Modern Introduction to Dynamical Systems

This text is a comprehensive introduction to the modern theory of dynamical systems, covering abstract and practical concepts, with prerequisite knowledge in calculus, linear algebra, and basic differential equations. It provides the foundation for further self-study and exploration of current research areas in dynamical systems.

Format: Paperback / softback
Length: 432 pages
Publication date: 28 June 2018
Publisher: Oxford University Press

This text serves as a comprehensive introduction to the modern theory of dynamical systems, a rigorous analysis-based course textbook that delves into the fundamental tools, techniques, theory, and development of mathematical modeling, encompassing both abstract and practical concepts. It employs both discrete and continuous ideas and examples, covering what is commonly referred to as the modern theory of dynamics. Prior knowledge is limited to calculus, linear algebra, and basic differential equations, with all higher-level analysis, geometry, and algebra being introduced as necessary throughout the text. Completing this text from start to finish will equip the diligent reader with the necessary tools, vocabulary, and conceptual foundation to pursue further self-study and embark on exploring current areas of active research in dynamical systems.

Dynamical systems are mathematical models that describe the behavior of physical systems over time. They are used in a wide range of fields, including physics, engineering, biology, and social sciences, to understand and predict complex phenomena.

The modern theory of dynamical systems is a branch of mathematics that emerged in the late 19th and early 20th centuries. It is rooted in the work of mathematicians such as Leonhard Euler, Isaac Newton, and Henri Poincaré, who studied the motion of objects under the influence of forces.

Dynamical systems can be described using a variety of mathematical tools, including differential equations, linear algebra, and complex analysis. These tools allow us to analyze the behavior of systems and predict their future behavior based on their initial conditions and the forces acting on them.

One of the key concepts in dynamical systems is the notion of stability. A system is stable if it remains in a particular state or region of space for all time, regardless of the forces acting on it. A system is unstable if it is prone to sudden changes in its behavior, such as oscillations or chaotic behavior.

Dynamical systems can be classified into different types, depending on their behavior. Some examples include:

Ordinary differential equations (ODEs): ODEs describe the motion of a system of ordinary differential equations, which are equations that involve the derivatives of a function with respect to time. ODEs are used to model a wide range of physical systems, including fluids, elasticity, and electromagnetism.

Nonlinear differential equations (NDEs): NDEs describe the motion of a system of differential equations that are not linear. NDEs are used to model complex systems, such as the behavior of planets, stars, and galaxies.

Stochastic differential equations (SDEs): SDEs describe the motion of a system of differential equations that are influenced by random variables. SDEs are used to model financial markets, biological systems, and other systems that are subject to random fluctuations.

Dynamical systems can also be used to study the behavior of complex systems, such as social networks, ecosystems, and the brain. These systems are often characterized by feedback loops, where the behavior of one part of the system affects the behavior of other parts.

The study of dynamical systems has led to many important discoveries and advancements in various fields. For example, the study of differential equations has led to the development of calculus, which is a fundamental tool in mathematics and physics. The study of chaos theory has led to the development of new technologies, such as chaos control and chaos synchronization.

Dynamical systems are also important in the field of computer science, where they are used to model complex systems, such as computer networks, artificial intelligence, and social networks. These systems are often characterized by feedback loops, where the behavior of one part of the system affects the behavior of other parts.

In conclusion, dynamical systems are a fundamental tool in the study of physical systems and the behavior of complex systems. The modern theory of dynamical systems has led to many important discoveries and advancements in various fields, and it continues to be an active area of research. By understanding the principles of dynamical systems, we can better understand and predict the behavior of the world around us.

Weight: 690g
Dimension: 233 x 155 x 21 (mm)
ISBN-13: 9780198743279