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Elements of Differentiable Dynamics and Bifurcation Theory
978-0126017106 0126017107
187 pages, Hardcover
First published January 1, 1989
7 people want to read
About the author
David Ruelle
19 books15 followersDavid Pierre Ruelle is a Belgian mathematical physicist, naturalized French. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term strange attractor, and developed a new theory of turbulence.
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Displaying 1 of 1 review
December 10, 2020
Inspired by a NeurIPs 2020 paper on non-stochastic control systems from Princeton University, I am intrigued by this topic of "differentiable dynamics."
Differentiable dynamics has the following implications/characteristics:
1) Example formulation:
x_{t+1} = A x_t + B u_t + w_t, where u_t is the action, and w_t is noise,
together with a reward/regret function to evaluate x_{t+1} versus x*_{t+1}.
2) It is prevalent in control systems, reinforcement learning, Markov chain, etc.
3) It belongs more to mathematical research, than to physics or applied science.
Basic concepts:
1) Manifold: the environment that provides feedbacks on how x changes after the action u, e.g., Banach spaces.
2) Differential dynamics: the time evolution of a natural system by a differential equation:
dx / dt = X(x), where X is an operator.
3) Common assumptions: e.g., the manifold is differentiable.
4) Fixed points & Attracting sets.
Content of the book:
Part 1) Concepts and notations, incl. manifolds, fixed points, periodic orbits, etc.
Part 2) Bifurcations, incl., attracting sets
Part 3) A collection of appendices.
More readings:
- A related NeurIPS workshop: https://neurips.cc/virtual/2020/prote...
- An example NeurIPS paper on non-stochastic control: https://arxiv.org/pdf/2008.05523.pdf
Differentiable dynamics has the following implications/characteristics:
1) Example formulation:
x_{t+1} = A x_t + B u_t + w_t, where u_t is the action, and w_t is noise,
together with a reward/regret function to evaluate x_{t+1} versus x*_{t+1}.
2) It is prevalent in control systems, reinforcement learning, Markov chain, etc.
3) It belongs more to mathematical research, than to physics or applied science.
Basic concepts:
1) Manifold: the environment that provides feedbacks on how x changes after the action u, e.g., Banach spaces.
2) Differential dynamics: the time evolution of a natural system by a differential equation:
dx / dt = X(x), where X is an operator.
3) Common assumptions: e.g., the manifold is differentiable.
4) Fixed points & Attracting sets.
Content of the book:
Part 1) Concepts and notations, incl. manifolds, fixed points, periodic orbits, etc.
Part 2) Bifurcations, incl., attracting sets
Part 3) A collection of appendices.
More readings:
- A related NeurIPS workshop: https://neurips.cc/virtual/2020/prote...
- An example NeurIPS paper on non-stochastic control: https://arxiv.org/pdf/2008.05523.pdf
Displaying 1 of 1 review