Orbit (Department of Mechanics)


August 8, 2022

A trajectory in a dynamical system is a set of states that pass through certain initial conditions and are determined according to the rules of time evolution of the system. Geometrically, the trajectory is a sequence of points in phase space in discrete dynamical systems, and a curve in phase space in continuous dynamical systems. More generally, it is synonymous with an orbit determined by group action. Investigating the properties of orbitals is one of the main interests in the field of dynamical systems.


Discrete Dynamical Systems

Let the independent variable be t and the m dependent variables be (x1, x2, …, xm)⊤ X. In dynamical systems, the independent variable t is called time. We call the space of X the phase space, and let it be the m-dimensional real number space Rm ∋ X. If we treat time as discrete, then t ∈ Z, we call it a discrete dynamical system. In discrete dynamical systems, time is expressed as n. Given a map f(X) defining a discrete dynamical system, by repeatedly applying f to a point X ∈ Rm, X , f ( X ) , f 2 ( X ) , f 3 ( X ) , … , f n ( X ) , … {\displaystyle X,\ f(X),\ f^{2}(X),\ f^{3}(X),\dotsc ,\ f^{n}(X),\dotsc } A sequence of numbers is obtained. This set is called the trajectory of the discrete dynamical system. where fn means apply f n times, and if m 3, then f3 f(f(f(X))). Geometrically, the trajectory of a discrete dynamical system is drawn as a sequence of points in phase space. The first given point X is called the initial condition or initial value, and is denoted by X0. We denote the orbit starting at X0 by O(X0). As above, the orbits are obtained by repeatedly applying the map f to X0. Precisely this is a non-negative integer time, called the forward orbit or positive half-orbit, O + ( X 0 ) { X ∈ R. m ∣ X f n ( X 0 ) , n 0 , 1 , 2 , … } {\displaystyle O_{+}(X_{0})\{X\in \mathbb {R} ^{m}\mid Xf^{n}(X_{0}),\ n0,1 ,2,\dotsc\}} is given by If f is a homeomorphism and the inverse map f−1 can be defined, then the time-reversed backward orbit or the negative half-orbit O − ( X 0 ) { X ∈ R. m ∣ X f n ( X 0 ) , n 0 , − 1 , − 2 , … } {\displaystyle O_{-}(X_{0})\{X\in \mathbb {R} ^{m}\mid Xf^{n}(X_{0}),\ n0,- 1,-2,\dotsc\}} is given. Furthermore, the set of forward and backward trajectories O ( X