Departamento de Ciencias Exactas y Tecnología, Centro Universitario de los Lagos, Universidad de Guadalajara, Enrique Díaz de León 1144, Colonia Paseos de la Monta na, 47460 Lagos de Moreno, Jalisco, Mexico.
Center for Biomedical Technology, Universidad Politécnica de Madrid, Campus de Montegancedo, Pozuelo de Alarcón, 28223 Madrid, Spain.
Chaos. 2023 Jul 1;33(7). doi: 10.1063/5.0141054.
We study the dynamics of multistable coexisting rotating waves that propagate along a unidirectional ring consisting of coupled double-well Duffing oscillators with different numbers of oscillators. By employing time series analysis, phase portraits, bifurcation diagrams, and basins of attraction, we provide evidence of multistability on the route from coexisting stable equilibria to hyperchaos via a sequence of bifurcations, including the Hopf bifurcation, torus bifurcations, and crisis bifurcations, as the coupling strength is increased. The specific bifurcation route depends on whether the ring comprises an even or odd number of oscillators. In the case of an even number of oscillators, we observe the existence of up to 32 coexisting stable fixed points at relatively weak coupling strengths, while a ring with an odd number of oscillators exhibits 20 coexisting stable equilibria. As the coupling strength increases, a hidden amplitude death attractor is born in an inverse supercritical pitchfork bifurcation in the ring with an even number of oscillators, coexisting with various homoclinic and heteroclinic orbits. Additionally, for stronger coupling, amplitude death coexists with chaos. Notably, the rotating wave speed of all coexisting limit cycles remains approximately constant and undergoes an exponential decrease as the coupling strength is increased. At the same time, the wave frequency varies among different coexisting orbits, exhibiting an almost linear growth with the coupling strength. It is worth mentioning that orbits originating from stronger coupling strengths possess higher frequencies.
我们研究了由具有不同数量振荡器的耦合双阱杜芬振子组成的单向环中传播的多稳态共存旋转波的动力学。通过时间序列分析、相图、分岔图和吸引域，我们提供了证据表明，随着耦合强度的增加，从共存稳定平衡点到超混沌的多稳定性路径包括Hopf 分岔、环面分岔和危机分岔，其中环由偶数或奇数个振荡器组成。在偶数个振荡器的情况下，我们观察到在相对较弱的耦合强度下存在多达 32 个共存稳定的固定点，而奇数个振荡器的环则存在 20 个共存稳定的平衡点。随着耦合强度的增加，在偶数个振荡器的环中，隐藏的振幅死亡吸引子在逆超临界叉形分岔中产生，与各种同宿和异宿轨道共存。此外，对于更强的耦合，振幅死亡与混沌共存。值得注意的是，所有共存极限环的旋转波速度保持大致恒定，并随着耦合强度的增加呈指数下降。同时，不同共存轨道的波频率不同，与耦合强度呈几乎线性增长。值得一提的是，起源于较强耦合强度的轨道具有较高的频率。
