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| '''Linear dynamical systems''' are [[dynamical systems]] whose evaluation functions are [[linear]]. While dynamical systems in general do not have [[closed-form expression|closed-form solutions]], linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the equilibrium points of the system and approximating it as a linear system around each such point.
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| ==Introduction==
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| In a linear dynamical system, the variation of a state vector
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| (an <math>N</math>-dimensional [[vector space|vector]] denoted <math>\mathbf{x}</math>) equals a constant matrix
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| (denoted <math>\mathbf{A}</math>) multiplied by
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| <math>\mathbf{x}</math>. This variation can take two forms: either
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| as a [[flow (mathematics)|flow]], in which <math>\mathbf{x}</math> varies
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| continuously with time
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| :<math>
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| \frac{d}{dt} \mathbf{x}(t) = \mathbf{A} \cdot \mathbf{x}(t)
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| </math>
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| or as a mapping, in which
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| <math>\mathbf{x}</math> varies in [[discrete time|discrete]] steps
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| :<math> | |
| \mathbf{x}_{m+1} = \mathbf{A} \cdot \mathbf{x}_{m}
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| </math>
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| These equations are linear in the following sense: if
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| <math>\mathbf{x}(t)</math> and <math>\mathbf{y}(t)</math>
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| are two valid solutions, then so is any [[linear combination]]
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| of the two solutions, e.g.,
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| <math>\mathbf{z}(t) \ \stackrel{\mathrm{def}}{=}\ \alpha \mathbf{x}(t) + \beta \mathbf{y}(t)</math>
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| where <math>\alpha</math> and <math>\beta</math>
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| are any two [[scalar (mathematics)|scalars]]. The matrix <math>\mathbf{A}</math>
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| need not be [[Symmetry in mathematics#Symmetry in linear algebra|symmetric]].
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| Linear dynamical systems can be solved exactly, in contrast to most nonlinear ones. Occasionally, a nonlinear system can be solved exactly by a change of variables to a linear system. Moreover, the solutions of (almost) any nonlinear system can be well-approximated by an equivalent linear system near its [[fixed point (mathematics)|fixed points]]. Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems.
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| ==Solution of linear dynamical systems==
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| If the initial vector <math>\mathbf{x}_{0} \ \stackrel{\mathrm{def}}{=}\ \mathbf{x}(t=0)</math>
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| is aligned with a [[right eigenvector]] <math>\mathbf{r}_{k}</math> of | |
| the [[matrix (mathematics)|matrix]] <math>\mathbf{A}</math>, the dynamics are simple
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| :<math> | |
| \frac{d}{dt} \mathbf{x}(t) =
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| \mathbf{A} \cdot \mathbf{r}_{k} = \lambda_{k} \mathbf{r}_{k}
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| </math>
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| where <math>\lambda_{k}</math> is the corresponding [[eigenvalue]];
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| the solution of this equation is
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| :<math>
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| \mathbf{x}(t) =
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| \mathbf{r}_{k} e^{\lambda_{k} t}
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| </math>
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| as may be confirmed by substitution.
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| If <math>\mathbf{A}</math> is [[diagonalizable matrix|diagonalizable]], then any vector in an <math>N</math>-dimensional space can be represented by a linear combination of the right and [[left eigenvector]]s (denoted <math>\mathbf{l}_{k}</math>) of the matrix <math>\mathbf{A}</math>.
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| :<math>
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| \mathbf{x}_{0} =
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| \sum_{k=1}^{N}
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| \left( \mathbf{l}_{k} \cdot \mathbf{x}_{0} \right)
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| \mathbf{r}_{k}
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| </math>
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| Therefore, the general solution for <math>\mathbf{x}(t)</math> is
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| a linear combination of the individual solutions for the right
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| eigenvectors
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| :<math>
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| \mathbf{x}(t) =
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| \sum_{k=1}^{n}
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| \left( \mathbf{l}_{k} \cdot \mathbf{x}_{0} \right)
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| \mathbf{r}_{k} e^{\lambda_{k} t}
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| </math>
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| Similar considerations apply to the discrete mappings.
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| == Classification in two dimensions ==
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| [[Image:LinDynSysTraceDet.jpg|right|thumb|400px|Classification of 2D fixed point according to the trace and the determinant of the Jacobian matrix.]]
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| The roots of the [[characteristic polynomial]] det('''A''' - λ'''I''') are the eigenvalues of '''A'''. The sign and relation of these roots, <math>\lambda_n</math>, to each other may be used to determine the stability of the dynamical system
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| :<math> | |
| \frac{d}{dt} \mathbf{x}(t) = \mathbf{A} \mathbf{x}(t).
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| </math>
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| For a 2-dimensional system, the characteristic polynomial is of the form <math>\lambda^2-\tau\lambda+\Delta=0</math> where <math>\tau</math> is the [[trace (linear algebra)|trace]] and <math>\Delta</math> is the [[determinant]] of '''A'''. Thus the two roots are in the form:
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| :<math>\lambda_1=\frac{\tau+\sqrt{\tau^2-4\Delta}}{2}</math>
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| :<math>\lambda_2=\frac{\tau-\sqrt{\tau^2-4\Delta}}{2}</math>
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| Note also that <math>\Delta=\lambda_1\lambda_2</math> and <math>\tau=\lambda_1+\lambda_2</math>. Thus if <math>\Delta<0</math> then the eigenvalues are of opposite sign, and the fixed point is a saddle. If <math>\Delta>0</math> then the eigenvalues are of the same sign. Therefore if <math>\tau>0</math> both are positive and the point is unstable, and if <math>\tau<0</math> then both are negative and the point is stable. The [[discriminant]] will tell you if the point is nodal or spiral (i.e. if the eigenvalues are real or complex).
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| <!--
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| ==Linear systems and higher-order differential equations==
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| ==Linearizing nonlinear systems==
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| -->
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| ==See also==
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| * [[Linear system]]
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| * [[Dynamic systems]]
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| * [[List of dynamical system topics]]
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| <!--
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| ==External links==
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| -->
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| [[Category:Dynamical systems]]
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