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
science, a nonlinear system is a
system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
in which the change of the output is not
proportional
Proportionality, proportion or proportional may refer to:
Mathematics
* Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant
* Ratio, of one quantity to another, especially of a part compare ...
to the change of the input. Nonlinear problems are of interest to
engineers,
biologist
A biologist is a scientist who conducts research in biology. Biologists are interested in studying life on Earth, whether it is an individual cell, a multicellular organism, or a community of interacting populations. They usually specialize in ...
s,
physicists,
mathematicians, and many other
scientists because most systems are inherently nonlinear in nature. Nonlinear
dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler
linear systems.
Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
s in which the unknowns (or the unknown functions in the case of
differential equations) appear as variables of a
polynomial of degree higher than one or in the argument of a
function which is not a polynomial of degree one.
In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a
linear combination of the unknown
variables or
functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.
As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (
linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as
solitons,
chaos, and
singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble
random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.
Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:
Definition
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 ...
, a
linear map (or ''linear function'')
is one which satisfies both of the following properties:
*Additivity or
superposition principle
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
:
*Homogeneity:
Additivity implies homogeneity for any
rational ''α'', and, for
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s, for any
real ''α''. For a
complex ''α'', homogeneity does not follow from additivity. For example, an
antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle
:
An equation written as
:
is called linear if
is a linear map (as defined above) and nonlinear otherwise. The equation is called ''homogeneous'' if
.
The definition
is very general in that
can be any sensible mathematical object (number, vector, function, etc.), and the function
can literally be any
mapping, including integration or differentiation with associated constraints (such as
boundary values
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 ...
). If
contains
differentiation with respect to
, the result will be a
differential equation.
Nonlinear algebraic equations
Nonlinear
algebraic equations, which are also called ''
polynomial equations'', are defined by equating
polynomials (of degree greater than one) to zero. For example,
:
For a single polynomial equation,
root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see
Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these
systems of polynomial equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations where the are polynomials in several variables, say , over some field .
A ''solution'' of a polynomial system is a set of values for the s ...
are now well understood and efficient methods exist for solving them.
Nonlinear recurrence relations
A nonlinear
recurrence relation defines successive terms of a
sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the
logistic map
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
and the relations that define the various
Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related
nonlinear system identification and analysis procedures.
[Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013] These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.
Nonlinear differential equations
A
system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
of
differential equations is said to be nonlinear if it is not a
system of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three ...
. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the
Navier–Stokes equations in fluid dynamics and the
Lotka–Volterra equations in biology.
One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of
linearly independent solutions can be used to construct general solutions through the
superposition principle
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
. A good example of this is one-dimensional heat transport with
Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.
Ordinary differential equations
First order
ordinary differential equations are often exactly solvable by
separation of variables, especially for autonomous equations. For example, the nonlinear equation
:
has
as a general solution (and also
as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as
:
and the left-hand side of the equation is not a linear function of
and its derivatives. Note that if the
term were replaced with
, the problem would be linear (the
exponential decay
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
problem).
Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield
closed-form solutions, though implicit solutions and solutions involving
nonelementary integral
In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an ''elementary function'' (i.e. a function constructed from a finite number of quotients of constan ...
s are encountered.
Common methods for the qualitative analysis of nonlinear ordinary differential equations include:
*Examination of any
conserved quantities
In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant (mathematics), constant along each trajectory of the system.
Not all systems have conserved quantities, and c ...
, especially in
Hamiltonian systems
*Examination of dissipative quantities (see
Lyapunov function) analogous to conserved quantities
*Linearization via
Taylor expansion
*Change of variables into something easier to study
*
Bifurcation theory
*
Perturbation methods (can be applied to algebraic equations too)
*Existence of solutions of Finite-Duration, which can happen under specific conditions for some non-linear ordinary differential equations.
Partial differential equations
The most common basic approach to studying nonlinear
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly linear). Sometimes, the equation may be transformed into one or more
ordinary differential equations, as seen in
separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.
Another common (though less mathematical) tactic, often exploited in fluid and heat mechanics, is to use
scale analysis to simplify a general, natural equation in a certain specific
boundary value problem
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 t ...
. For example, the (very) nonlinear
Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.
Other methods include examining the
characteristics and using the methods outlined above for ordinary differential equations.
Pendula

A classic, extensively studied nonlinear problem is the dynamics of a frictionless
pendulum under the influence of
gravity. Using
Lagrangian mechanics, it may be shown
David Tong: Lectures on Classical Dynamics
/ref> that the motion of a pendulum can be described by the dimensionless nonlinear equation
:
where gravity points "downwards" and is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use as an integrating factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calc ...
, which would eventually yield
:
which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral
In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an ''elementary function'' (i.e. a function constructed from a finite number of quotients of constan ...
(nonelementary unless ).
Another way to approach the problem is to linearize any nonlinearity (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at , called the small angle approximation, is
:
since for . This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at , corresponding to the pendulum being straight up:
:
since for . The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.
One more interesting linearization is possible around , around which :
:
This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.
Types of nonlinear dynamic behaviors
* Amplitude death – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system
* Chaos – values of a system cannot be predicted indefinitely far into the future, and fluctuations are aperiodic
* Multistability – the presence of two or more stable states
* Solitons – self-reinforcing solitary waves
*Limit cycles
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity ...
– asymptotic periodic orbits to which destabilized fixed points are attracted.
* Self-oscillations – feedback oscillations taking place in open dissipative physical systems.
Examples of nonlinear equations
* Algebraic Riccati equation
*Ball and beam
The ball and beam system consists of a long beam which can be tilted by a servo or electric motor together with a ball
rolling back and forth on top of the beam.
It is a popular textbook example in control theory.
The significance of the ball and ...
system
* Bellman equation for optimal policy
* Boltzmann equation
* Colebrook equation
* General relativity
* Ginzburg–Landau theory
* Ishimori equation
* Kadomtsev–Petviashvili equation
* Korteweg–de Vries equation
* Landau–Lifshitz–Gilbert equation
*Liénard equation In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a second order differential equation, named after the French physicist Alfred-Marie Liénard.
During the development of radio an ...
* Navier–Stokes equations of fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
* Nonlinear optics
* Nonlinear Schrödinger equation
* Power-flow study
* Richards equation for unsaturated water flow
* Self-balancing unicycle
*Sine-Gordon equation
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of surfa ...
* Van der Pol oscillator
* Vlasov equation
See also
*Aleksandr Mikhailovich Lyapunov
Aleksandr Mikhailovich Lyapunov (russian: Алекса́ндр Миха́йлович Ляпуно́в, ; – 3 November 1918) was a Russian mathematician, mechanician and physicist. His surname is variously romanized as Ljapunov, Liapunov, Liap ...
* Dynamical system
*Feedback
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...
*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). For ...
* Linear system
* Mode coupling
*Vector soliton In physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation. Ordinary solitons maintain their shape but have effectively only one (scalar) polarization ...
* Volterra series
References
Further reading
*
*
*
*
*
External links
Command and Control Research Program (CCRP)
* ttp://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaosa
MIT's OpenCourseWare
(in MATLAB) a Database of Physical Systems
The Center for Nonlinear Studies at Los Alamos National Laboratory
{{Authority control
Dynamical systems
Concepts in physics