Towards Analytical Chaotic Evolutions in Brusselators

The Brusselator is a mathematical model for autocatalytic reaction, which was proposed by Ilya Prigogine and his collaborators at the Universit Libre de Bruxelles. The dynamics of the Brusselator gives an oscillating reaction mechanism for an autocatalytic, oscillating chemical reaction. The Brusselator is a slow-fast oscillating chemical reaction system. The traditional analytical methods cannot provide analytical solutions of such slow-fast oscillating reaction, and numerical simulations cannot provide a full picture of periodic evolutions in the Brusselator. In this book, the generalized harmonic balance methods are employed for analytical solutions of periodic evolutions of the Brusselator with a harmonic diffusion. The bifurcation tree of period-1 motion to chaos of the Brusselator is presented through frequency-amplitude characteristics, which be measured in frequency domains. Two main results presented in this book are:

- analytical routes of periodical evolutions to chaos and

- independent period-(2�� + 1) evolution to chaos.

This book gives a better understanding of periodic evolutions to chaos in the slow-fast varying Brusselator system, and the bifurcation tree of period-1 evolution to chaos is clearly demonstrated, which can help one understand routes of periodic evolutions to chaos in chemical reaction oscillators. The slow-fast varying systems extensively exist in biological systems and disease dynamical systems. The methodology presented in this book can be used to investigate the slow-fast varying oscillating motions in biological systems and disease dynamical systems for a better understanding of how infectious diseases spread.

布魯塞爾反應器是一個自催化反應的數學模型，由伊利亞·普里戈金和他在布魯塞爾自由大學的合作者提出。布魯塞爾反應器的動力學提供了一個自催化、振盪的化學反應機制。布魯塞爾反應器是一個慢-快振盪的化學反應系統。傳統的分析方法無法提供這種慢-快振盪反應的解析解，而數值模擬也無法提供布魯塞爾反應器中週期演化的完整圖像。本書使用廣義諧波平衡方法對具有諧波擴散的布魯塞爾反應器週期演化進行解析解。通過頻率-振幅特性，展示了布魯塞爾反應器從週期1運動到混沌的分叉樹。本書提出的兩個主要結果是：週期演化到混沌的解析路徑和獨立週期-(2 + 1)演化到混沌。本書更好地理解了慢-快變化的布魯塞爾反應器系統中的週期演化到混沌，並清晰展示了週期1演化到混沌的分叉樹，這有助於理解化學反應振盪器中週期演化到混沌的路徑。慢-快變化的系統廣泛存在於生物系統和疾病動力學系統中。本書中提出的方法可以用於研究生物系統和疾病動力學系統中慢-快變化的振盪運動，以更好地理解傳染病的傳播方式。