NUL
发表于 2025-3-28 16:30:41
Classification of the Seven Elementary Catastrophes,see that for a nonlinear system of four (or fewer) (dimensionless) control parameters the solutions can undergo only seven different types of qualitative change (catastrophes). Many systems found in nature can be classified in a general way within this framework irrespective of our knowledge of the
重力
发表于 2025-3-28 18:58:38
Universal Amplitude Equation in the Neighborhood of a Hopf Bifurcation,ependent of the system itself). We first derive the universal equation using a concrete example, and then from symmetries reveal and explain its universal character. We introduce a few more general concepts, such as that of the limit cycle associated with the Hopf bifurcation, and will show that a t
Dysarthria
发表于 2025-3-29 02:25:10
http://reply.papertrans.cn/24/2315/231426/231426_43.png
DAMN
发表于 2025-3-29 06:49:48
Introduction to Chaos,ree possible scenarios for transition toward chaos, and introduce concepts useful for the study of the said chaos, namely the strange attractor, the Poincaré section, the fractal dimension, and self-similarity. We will discuss the difference between randomness and chaos and show that chaos, synonymo
ABOUT
发表于 2025-3-29 11:01:18
Pattern Formation in One Dimension, animals like jaguars and leopards. Two problems will be discussed in detail: the Turing chemical instability, and hydrodynamic convection. We will discuss the relevance of this chemical instability evidenced in diverse forms as it appears in nature, such as in the patterns of seashells, tropical fi
NADIR
发表于 2025-3-29 11:44:43
http://reply.papertrans.cn/24/2315/231426/231426_46.png
协奏曲
发表于 2025-3-29 19:13:09
Fronts between Domains and Invasion of One State by Another,, instead, begin as a localized pattern which then propagates until it has taken over the rest of the system, rather like a forest fire. We will see how the latter (and more usual) case boasts of an invasion described by an infinitude of possible propagation speeds, and show how the system settles o
繁重
发表于 2025-3-29 21:42:34
http://reply.papertrans.cn/24/2315/231426/231426_48.png
Diuretic
发表于 2025-3-30 03:40:43
Two Dimensional Patterns,n the equations describing systems in the neighborhood of the instability threshold giving rise to these patterns. The equations are universal. We will explain why most ordered patterns found in nature have a honeycomb or hexagonal shape, and discuss the stability of different structures.
quiet-sleep
发表于 2025-3-30 05:22:21
Wavelength Selection,More specifically, we will discuss wavelength selection for patterns, and then summarize different mechanisms, such as boundary effects, the effect of the invasion of a localized pulse or of a front, and the effect of defects. We will see that defects provide a robust wavelength selection mechanism.