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In particular, a C k-atlas that is C 0-compatible with a C 0-atlas that defines a topological manifold is said to determine a C k differential structure on the topological manifold. The C k equivalence classes of such atlases are the distinct C k differential structures of the manifold. Each distinct differential structure is determined by a ...
A real dynamical system, real-time dynamical system, continuous time dynamical system, or flow is a tuple (T, M, Φ) with T an open interval in the real numbers R, M a manifold locally diffeomorphic to a Banach space, and Φ a continuous function. If Φ is continuously differentiable we say the system is a differentiable dynamical system.
A topological manifold that is in the image of is said to "admit a differentiable structure", and the fiber over a given topological manifold is "the different differentiable structures on the given topological manifold". Thus given two categories, the two natural questions are:
The internal state variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time. [13] The minimum number of state variables required to represent a given system, , is usually equal to the order of the system's defining differential equation, but not necessarily.
Most differentiable programming frameworks work by constructing a graph containing the control flow and data structures in the program. [7] Attempts generally fall into two groups: Static, compiled graph-based approaches such as TensorFlow, [note 1] Theano, and MXNet.
Assume the system in question has the rotation group SO(3) as a symmetry, meaning that the Hamiltonian operator ^ commutes with the action of SO(3) on the wave function . (One important example of such a system is the hydrogen atom , which has a spherically symmetric potential.)
In fact, the level surfaces for this system are all planes in R 3 of the form x − y + z = C, for C a constant. The second observation is that, once the level surfaces are known, all solutions can then be given in terms of an arbitrary function. Since the value of a solution f on a level surface is constant by definition, define a function C(t ...
If the system is dissipative, the sum of Lyapunov exponents is negative. If the system is a flow and the trajectory does not converge to a single point, one exponent is always zero—the Lyapunov exponent corresponding to the eigenvalue of with an eigenvector in the direction of the flow.