Spiral waves are typical dissipative structures commonly observed in distributed systems in physics, chemistry, and biology. Spiral waves generally exist in excitable and oscillatory systems having spatial extensions as well as spiral tips that serve as the oscillation sources. The study of the dynamics of spiral waves has gained momentum in recent years due to its extensive applications. These experimental and simulation-based studies have led to the discovery of many spiral wave patterns with novel structures. Among them, super-spiral wave patterns with spiral structure beyond the original spiral structure have attracted significant attention. They include over-spiral spiral waves caused by multi-period oscillation of local points and super-multi armed and segmented (SMAS) spiral patterns comprising of several super-arms sharing the same spiral tip with the original spiral. Alternatively, line-defect spiral waves mainly caused by period-doubling bifurcation have also been extensively researched in the literature. Nevertheless, the spiral wave pattern containing both super-critical structures and line defects have not been found to date, despite its potential applications in several fields.
In an effort to bridge the existing research gap, a team of researchers at the University of Shanghai for Science and Technology: Dr. Jian Gao, Professor Changgui Gu, Professor Huijie Yang, and Professor Tongfeng Weng reported a new type of bi-stable spiral wave, known as bistable Ying-Yang (BYY) spiral wave, containing both super-spiral structures and line defects. This novel spiral wave caused by the pitchfork bifurcation of a limit cycle of a local dynamic behavior was reported in a single-period oscillatory medium. Their research work is currently published in the research journal, Communications in Nonlinear Science and Numerical Simulation.
Briefly, the authors commenced their research work by exploring the dynamics of spatially distributed systems, governed by reaction-diffusion equations, where the homogeneous system exhibits pitchfork bifurcation. The new spiral wave was characterized by a slow-moving line defect passing through the spiral center. A superposition principle of disturbance intensity vectors was employed to explain the dynamic behavior of the spiral wave in an oscillatory medium. Besides, an additional parameter c was introduced to investigate the structure of the spiral wave after the local oscillation had undergone the pitchfork bifurcation.
The author observed that the local dynamic behavior underwent a pitchfork bifurcation when the parameter c was increased beyond the threshold. Results showed that the superposition principle of disturbances intensity vector provided a new approach for studying the formation of patterns in oscillation media. For instance, the disturbance on the local phase waves due to line-defects could be superposed with the disturbance emanating from the spiral center and analyzed to obtain more insights on the dynamic behavior of the system. Moreover, it was worth noting that the predicted spiral wave exists in nonsymbiotic ecosystems comprising two or more species and can thus be utilized in various ecological studies.
In summary, the study is the first to report a type of bi-stable spiral wave caused by pitchfork bifurcation of local dynamic behaviors in a single-period oscillatory medium. The authors demonstrated the significance of the superposition principle of disturbance intensity vectors in studying the dynamic formation of different patterns. In a statement to Advances in Engineering, the authors said that the predicted bi-stable spiral wave would be particularly useful for researches about ecological systems comprising of nonsymbiotic species.
Gao, J., Gu, C., Yang, H., & Weng, T. (2020). A type of bi-stable spiral wave in a single-period oscillatory medium. Communications in Nonlinear Science and Numerical Simulation, 85, 105233.