Good website, with poems in Chinese, pinyin and commented translation. Well done!
100 poems website 100tangpoems.wordpress.com/2023/02/27/pure-and-fair/

In dynamical systems, "theorems" refer to established results that describe the behavior of systems over time under certain conditions. Dynamical systems are mathematical models used to describe the evolution of points in a given space according to specific rules, often represented by differential equations or discrete mappings.

Self-organization refers to a process in which a system spontaneously organizes itself without external direction or intervention. This phenomenon occurs in various fields, including biology, physics, chemistry, social sciences, and computer science. Key characteristics of self-organization include: 1. **Complex Interactions**: Individual components of the system interact in simple ways, leading to complex group behavior or structures.

Nonlinear systems refer to mathematical models or systems of equations where the output is not directly proportional to the input. In contrast to linear systems, where a change in input produces a proportional change in output, nonlinear systems exhibit more complex behavior. Here are some key characteristics of nonlinear systems: 1. **Mathematical Representation**: Nonlinear systems can be described by nonlinear differential equations, polynomial equations, or other complex functions where the variables are raised to powers other than one or are multiplied together.

Hidden oscillation typically refers to a phenomenon in systems that exhibit oscillatory behavior that may not be immediately observable or detectable. This term can be applied in various fields, such as physics, engineering, biology, and even in social sciences, where systems can undergo oscillatory dynamics without clear or visible cycles. For example, in control systems or electronic circuits, hidden oscillations might occur due to nonlinearities or other complex interactions that mask the periodic behavior.

Chaotic maps are mathematical functions or systems that exhibit chaos, which is a complex and unpredictable behavior that arises in certain dynamical systems. These maps are often studied in the context of chaos theory, where small changes in initial conditions can lead to significantly different outcomes, a phenomenon popularly known as the "butterfly effect." Key characteristics of chaotic maps include: 1. **Nonlinearity**: Most chaotic systems are nonlinear, meaning that their relationships cannot be described with simple linear equations.

Bifurcation theory is a branch of mathematics and dynamical systems that studies changes in the structure of a system's solutions as parameters vary. In simpler terms, it examines how small changes in the parameters of a system can lead to significant changes in its behavior or structure, often resulting in the creation or annihilation of stable states or periodic solutions. Key concepts in bifurcation theory include: 1. **Equilibrium Points**: These are the steady-state solutions of a dynamical system.

Astronomical dynamical systems is a field of study in celestial mechanics that focuses on the motion of celestial bodies under the influence of gravitational forces. It combines concepts from physics, mathematics, and astronomy to understand how objects in space, such as planets, moons, asteroids, and stars, interact with each other and evolve over time. Key aspects of astronomical dynamical systems include: 1. **Orbital Mechanics**: This involves the study of the orbits of celestial bodies.

Ergodic theory is a branch of mathematics that studies the long-term average behavior of dynamical systems. It originated in the context of statistical mechanics and has applications in various fields, including probability theory, statistics, and even areas of physics and number theory. At its core, ergodic theory investigates how a system evolves over time and how its states are distributed in space.

Dynamical systems theorists are researchers and mathematicians who study dynamical systems, which are mathematical models that describe how a point in a given space evolves over time according to a set of defined rules. These systems can be linear or nonlinear and can be applied to various fields, including physics, biology, economics, engineering, and more.

In discrete mathematics, a theorem refers to a statement that has been proven to be true based on previously established statements such as axioms, definitions, and other theorems. Theorems are integral to the field as they form the backbone of mathematical reasoning and structure. ### Key Components of Theorems in Discrete Mathematics: 1. **Definitions**: Before proving a theorem, precise definitions of terms involved are necessary to ensure clarity and avoid ambiguity.

Discrete geometry is a branch of mathematics that studies geometric objects and properties in a discrete setting, as opposed to continuous geometry. It focuses on structures that are made up of distinct, separate elements rather than continuous shapes or surfaces. This can include the study of points, lines, polygons, polyhedra, and more complex shapes, particularly in finite or countable settings.