COMPLEX BURSTING OSCILLATION STRUCTURES IN A TWO-DIMENSIONAL NON-AUTONOMOUS DISCRETE SYSTEM
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摘要: 簇发振荡是多时间尺度系统复杂动力学行为的典型代表，簇发振荡的动力学机制与分类问题是簇发研究的重要问题之一，但当前学者们所揭示的簇发振荡的结构大多较为简单.研究以非自治离散Duffing系统为例，探讨具有复杂分岔结构的新型簇发振荡模式，并将其分为两大类，一类经由Fold分岔所诱发的对称式簇发，另一类经由延迟倍周期分岔所诱发的非对称式簇发.快子系统的分岔表现为典型的含有两个Fold分岔点的S形不动点曲线，其上、下稳定支可经由倍周期（即Flip）分岔通向混沌.当非自治项（即慢变量）穿越Fold分岔点时，系统的轨线可以向上、下稳定支的各种吸引子（例如，周期轨道和混沌）进行转迁，因此得到了经由Fold分岔所诱发的各种对称式簇发；而当非自治项无法穿越Fold分岔点，但可以穿越Flip分岔点时，系统产生了延迟Flip分岔现象.基于此，得到了经由延迟Flip分岔所诱发的各种非对称簇发.特别地，文中所报道的簇发振荡模式展现出复杂的反向Flip分岔结构.研究结果表明，这与非自治项缓慢地反向穿越快子系统的Flip分岔点有关.研究结果丰富了离散系统簇发的动力学机理和分类.
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关键词:
- 离散Duffing系统 /
- 复杂的簇发振荡模式 /
- 延迟Flip分岔 /
- 反向Flip分岔结构
Abstract: Bursting oscillations are the archetypes of complex dynamical behaviors in systems with multiple time scales, and the problem related to dynamical mechanisms and classifications of bursting oscillations is one of the important problems in bursting research.However, up to now, most of the structures of bursting revealed by researchers are relatively simple.In this paper, we take the non-autonomous discrete Duffing system as an example to explore novel bursting patterns with complex bifurcation structures, which are divided into two groups, i.e., symmetric bursting induced by Fold bifurcations and asymmetric bursting induced by delayed Flip bifurcations.Typically, the fast subsystem exhibits an Sshaped fixed point curve with two Fold points, and the stable upper and lower branches evolve into chaos by a cascade of Flip bifurcations.When the non-autonomous term (i.e., the slow variable) passes through Fold points, transitions to various attractors (e.g., periodic orbits and chaos) on the stable branches may take place, which accounts for the appearance of Fold-bifurcation-induced symmetric bursting patterns.If the non-autonomous term is not able to pass through Fold points, but to go through Flip points, delayed Flip bifurcations can be observed.Based on this, delayed-Flip-bifurcation-induced asymmetric bursting patterns are obtained.In particular, the bursting patterns reported here exhibit complex structures containing inverse Flip bifurcations, which has been found to be related to the fact that the nonautonomous term slowly passes through Flip points of the fast subsystem in an inverse way.Our results enrich dynamical mechanisms and classifications of bursting in discrete systems. -
图 1 快子系统关于单参数$\beta $的分岔图
Figure 1. One parameter bifurcation diagram of the fast sub-system with respect to $\beta $
图 2 快子系统在($\beta,b$)平面上的分岔集
Figure 2. Bifurcation set of the fast system on the plane ($\beta ,b$)
图 3 映射(2)关于参数 $\beta $ 的典型单参数分岔图
Figure 3. Typical one parameter bifurcation diagrams of the map (2) with respect to $\beta $
图 5 对称式"Fold/级联反向 Flip"簇发,$b =0.215$,$Z_n =0.4\cos(0.001 n)$
Figure 5. Symmetric "Fold/a cascade of inverse Flip" bursting,$b =0.215$,$Z_n =0.4\cos(0.001 n)$
图 6 另一种混沌簇发,$a =1.975$,$b =0$,$Z_n = 0.38 \cos(0.001 n) $
Figure 6. Another chaotic bursting,where $a =1.975$,$b =0$,$Z_n = 0.38\cos(0.001 n ) $
图 7 $ b =0.2$时快子系统关于 $\beta $ 的分岔图
Figure 7. Bifurcation diagram of fast subsystem with $\beta $ for $b =0.2$
图 8 "延迟Flip/反向Flip"簇发，$b =0.2$,$Z_n =0.27\cos(0.001 n)$
Figure 8. "Delayed F lip/inverse Flip" bursting,$b =0.2$,$Z_n =0.27\cos(0.001n)$
图 9 "双重延迟 Flip/双重反向 Flip"簇发,$ b =0.2$,$Z_n =0.34\cos(0.001 n)$
Figure 9. "Delayed double Flip/double inverse Flip" bursting,$b =0.2$,$Z_n =0.34\cos(0.001 n)$
图 10 "多重延迟 Flip/多重反向Flip"簇发,$ b =0.2$,$Z_n =0.35\cos(0.001 n)$
Figure 10. "Delayed multiple Flip/inverse multiple Flip" bursting,$b =0.2$,$Z_n =0.35\cos(0.001 n)$
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