赖于耦合强度，这意味着在一定的耦合强度水平下，耦合分数阶 Stuart-Landau 振荡器的振荡稳定性最差。
