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 ...
and particularly in
dynamic systems
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 in a p ...
, an initial condition, in some contexts called a seed value,
is a value of an evolving
variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
at some point in time designated as the initial time (typically denoted ''t'' = 0). For a system of
order ''k'' (the number of time lags in
discrete time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
, or the order of the largest derivative in
continuous time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
) and
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
''n'' (that is, with ''n'' different evolving variables, which together can be denoted by an ''n''-dimensional
coordinate vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensiona ...
), generally ''nk'' initial conditions are needed in order to trace the system's variables forward through time.
In both
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s in continuous time and
difference equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s in discrete time, initial conditions affect the value of the dynamic variables (
state variable
A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of a ...
s) at any future time. In continuous time, the problem of finding a
closed form solution
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th root ...
for the state variables as a function of time and of the initial conditions is called the
initial value problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or oth ...
. A corresponding problem exists for discrete time situations. While a closed form solution is not always possible to obtain, future values of a discrete time system can be found by iterating forward one time period per iteration, though rounding error may make this impractical over long horizons.
Linear system
Discrete time
A linear
matrix difference equation
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. The order of the eq ...
of the homogeneous (having no constant term) form
has closed form solution
predicated on the vector
of initial conditions on the individual variables that are stacked into the vector;
is called the vector of initial conditions or simply the initial condition, and contains ''nk'' pieces of information, ''n'' being the dimension of the vector ''X'' and ''k'' = 1 being the number of time lags in the system. The initial conditions in this linear system do not affect the qualitative nature of the future behavior of the state variable ''X''; that behavior is
stable
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
or unstable based on the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of the matrix ''A'' but not based on the initial conditions.
Alternatively, a dynamic process in a single variable ''x'' having multiple time lags is
:
Here the dimension is ''n'' = 1 and the order is ''k'', so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, is ''nk'' = ''k''. Again the initial conditions do not affect the qualitative nature of the variable's long-term evolution. The solution of this equation is found by using its
characteristic equation to obtain the latter's ''k'' solutions, which are the
characteristic value
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s
for use in the solution equation
:
Here the constants
are found by solving a system of ''k'' different equations based on this equation, each using one of ''k'' different values of ''t'' for which the specific initial condition
Is known.
Continuous time
A differential equation system of the first order with ''n'' variables stacked in a vector ''X'' is
:
Its behavior through time can be traced with a closed form solution conditional on an initial condition vector
. The number of required initial pieces of information is the dimension ''n'' of the system times the order ''k'' = 1 of the system, or ''n''. The initial conditions do not affect the qualitative behavior (stable or unstable) of the system.
A single ''k''
th order linear equation in a single variable ''x'' is
:
Here the number of initial conditions necessary for obtaining a closed form solution is the dimension ''n'' = 1 times the order ''k'', or simply ''k''. In this case the ''k'' initial pieces of information will typically not be different values of the variable ''x'' at different points in time, but rather the values of ''x'' and its first ''k'' – 1 derivatives, all at some point in time such as time zero. The initial conditions do not affect the qualitative nature of the system's behavior. The
characteristic equation of this dynamic equation is
whose solutions are the
characteristic value
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s
these are used in the solution equation
:
This equation and its first ''k'' – 1 derivatives form a system of ''k'' equations that can be solved for the ''k'' parameters
given the known initial conditions on ''x'' and its ''k'' – 1 derivatives' values at some time ''t''.
Nonlinear systems
Nonlinear system
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
s can exhibit a substantially richer variety of behavior than linear systems can. In particular, the initial conditions can affect whether the system diverges to infinity or whether it
converges to one or another
attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
of the system. Each attractor, a (possibly disconnected) region of values that some dynamic paths approach but never leave, has a (possibly disconnected)
basin of attraction
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
such that state variables with initial conditions in that basin (and nowhere else) will evolve toward that attractor. Even nearby initial conditions could be in basins of attraction of different attractors (see for example
Newton's method#Basins of attraction).
Moreover, in those nonlinear systems showing
chaotic behavior
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have c ...
, the evolution of the variables exhibits
sensitive dependence on initial conditions: the iterated values of any two very nearby points on the same
strange attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
, while each remaining on the attractor, will diverge from each other over time. Thus even on a single attractor the precise values of the initial conditions make a substantial difference for the future positions of the iterates. This feature makes accurate
simulation
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...
of future values difficult, and impossible over long horizons, because stating the initial conditions with exact precision is seldom possible and because rounding error is inevitable after even only a few iterations from an exact initial condition.
Empirical laws and initial conditions
See also
*
Boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
*
Initialization vector
In cryptography, an initialization vector (IV) or starting variable (SV) is an input to a cryptographic primitive being used to provide the initial state. The IV is typically required to be random or pseudorandom, but sometimes an IV only needs to ...
, in cryptography
References
External links
* {{wikiquote-inline
Recurrence relations
Differential equations