# Want a book/notes that cover this syllabus (dynamical systems).

• Lavabug
In summary, the conversation is about finding a self-contained book for self-study in mathematical physics, specifically in the topics of Lagrangian and Hamiltonian mechanics, non-linearity, chaos, and fractal geometry. The listener mentions that they are not interested in an encyclopedic book and would prefer something tailored for a "mathematical physics" course. They mention that they have a background in differential geometry and topology. The speaker recommends a book by Jose and Saletan or Scheck as well as one by Arnol'd for a comprehensive coverage of the desired topics. They also mention that the book by Guckenheimer may also be a good option.
Lavabug
I am looking for a book that covers these topics at a self-contained level for self-study (ie: a book designed for a short course on the subject or lecture notes):
-Develop Lagrangian and Hamiltonian mechanics for single particles and for fields;

-Understand the role of non-linearity in discrete and continuous equations of motion,
particularly through the development of phase space portraits, local stability analysis and
bifurcation diagrams;

-Show how non-linear classical mechanics can give rise to chaotic motion, and to describe the character of chaos; develop ideas of scale-invariance and fractal geometry.

Objectives

For Continuous Dynamical Systems, students should be able to:
-Derive the Lagrangian and Hamiltonian using generalised coordinates and momenta for simple mechanical systems;
-Derive the equations of energy, momentum and angular momentum conservation from symmetries of the Hamiltonian;
-Derive and manipulate Hamiltonians and Lagrangians for classical field theories, including electromagnetism;
-Derive and give a physical interpretation of Liouville‟s theorem in n dimensions;

-Determine the local and global stability of the equilibrium of a linear system;
-Find the equilibria and determine their local stability for one- and two-dimensional nonlinear systems;
-Give a qualitative analysis of the global phase portrait for simple one- and two-
dimensional systems;

-Give examples of the saddle-node, transcritical, pitchfork and Hopf bifurcations;
-Determine the type of bifurcation in one-dimensional real and complex systems;

For Discrete Dynamical Systems, students should be able to:
-Find equilibria and cycles for simple systems, and determine their stability;
-Describe period-doubling bifurcations for a general discrete system;
-Calculate the Lyapunov exponent of a given trajectory and interpret the result for
attracting and repelling trajectories;

-Give a qualitative description of the origin of chaotic behaviour in discrete systems;
-Understand the concept and define various properties of fractals

Things in bold are of most interest to me. I notice there are a lot of pure math books on this subject but I'm looking for something more tailored for a "mathematical physics" course and less encyclopedic, that doesn't require much background in differential geometry or topology (just had a course in GR that teaches the basics). Anything like this out there?

Would you say it is self-contained? Ie: Won't require learning the necessary differential geometry elsewhere?

They have this at my library but it's currently borrowed. This one caught my eye for the time being:

https://www.amazon.com/gp/product/0738204536/?tag=pfamazon01-20

how does it compare to Guckenheimer's?

Last edited by a moderator:
Classical Dynamics: A Contemporary Approach by Jose and Saletan or Scheck's Mechanics: From Newton's Laws to Deterministic Chaos should cover most of your list. Arnol'd's Catastrophe Theory should cover the rest.

Thank you for your question. I highly recommend the book "Dynamical Systems: An Introduction" by K.T. Alligood, T.D. Sauer, and J.A. Yorke. This book covers all the topics you mentioned in a self-contained manner, making it suitable for self-study. It also includes many examples and exercises to help you understand the concepts better.

Another great book is "Nonlinear Dynamics and Chaos" by Steven H. Strogatz. This book is more focused on the physical applications of dynamical systems and does not require a strong background in mathematics. It also includes many real-world examples and applications, making it a great choice for a mathematical physics course.

Both of these books are highly recommended by experts in the field and have been used in courses on dynamical systems at various universities. I believe they will meet your requirements and provide you with a solid understanding of the subject. Happy studying!

## 1. What is a dynamical system?

A dynamical system is a mathematical model that describes the behavior of a set of variables over time. It can be used to predict how a system will evolve and change in response to various inputs or conditions.

## 2. What topics are typically covered in a dynamical systems syllabus?

A dynamical systems syllabus may cover topics such as differential equations, stability and bifurcations, chaos theory, nonlinear dynamics, and applications in various fields such as physics, biology, and economics.

## 3. Can you recommend a specific book or notes for studying dynamical systems?

There are many excellent books and notes available for studying dynamical systems, and the best one for you will depend on your level of knowledge and specific interests. Some popular choices include "Differential Equations, Dynamical Systems, and an Introduction to Chaos" by Hirsch, Smale, and Devaney, and "Nonlinear Dynamics and Chaos" by Strogatz.

## 4. How can I apply dynamical systems in my research or work?

Dynamical systems have a wide range of applications in various fields, including biology, physics, engineering, economics, and more. Some examples of how dynamical systems can be applied include modeling population growth, predicting the behavior of chaotic systems, and understanding the dynamics of chemical reactions.

## 5. Are there any online resources or lectures available for learning about dynamical systems?

Yes, there are many online resources and lectures available for learning about dynamical systems. Some universities offer free online courses on the subject, and there are also many video lectures and tutorials available on platforms like YouTube and Coursera.

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