Logo
Pattern

Discover published sets by community

Explore tens of thousands of sets crafted by our community.

Phase Space Analysis

18

Flashcards

0/18

Still learning
StarStarStarStar

Hamilton's Equations

StarStarStarStar

Hamilton's equations describe the evolution of a physical system in Hamiltonian mechanics. It consists of two sets of equations:

q˙=Hp,\dot{q} = \frac{\partial H}{\partial p},
p˙=Hq\dot{p} = -\frac{\partial H}{\partial q}
, where qq are generalized coordinates, pp are conjugate momenta, and HH is the Hamiltonian.

StarStarStarStar

Trajectory

StarStarStarStar

In phase space, a trajectory is a path that represents the state of the system evolving over time. It is determined by the initial conditions and the system's equations of motion.

StarStarStarStar

Attractor

StarStarStarStar

An attractor is a set of numerical values toward which a system tends to evolve. It can be a point, curve, manifold, or more complicated structure.

StarStarStarStar

Homoclinic Orbit

StarStarStarStar

A homoclinic orbit is a trajectory in phase space that joins a saddle point to itself. It is a connection between the stable and unstable manifolds of the same point.

StarStarStarStar

Pendulum Phase Space

StarStarStarStar

For a simple pendulum, phase space is represented by angular displacement and angular momentum. The phase space plots show the motion as trajectories which, for small oscillations, are closed orbits around a fixed point.

StarStarStarStar

Bifurcation

StarStarStarStar

Bifurcation is a qualitative change in the structure of a system's phase space, often as a parameter is varied. It marks the appearance of a new system behavior.

StarStarStarStar

Phase Space

StarStarStarStar

A phase space is a mathematical space with coordinates representing all possible states of a system. Each point in this space corresponds to one possible state of the system.

StarStarStarStar

Strange Attractor

StarStarStarStar

A strange attractor is an attractor that has a fractal structure. This is typical in chaotic systems, where it exhibits sensitive dependence on initial conditions.

StarStarStarStar

Center Manifold

StarStarStarStar

A center manifold is an invariant manifold in a dynamical system containing trajectories that neither exponentially grow nor decay. They often separate regions of different behavior.

StarStarStarStar

Lyapunov Function

StarStarStarStar

A Lyapunov function is a scalar function that decreases along system trajectories and is used to assess the stability of equilibrium points in phase space.

StarStarStarStar

Ergodicity

StarStarStarStar

Ergodicity describes a system where, in the long term, every state is equally probable. Any one trajectory will visit every possible state given sufficient time.

StarStarStarStar

Fixed Point

StarStarStarStar

A fixed point in a dynamical system is a point in phase space where the system does not change in time. For a continuous system, it is where the derivatives are zero.

StarStarStarStar

Heteroclinic Orbit

StarStarStarStar

A heteroclinic orbit is a trajectory in phase space that connects two different saddle points. It represents a path from the unstable manifold of one point to the stable manifold of another.

StarStarStarStar

Poincaré Map

StarStarStarStar

A Poincaré map is a specific type of recurrence plot used in the study of dynamical systems to analyze the behavior of orbits by looking at their intersections with a lower-dimensional subspace.

StarStarStarStar

Limit Cycle

StarStarStarStar

A limit cycle is a closed trajectory in phase space that represents a periodic orbit. Systems that exhibit this behavior are oscillatory.

StarStarStarStar

Stability

StarStarStarStar

Stability in dynamical systems refers to whether a system returns to a fixed point after a small perturbation. Stable if it returns, unstable if it diverges.

StarStarStarStar

Hamiltonian System

StarStarStarStar

A Hamiltonian system is a dynamical system governed by Hamilton's equations. The Hamiltonian function represents the total energy of the system.

StarStarStarStar

Floquet Theory

StarStarStarStar

Floquet theory analyzes the stability of solutions to differential equations with periodic coefficients. It extends the concept of eigenvectors and eigenvalues to periodic systems.

Know
0
Still learning
Click to flip
Know
0
Logo

© Hypatia.Tech. 2024 All rights reserved.