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`!Linear dynamical systems`! are `F33f`_`[dynamical systems`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dynamical_systems]`_`f whose `F33f`_`[evolution functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Evolution_function]`_`f are `F33f`_`[linear`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear]`_`f. While dynamical systems, in general, do not have `F33f`_`[closed-form solutions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Closed-form_expression]`_`f, 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 `F33f`_`[equilibrium points`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Equilibrium_points]`_`f of the system and approximating it as a linear system around each such point.

>>Contents

• `F0af`_`[Introduction`#introduction]`_`f

• `F0af`_`[Solution of linear dynamical systems`#solution-of-linear-dynamical-systems]`_`f

• `F0af`_`[Classification in two dimensions`#classification-in-two-dimensions]`_`f

• `F0af`_`[See also`#see-also]`_`f

-─

>>Introduction

In a linear dynamical system, the variation of a state vector (an N {\\displaystyle N} -dimensional `F33f`_`[vector`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vector_space]`_`f denoted x {\\displaystyle \\mathbf {x} } ) equals a constant matrix (denoted A {\\displaystyle \\mathbf {A} } ) multiplied by x {\\displaystyle \\mathbf {x} } . This variation can take two forms: either as a `F33f`_`[flow`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Flow_(mathematics)]`_`f, in which x {\\displaystyle \\mathbf {x} } varies continuously with time

d d t x ( t ) = A x ( t ) {\\displaystyle {\\frac {d}{dt}}\\mathbf {x} (t)=\\mathbf {A} \\mathbf {x} (t)}

or as a mapping, in which x {\\displaystyle \\mathbf {x} } varies in `F33f`_`[discrete`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Discrete_time]`_`f steps

x m + 1 = A x m {\\displaystyle \\mathbf {x} _{m+1}=\\mathbf {A} \\mathbf {x} _{m}}

These equations are linear in the following sense: if x ( t ) {\\displaystyle \\mathbf {x} (t)} and y ( t ) {\\displaystyle \\mathbf {y} (t)} are two valid solutions, then so is any `F33f`_`[linear combination`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_combination]`_`f of the two solutions, e.g., z ( t ) = d e f α α x ( t ) + β β y ( t ) {\\displaystyle \\mathbf {z} (t)\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\alpha \\mathbf {x} (t)+\\beta \\mathbf {y} (t)} where α α {\\displaystyle \\alpha } and β β {\\displaystyle \\beta } are any two `F33f`_`[scalars`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Scalar_(mathematics)]`_`f. The matrix A {\\displaystyle \\mathbf {A} } need not be `F33f`_`[symmetric`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Symmetry_in_mathematics]`_`f.

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 `F33f`_`[fixed points`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fixed_point_(mathematics)]`_`f. Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems.

>>Solution of linear dynamical systems

If the initial vector x 0 = d e f x ( t = 0 ) {\\displaystyle \\mathbf {x} _{0}\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\mathbf {x} (t=0)} is aligned with a `F33f`_`[right eigenvector`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Right_eigenvector]`_`f r k {\\displaystyle \\mathbf {r} _{k}} of the `F33f`_`[matrix`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Matrix_(mathematics)]`_`f A {\\displaystyle \\mathbf {A} } , the dynamics are simple

d d t x ( t ) = A r k = λ λ k r k {\\displaystyle {\\frac {d}{dt}}\\mathbf {x} (t)=\\mathbf {A} \\mathbf {r} _{k}=\\lambda _{k}\\mathbf {r} _{k}}

where λ λ k {\\displaystyle \\lambda _{k}} is the corresponding `F33f`_`[eigenvalue`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Eigenvalue]`_`f; the solution of this equation is

x ( t ) = r k e λ λ k t {\\displaystyle \\mathbf {x} (t)=\\mathbf {r} _{k}e^{\\lambda _{k}t}}

as may be confirmed by substitution.

If A {\\displaystyle \\mathbf {A} } is `F33f`_`[diagonalizable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Diagonalizable_matrix]`_`f, then any vector in an N {\\displaystyle N} -dimensional space can be represented by a linear combination of the right and `F33f`_`[left eigenvectors`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Left_eigenvector]`_`f (denoted l k {\\displaystyle \\mathbf {l} _{k}} ) of the matrix A {\\displaystyle \\mathbf {A} } .

x 0 = ∑ ∑ k = 1 N ( l k ⋅ ⋅ x 0 ) r k {\\displaystyle \\mathbf {x} _{0}=\\sum _{k=1}^{N}\\left(\\mathbf {l} _{k}\\cdot \\mathbf {x} _{0}\\right)\\mathbf {r} _{k}}

Therefore, the general solution for x ( t ) {\\displaystyle \\mathbf {x} (t)} is a linear combination of the individual solutions for the right eigenvectors

x ( t ) = ∑ ∑ k = 1 n ( l k ⋅ ⋅ x 0 ) r k e λ λ k t {\\displaystyle \\mathbf {x} (t)=\\sum _{k=1}^{n}\\left(\\mathbf {l} _{k}\\cdot \\mathbf {x} _{0}\\right)\\mathbf {r} _{k}e^{\\lambda _{k}t}}

Similar considerations apply to the discrete mappings.

>>Classification in two dimensions

The roots of the `F33f`_`[characteristic polynomial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Characteristic_polynomial]`_`f det(`!A`! - λ`!I`!) are the eigenvalues of `!A`!. The sign and relation of these roots, λ λ n {\\displaystyle \\lambda _{n}} , to each other may be used to determine the stability of the dynamical system

d d t x ( t ) = A x ( t ) . {\\displaystyle {\\frac {d}{dt}}\\mathbf {x} (t)=\\mathbf {A} \\mathbf {x} (t).}

For a 2-dimensional system, the characteristic polynomial is of the form λ λ 2 − − τ τ λ λ + Δ Δ = 0 {\\displaystyle \\lambda ^{2}-\\tau \\lambda +\\Delta =0} where τ τ {\\displaystyle \\tau } is the `F33f`_`[trace`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Trace_(linear_algebra)]`_`f and Δ Δ {\\displaystyle \\Delta } is the `F33f`_`[determinant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Determinant]`_`f of `!A`!. Thus the two roots are in the form:

λ λ 1 = τ τ + τ τ 2 − − 4 Δ Δ 2 {\\displaystyle \\lambda _{1}={\\frac {\\tau +{\\sqrt {\\tau ^{2}-4\\Delta }}}{2}}}

λ λ 2 = τ τ − − τ τ 2 − − 4 Δ Δ 2 {\\displaystyle \\lambda _{2}={\\frac {\\tau -{\\sqrt {\\tau ^{2}-4\\Delta }}}{2}}} ,

and Δ Δ = λ λ 1 λ λ 2 {\\displaystyle \\Delta =\\lambda _{1}\\lambda _{2}} and τ τ = λ λ 1 + λ λ 2 {\\displaystyle \\tau =\\lambda _{1}+\\lambda _{2}} . Thus if Δ Δ < 0 {\\displaystyle \\Delta <0} then the eigenvalues are of opposite sign, and the fixed point is a saddle. If Δ Δ > 0 {\\displaystyle \\Delta >0} then the eigenvalues are of the same sign. Therefore, if τ τ > 0 {\\displaystyle \\tau >0} both are positive and the point is unstable, and if τ τ < 0 {\\displaystyle \\tau <0} then both are negative and the point is stable. The `F33f`_`[discriminant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Discriminant]`_`f will tell you if the point is nodal or spiral (i.e. if the eigenvalues are real or complex).

>>See also

• `F33f`_`[Linear system`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_system]`_`f

• `F33f`_`[Dynamical system`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dynamical_system]`_`f

• `F33f`_`[List of dynamical system topics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=List_of_dynamical_system_topics]`_`f

• `F33f`_`[Matrix differential equation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Matrix_differential_equation]`_`f

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