In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in the study of
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s, an orbit is a collection of points related by the
evolution function of the dynamical system. It can be understood as the subset of
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
covered by the trajectory of the dynamical system under a particular set of
initial condition
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). Fo ...
s, as the system evolves. As a phase space trajectory is uniquely determined for any given set of phase space coordinates, it is not possible for different orbits to intersect in phase space, therefore the set of all orbits of a dynamical system is a
partition of the phase space. Understanding the properties of orbits by using
topological methods is one of the objectives of the modern theory of dynamical systems.
For
discrete-time dynamical systems, the orbits are
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
s; for
real dynamical systems, the orbits are
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s; and for
holomorphic dynamical systems, the orbits are
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s.
Definition
Given a dynamical system (''T'', ''M'', Φ) with ''T'' a
group, ''M'' a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
and Φ the evolution function
:
where
with
we define
:
then the set
:
is called orbit through ''x''. An orbit which consists of a single point is called constant orbit. A non-constant orbit is called closed or periodic if there exists a
in
such that
:
.
Real dynamical system
Given a real dynamical system (''R'', ''M'', Φ), ''I''(''x'') is an open interval in the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, that is
. For any ''x'' in ''M''
:
is called positive semi-orbit through ''x'' and
:
is called negative semi-orbit through ''x''.
Discrete time dynamical system
For discrete time dynamical system :
forward orbit of x is a set :
:
backward orbit of x is a set :
:
and orbit of x is a set :
:
where :
*
is an evolution function
which is here an
iterated function,
* set
is dynamical space,
*
is number of iteration, which is
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
and
*
is initial state of system and
Usually different notation is used :
*
is written as
*
where
is
in the above notation.
General dynamical system
For a general dynamical system, especially in homogeneous dynamics, when one has a "nice" group
acting on a probability space
in a measure-preserving way, an orbit
will be called periodic (or equivalently, closed) if the stabilizer
is a lattice inside
.
In addition, a related term is a bounded orbit, when the set
is pre-compact inside
.
The classification of orbits can lead to interesting questions with relations to other mathematical areas, for example the Oppenheim conjecture (proved by Margulis) and the Littlewood conjecture (partially proved by Lindenstrauss) are dealing with the question whether every bounded orbit of some natural action on the homogeneous space
is indeed periodic one, this observation is due to Raghunathan and in different language due to Cassels and Swinnerton-Dyer . Such questions are intimately related to deep measure-classification theorems.
Notes
It is often the case that the evolution function can be understood to compose the elements of a
group, in which case the
group-theoretic orbits of the
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
are the same thing as the dynamical orbits.
Examples
Critical orbit 3d.png, Critical orbit of discrete dynamical system based on complex quadratic polynomial. It tends to weakly attracting fixed point with multiplier=0.99993612384259
Julia set p(z)= z^3+(1.0149042485835864102+0.10183008497976470119i)*z; (zoom).png, critical orbit tends to weakly attracting point. One can see spiral from attracting fixed point to repelling fixed point ( z= 0) which is a place with high density of level curves.
* The orbit of an
equilibrium point
In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation.
Formal definition
The point \tilde\in \mathbb^n is an equilibrium point for the differential equation
:\frac = \ ...
is a constant orbit.
Stability of orbits
A basic classification of orbits is
* constant orbits or fixed points
* periodic orbits
* non-constant and non-periodic orbits
An orbit can fail to be closed in two ways.
It could be an asymptotically periodic orbit if it
converges to a periodic orbit. Such orbits are not closed because they never truly repeat, but they become arbitrarily close to a repeating orbit.
An orbit can also be
chaotic
Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
. These orbits come arbitrarily close to the initial point, but fail to ever converge to a periodic orbit. They exhibit
sensitive dependence on initial conditions, meaning that small differences in the initial value will cause large differences in future points of the orbit.
There are other properties of orbits that allow for different classifications. An orbit can be
hyperbolic if nearby points approach or diverge from the orbit exponentially fast.
See also
*
Wandering set
*
Phase space method
*
Cobweb plot or Verhulst diagram
*
Periodic points of complex quadratic mappings and multiplier of orbit
*
Orbit portrait
In mathematics, an orbit portrait is a combinatorial tool used in complex dynamics for understanding the behavior of one-complex dimensional quadratic maps.
In simple words one can say that it is :
* a list of external angles for which rays lan ...
References
*
*
* {{cite book , last=Perko , first=Lawrence , chapter=Periodic Orbits, Limit Cycles and Separatrix Cycles , title=Differential Equations and Dynamical Systems , location=New York , publisher=Springer , edition=Third , year=2001 , pages=202–211 , isbn=0-387-95116-4 , chapter-url=https://books.google.com/books?id=VFnSBwAAQBAJ&pg=PA202
Dynamical systems
Group actions (mathematics)