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This article describes the use of the term nonlinearity in mathematics. For other meanings, see nonlinearity (disambiguation).
In
mathematics, a
nonlinear system is a system which is not
linear system i.e. a system which does not satisfy the superposition principle. Less technically, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear sum of independent components. A
homogenous system, which is linear apart from the presence of a function of the
independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system as long as a particular solution is known.
Generally, nonlinear problems are difficult (if possible) to solve and are much less understandable than linear problems. Even if not exactly solvable, the outcome of a linear problem is rather predictable, while the outcome of a nonlinear is inherently not.
Nonlinear problems are of interest to physicists and
mathematicians because most physical systems are inherently nonlinear in nature. Physical examples of linear systems are not very common. Nonlinear equations are difficult to solve and give rise to interesting phenomena such as
chaos theory. The weather is famously nonlinear, where simple changes in one part of the system produce complex effects throughout.
Definition
In mathematics, a
linear function (mathematics) (or map) f(x) is one which satisfies both of the following properties:
Additivity: f(x + y) = f(x) + f(y)\,
Homogeneity: f(\alpha x) = \alpha f(x)\,
An equation written as
f(x) = C\,
is called
linear if f(x) is
linear map (as defined above) and
nonlinear otherwise. Note that x does not need to be a scalar (can be a vector,
function (mathematics), etc), and that C must not depend on x. The equation is called
homogeneous if C = 0.
Nonlinear algebraic equations
Generally, nonlinear
algebraic problems are often exactly solvable, and if not they usually can be thoroughly understood through qualitative analysis and
numerical analysis analysis. As an example, the equation
x^2 + x - 1 = 0\,
may be written as
f(x) = C \quad \mbox{where} \quad f(x) = x^2 + x \quad \mbox{and} \quad C = 1\,
and is nonlinear because f(x) satisfies neither additivity nor homogeneity (the nonlinearity is due to the x^2). Though nonlinear, this simple example may be solved exactly (via the
quadratic formula) and is very well understood. On the other hand, the nonlinear equation
x^5 + x - 1 = 0\,
is not exactly solvable (see quintic equation), though it may be qualitatively analyzed and is well understood, for example through making a graph and examining the
root (mathematics)s of f(x) - C = 0.
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 and the relations that define the various
Hofstadter sequences.
Nonlinear differential equations
Problems involving nonlinear
differential equations are extremely diverse, and methods of solution or analysis are very problem dependent.
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. 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
\frac{d u}{d x} = -u^2\,
will easily yield u = (x + C)^{-1} as a general solution which happens to be simpler then the solution to the linear equation du/dx = -u. The equation is nonlinear because it may be written as
\frac{d u}{d x} + u^2=0\,
and the left-hand side of the equation is not a linear function of
u and its derivatives. Note that if the
u² term were replaced with
u, the problem would be linear (the
exponential decay 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 integrals are encountered.
Common methods for the qualitative analysis of nonlinear ordinary differential equations include:
- Examination of any conserved quantities, 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 theory methods (can be applied to algebraic equations too).
Partial differential equations
The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in the
similarity transform or
separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable or not.
Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use
Scale analysis (mathematics) to simplify a general, natural equation in a certain specific
boundary value problem. 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
Method of characteristics and using the methods outlined above for ordinary differential equations.
Example: pendulum
A classic, extensively studied nonlinear problem is the dynamics of a
Pendulum (mathematics). Using Lagrangian mechanics, it may be shown David Tong: Lectures on Classical Dynamics that the motion of a pendulum can be described by the dimensionless nonlinear equation
\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0\,
where gravity is "down" and \theta is as shown in the figure at right. One approach to "solving" this equation is to use \scriptstyle \frac{d \theta}{d t} as an integrating factor, which would eventually yield
\int \frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)--> = t + C_1\,
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 (nonelementary even if C_0 = 0).
Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at \theta = 0, called the small angle approximation, is
\frac{d^2 \theta}{d t^2} + \theta = 0\,
since \sin(\theta) \approx \theta for \theta \approx 0. This is a
simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at \theta = \pi, corresponding to the pendulum being straight up:
\frac{d^2 \theta}{d t^2} + \pi - \theta = 0\,
since \sin(\theta) \approx \pi - \theta for \theta \approx \pi. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that |\theta| 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 \theta = \pi/2, around which \sin(\theta) \approx 1:
\frac{d^2 \theta}{d t^2} + 1 = 0\,
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.
Metaphorical Use
Engineers often use the term
non-linear to refer to irrational behavior, with the implication that the person who has become non-linear is on the edge of losing control or even having a
nervous breakdown. Sometimes a person with
bipolar disorder is described as non-linear.
Types of Nonlinear Behaviors
- Indeterminism - The behavior of a system cannot be predicted.
- Multistability - Alternating between two or more exclusive states.
- Aperiodic Oscillations - Functions that do not repeat values after some period.
Examples of nonlinear equations
See also
Bibliography
| last = Kreyszig
| first = Erwin
| authorlink = Erwin Kreyszig
| year = 1998
| title = Advanced Engineering Mathematics
| publisher = Wiley
| id = ISBN 0-471-15496-2
-->
| last = Khalil
| first = Hassan K.
| year = 2001
| title = Nonlinear Systems
| publisher = Prentice Hall
| id = ISBN 0-13-067389-7
-->
| author= Diederich Hinrichsen and Anthony J. Pritchard| year= 2005| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness| publisher= Springer Verlag| id=ISBN 0-978-3-540-441250-->
| last = Sontag
| first = Eduardo
| authorlink = Eduardo D. Sontag
| year = 1998
| title = Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition
| publisher = Springer
| id = ISBN 0-387-984895
-->
External links
- A collection of non-linear models and demo applets (in Monash University's Virtual Lab)
- Command and Control Research Program (CCRP)
- New England Complex Systems Institute: Concepts in Complex Systems
- Nonlinear Dynamics I: Chaos at MIT's OpenCourseWare
- Nonlinear Models Nonlinear Model Database of Physical Systems (MATLAB)
- The Center for Nonlinear Studies at Los Alamos National Laboratory
This article describes the use of the term nonlinearity in mathematics. For other meanings, see nonlinearity (disambiguation).
In
mathematics, a
nonlinear system is a system which is not
linear system i.e. a system which does not satisfy the
superposition principle. Less technically, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear sum of independent components. A
homogenous system, which is linear apart from the presence of a function of the
independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system as long as a particular solution is known.
Generally, nonlinear problems are difficult (if possible) to solve and are much less understandable than linear problems. Even if not exactly solvable, the outcome of a linear problem is rather predictable, while the outcome of a nonlinear is inherently not.
Nonlinear problems are of interest to
physicists and mathematicians because most physical systems are inherently nonlinear in nature. Physical examples of linear systems are not very common. Nonlinear equations are difficult to solve and give rise to interesting phenomena such as chaos theory. The weather is famously nonlinear, where simple changes in one part of the system produce complex effects throughout.
Definition
In mathematics, a
linear function (mathematics) (or map) f(x) is one which satisfies both of the following properties:
Additivity: f(x + y) = f(x) + f(y)\,
Homogeneity: f(\alpha x) = \alpha f(x)\,
An equation written as
f(x) = C\,
is called
linear if f(x) is linear map (as defined above) and
nonlinear otherwise. Note that x does not need to be a scalar (can be a vector,
function (mathematics), etc), and that C must not depend on x. The equation is called
homogeneous if C = 0.
Nonlinear algebraic equations
Generally, nonlinear
algebraic problems are often exactly solvable, and if not they usually can be thoroughly understood through qualitative analysis and numerical analysis analysis. As an example, the equation
x^2 + x - 1 = 0\,
may be written as
f(x) = C \quad \mbox{where} \quad f(x) = x^2 + x \quad \mbox{and} \quad C = 1\,
and is nonlinear because f(x) satisfies neither additivity nor homogeneity (the nonlinearity is due to the x^2). Though nonlinear, this simple example may be solved exactly (via the quadratic formula) and is very well understood. On the other hand, the nonlinear equation
x^5 + x - 1 = 0\,
is not exactly solvable (see
quintic equation), though it may be qualitatively analyzed and is well understood, for example through making a graph and examining the root (mathematics)s of f(x) - C = 0.
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 and the relations that define the various Hofstadter sequences.
Nonlinear differential equations
Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are very problem dependent.
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. 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
\frac{d u}{d x} = -u^2\,
will easily yield u = (x + C)^{-1} as a general solution which happens to be simpler then the solution to the linear equation du/dx = -u. The equation is nonlinear because it may be written as
\frac{d u}{d x} + u^2=0\,
and the left-hand side of the equation is not a linear function of
u and its derivatives. Note that if the
u² term were replaced with
u, the problem would be linear (the exponential decay 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 integrals are encountered.
Common methods for the qualitative analysis of nonlinear ordinary differential equations include:
- Examination of any conserved quantities, 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 theory methods (can be applied to algebraic equations too).
Partial differential equations
The most common basic approach to studying nonlinear
partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in the similarity transform or separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable or not.
Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use
Scale analysis (mathematics) to simplify a general, natural equation in a certain specific boundary value problem. 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 Method of characteristics and using the methods outlined above for ordinary differential equations.
Example: pendulum
A classic, extensively studied nonlinear problem is the dynamics of a
Pendulum (mathematics). Using
Lagrangian mechanics, it may be shown David Tong: Lectures on Classical Dynamics that the motion of a pendulum can be described by the
dimensionless nonlinear equation
\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0\,
where gravity is "down" and \theta is as shown in the figure at right. One approach to "solving" this equation is to use \scriptstyle \frac{d \theta}{d t} as an integrating factor, which would eventually yield
\int \frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)--> = t + C_1\,
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 (nonelementary even if C_0 = 0).
Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at \theta = 0, called the small angle approximation, is
\frac{d^2 \theta}{d t^2} + \theta = 0\,
since \sin(\theta) \approx \theta for \theta \approx 0. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at \theta = \pi, corresponding to the pendulum being straight up:
\frac{d^2 \theta}{d t^2} + \pi - \theta = 0\,
since \sin(\theta) \approx \pi - \theta for \theta \approx \pi. The solution to this problem involves
hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that |\theta| 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 \theta = \pi/2, around which \sin(\theta) \approx 1:
\frac{d^2 \theta}{d t^2} + 1 = 0\,
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.
Metaphorical Use
Engineers often use the term
non-linear to refer to irrational behavior, with the implication that the person who has become non-linear is on the edge of losing control or even having a nervous breakdown. Sometimes a person with bipolar disorder is described as non-linear.
Types of Nonlinear Behaviors
- Indeterminism - The behavior of a system cannot be predicted.
- Multistability - Alternating between two or more exclusive states.
- Aperiodic Oscillations - Functions that do not repeat values after some period.
Examples of nonlinear equations
See also
Bibliography
| last = Kreyszig
| first = Erwin
| authorlink = Erwin Kreyszig
| year = 1998
| title = Advanced Engineering Mathematics
| publisher = Wiley
| id = ISBN 0-471-15496-2
-->
| last = Khalil
| first = Hassan K.
| year = 2001
| title = Nonlinear Systems
| publisher = Prentice Hall
| id = ISBN 0-13-067389-7
-->
| author=
Diederich Hinrichsen and Anthony J. Pritchard| year= 2005| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness| publisher= Springer Verlag| id=ISBN 0-978-3-540-441250-->
| last = Sontag
| first = Eduardo
| authorlink = Eduardo D. Sontag
| year = 1998
| title = Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition
| publisher = Springer
| id = ISBN 0-387-984895
-->
External links
- A collection of non-linear models and demo applets (in Monash University's Virtual Lab)
- Command and Control Research Program (CCRP)
- New England Complex Systems Institute: Concepts in Complex Systems
- Nonlinear Dynamics I: Chaos at MIT's OpenCourseWare
- Nonlinear Models Nonlinear Model Database of Physical Systems (MATLAB)
- The Center for Nonlinear Studies at Los Alamos National Laboratory
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