Sinopsis
To construct a mathematical model of a physical system, one must decide on the realm in which the model lives. Since it would be impossible to describe everything in the universe, a model must include only a limited number of variables. The set of values that these variables can take makes up the phase space of the model. In this book we will study systems for which the phase space is finite dimensional—that is, the state of the model can be described by the values of finitely many variables. Typically, the state of the system will be denoted by x and the phase space byM; sometimesM will be the Euclidean space Rn, and x a vector in that space; however, it is also common for the phase space to be a manifold. The main point is that for a given model with a phase spaceM, the modeler asserts that the system can be completely described by the variables x ∈ M together with a set of constants that define the parameters of the model. For example a simple, planar pendulum has a fixed length and mass and is acted on by a constant gravitational field. The values of these constants describe the parameters of the system. The phase space M consists of possible values of the pendulum’s position, represented by an angle, and of its angular velocity. ThusM is the two-dimensional cylinder, and the dynamics corresponds to smooth motion on M.
Content
- Introduction
- Linear Systems
- Existence and Uniqueness
- Dynamical Systems
- Invariant Manifolds
- The Phase Plane
- Chaotic Dynamics
- Bifurcation Theory
- Hamiltonian Dynamics
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