Turing Patterns
Self-organization in physical systems:
rhythms, patterns and chaos
24.11.2010
Turing Patterns by Miriam Daeubler
1
Existence of Turing Patterns and different types of
patterns will be discussed
1
Short Introduction to Turing Patterns
2
2
Linear Stability Analysis
6
3
Existence of Turing Patterns
11
4
Types of Turing Patterns in a System
20
5
Summary and Outlook
26
6
Literature
29
24.11.2010
Turing Patterns by Miriam Daeubler
2
Turing patterns can be used to describe evolution of cat
pelts
Leopard (left) and jaguar (right) pelt patterns at different ages of the animal
Model not
proven
experimentally
Source:
24.11.2010
R. T. Liu, S. S. Liaw, and P. K. Maini, Physical Review E 74, 011914 (2006)
Turing Patterns by Miriam Daeubler
3
Turing patterns are topologically different, within one
morphology different typical length scales exist
Experimentally observed patterns in chlorite-iodine-malonic acid reactions
Source:
24.11.2010
Ouyang/Swinney, Transition from a uniform state hexagonal and striped Turing patterns, Nature, Vol. 352, 610-611, 1991
Turing Patterns by Miriam Daeubler
4
The Turing model originally derived to explain
morphogenesis in biological systems
Some facts about Turing patterns
•
Model systems of two reacting and diffusing substances with concentrations u, v
𝜕𝑢
= 𝑓 𝑢, 𝑣 + 𝐷𝑢 ∆𝑢
𝜕𝑡
𝜕𝑣
= 𝑔 𝑢, 𝑣 + 𝐷𝑣 ∆𝑣
𝜕𝑡
•
•
•
The above system describes an activator-inhibitor system
• Activator: substance that stimulates the growth of the concentration of both
chemicals
• Inhibitor: substance that leads to a decrease in the concentrations
In 1952, Turing predicted that such a system can produce a stationary pattern, if the
inhibitor diffuses much faster than the activator (local activation with lateral inhibition).
Turing tried to explain morphogenesis, i.e. the development of shape or form in plants and
animals.
24.11.2010
Turing Patterns by Miriam Daeubler
5
Existence of Turing Patterns and different types of
patterns will be discussed
1
Short Introduction to Turing Patterns
2
2
Linear Stability Analysis
6
3
Existence of Turing Patterns
11
4
Types of Turing Patterns in a System
20
5
Summary and Outlook
26
6
Literature
29
24.11.2010
Turing Patterns by Miriam Daeubler
6
A diagonalizable coefficient matrix makes system of
linear differential equations easy to handle
A homogeneous linear system of ODE’s with constant coefficients
𝑦 ′ 𝑡 = 𝐴𝑦(𝑡), 𝐴 ∈ ℝ(𝑛𝑥𝑛)
•
•
•
For further discussion, assume that 𝑨 is diagonalizable
The eigenvalues are obtained by solving the secular equation
det 𝐴 − 𝜆 𝕀 = 0
In order to construct fundamental system choose eigenbasis of 𝐴 𝑣 𝑘 ∈ ℂ𝑛 , thus
𝑌 𝑡 = (𝑒 𝜆1 𝑡 𝑣 1 , . . , 𝑒 𝜆𝑛 𝑡 𝑣 𝑛 ), 𝜆𝑘 ∈ ℂ eigenvalues
•
The solution of the homogeneous system is
𝑦ℎ = 𝑌 𝑡 ∙ 𝐶 =
•
𝑛
𝜆𝑘 𝑡 𝑘
𝑣 ,
𝑘=1 𝐶𝑘 𝑒
𝐶𝑘 ∈ ℂ
Usually, one is only interested in real solutions. Complex eigenvalues and eigenvectors
always come in pairs, therefore, both can be combined to obtain only real solutions.
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Turing Patterns by Miriam Daeubler
7
Thanks to coordinate transform stability analysis only
needs to be derived once
Stability analysis in the simplest linear system (1/2)
Consider the following system with constant coefficients
𝑦 ′ 𝑡 = 𝐴𝑦 𝑡 + 𝑏, 𝐴 ∈ ℝ(𝑛𝑥𝑛)
•
•
•
For further discussion, assume that 𝐴 is diagonalizable
Let 𝑦 ∗ be a special solution of interest, usually a fixed point of the system
Transform coordinates such that
𝑧 𝑡 = 𝑦 𝑡 − 𝑦∗ 𝑡
𝑧 ′ 𝑡 = 𝐴𝑧(𝑡)
•
Thus, it is only necessary to assess the stability of 𝑧 ∗ = 0.
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Turing Patterns by Miriam Daeubler
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Stability analysis reduces to an eigenvalue problem
Stability analysis in the simplest linear system (2/2)
The stability is described by the initial value problem 𝑧 ′ 𝑡 = 𝐴𝑧 𝑡 and 𝑧 𝑡0 = 𝑧𝑜
with the solution
𝑧 𝑡 = 𝑌 𝑡 𝑌(𝑡0 )−1 𝑧𝑜
Theorem
The solution 𝑧 ∗ = 0 is
a) strictly stable, if and only if all eigenvalues of A have a real part smaller than zero:
𝑅𝑒 𝜆𝑘 < 0
b) stable, if and only if all eigenvalues of A have a real part smaller or equal zero:
𝑅𝑒 𝜆𝑘 ≤ 0
c) unstable, in every other case
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Turing Patterns by Miriam Daeubler
9
Under certain conditions even the stability analysis of a
nonlinear system reduces to an eigenvalue problem
Stability analysis in the nonlinear case
Consider the following nonlinear system of differential equations with 𝑓 being a nonlinear
function which can be continuously differentiated a sufficient number of times
𝑦 ′ 𝑡 = 𝑓(𝑦 𝑡 )
Assume 𝑦 ∗ = 0 is a fixed point with 𝑓 0 = 0.
Using Taylor expansion one obtains
𝑦 ′ 𝑡 = 𝐴 𝑦 𝑡 + 𝑔(𝑦 𝑡 )
𝐴 = 𝕁 𝑓(0), 𝑔 0 = 0
𝑔 𝑦 = 𝑜( 𝑦 )
It can be shown that if the shortened system of differential equations 𝒚′ 𝒕 = 𝑨 𝒚 𝒕 is either
strictly stable or unstable so is the nonlinear system.
24.11.2010
Turing Patterns by Miriam Daeubler
10
Existence of Turing Patterns and different types of
patterns will be discussed
1
Short Introduction to Turing Patterns
2
2
Linear Stability Analysis
6
3
Existence of Turing Patterns
11
4
Types of Turing Patterns in a System
20
5
Summary and Outlook
26
6
Literature
29
24.11.2010
Turing Patterns by Miriam Daeubler
11
The possible existence of Turing patterns can be derived
using stability analysis
Stability analysis of Turing model (1/7)
Let there be a steady, spatially homogeneous solution 𝑢
𝑣 = 𝑢0
𝑣𝑜 (fixed point)
𝜕𝑢
= 𝑓 𝑢, 𝑣 + 𝐷𝑢 ∆𝑢
𝜕𝑡
𝜕𝑣
= 𝑔 𝑢, 𝑣 + 𝐷𝑣 ∆𝑣
𝜕𝑡
Firstly, linearize the system of equations in the vicinity of the fixed point
𝜕 𝑈
𝑎 + 𝐷𝑢 ∆
=
𝑐
𝜕𝑡 𝑉
−𝑏
−𝑑 + 𝐷𝑣 ∆
𝑈
𝑉
With
𝜕𝑓
𝑎=
𝜕𝑢
c=
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𝜕𝑔
𝜕𝑢
𝑢0
𝑣0
𝜕𝑓
,𝑏 = −
𝜕𝑣
,𝑑 = −
𝑢0
𝑣0
𝜕𝑔
𝜕𝑣
𝑢0
𝑣0
𝑢0
𝑣0
Turing Patterns by Miriam Daeubler
Signs of a, b, c and d
chooses in such way
that u corresponds to
the activator, v to the
inhibitor
12
The instability shall be diffusion driven, the system must
be stable in the absence of diffusion
Stability analysis of Turing model (2/7)
Neglecting the diffusion terms the linearized system becomes
𝜕 𝑈
𝑎
=
𝑐
𝜕𝑡 𝑉
−𝑏
−𝑑
𝑈
𝑉
In order to test stability, one must solve the eigenvalue problem
−𝑏
−𝜆𝕀 =0
−𝑑
1
⟺ 𝜆± = 𝑎 − 𝑑 ± 𝑎 + 𝑑
2
det
𝑎
𝑐
2
− 4𝑏𝑐
For the fixed point to be strictly stable 𝜆± < 0. Thus,
𝑎<𝑑
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∧
𝑎𝑑 < 𝑏𝑐
Turing Patterns by Miriam Daeubler
13
In order to reduce the complexity of the PDE system use
Fourier integral transform
Stability analysis of Turing model (3/7)
Linearizing one obtained a system of PDEs
𝜕 𝑈
𝑎 + 𝐷𝑢 ∆
−𝑏
=
𝑐
−𝑑 + 𝐷𝑣 ∆
𝜕𝑡 𝑉
𝑈
𝑉
Integral transforms are standard way to reduce complex PDEs to simpler PDEs or even ODEs,
here use Fourier transform
𝑈𝑘 =
𝑈(𝑥 , 𝑡) 𝑒 −𝑖𝑘∙𝑥 𝑑3 𝑥
𝑉𝑘 =
𝑉(𝑥 , 𝑡) 𝑒 −𝑖𝑘∙𝑥 𝑑3 𝑥
Fourier transforms
describe a change in
basis, new basis
consists of
eigenfunctions of
Laplace operator
Fourier transforms of a function only exist if it is absolutely integrable
𝑈(𝑥 , 𝑡) 𝑑3 𝑥 < ∞
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Turing Patterns by Miriam Daeubler
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Properties of Fourier transforms needed to derive
system of differential equations in Fourier space
Stability analysis of Turing model (4/7)
Fourier transform of gradient of a function
𝛻𝑈 𝑥 , 𝑡 𝑒 −𝑖𝑘∙𝑥 𝑑3 𝑥 =
= 𝑖𝑘
𝛻 𝑈 𝑥 , 𝑡 𝑒 −𝑖𝑘∙𝑥 𝑑3 𝑥 −
𝑈 𝑥 , 𝑡 𝛻𝑒 −𝑖𝑘∙𝑥 𝑑3 𝑥
𝑈 𝑥 , 𝑡 𝑒 −𝑖𝑘∙𝑥 𝑑3 𝑥 = 𝑖𝑘 𝑈𝑘 (𝑘, 𝑡)
Fourier transform of Laplace of a function
∆𝑈 𝑥 , 𝑡 𝑒 −𝑖𝑘∙𝑥 𝑑3 𝑥 =
= 𝑖𝑘 ∙
𝛻 𝛻𝑈 𝑥 , 𝑡 𝑒 −𝑖𝑘∙𝑥 𝑑3 𝑥 −
𝛻𝑈 𝑥 , 𝑡 𝑒 −𝑖𝑘∙𝑥 𝑑3 𝑥 = 𝑖𝑘 ∙ 𝑖𝑘 𝑈𝑘 𝑘, 𝑡
𝛻𝑈 𝑥 , 𝑡 ∙ 𝛻𝑒 −𝑖𝑘∙𝑥 𝑑3 𝑥
= −𝑘 2 𝑈𝑘 (𝑘, 𝑡)
Fourier transform of derivative of a function with respect to time
𝜕𝑈 𝑥 , 𝑡 −𝑖𝑘∙𝑥 3
1
𝑒
𝑑 𝑥 = lim
𝑈 𝑥, 𝑡 + 𝑕 − 𝑈 𝑥, 𝑡
ℎ→0 𝑕
𝜕𝑡
1
ℎ→0 𝑕
= lim
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𝑈𝑘 𝑘, 𝑡 + 𝑕 − 𝑈𝑘 (𝑘, 𝑡) =
𝑒 −𝑖𝑘∙𝑥 𝑑3 𝑥
𝜕
𝑈 (𝑘, 𝑡)
𝜕𝑡 𝑘
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15
Fourier transforms reduce PDEs to ODEs and stability
analysis becomes an eigenvalue problem
Stability analysis of Turing model (5/7)
Fourier transform the system of PDEs
⟺
𝜕 𝑈 −𝑖𝑘∙𝑥 3
𝑒
𝑑 𝑥=
𝜕𝑡 𝑉
𝑎 + 𝐷𝑢 ∆
𝑐
𝜕 𝑈 −𝑖𝑘∙𝑥 3
𝑒
𝑑 𝑥=
𝜕𝑡 𝑉
𝑎 𝑈 + 𝐷𝑢 ∆𝑈
𝑐𝑈
−𝑏
−𝑑 + 𝐷𝑣 ∆
𝑈 −𝑖𝑘∙𝑥 3
𝑒
𝑑 𝑥
𝑉
−𝑏 𝑉
𝑒 −𝑖𝑘∙𝑥 𝑑3 𝑥
−𝑑 𝑉 + 𝐷𝑣 ∆𝑉
𝜕 𝑈𝑘
𝑎 𝑈𝑘 − 𝐷𝑢 𝑘 2 𝑈𝑘
−𝑏 𝑉𝑘
⟺
=
𝜕𝑡 𝑉𝑘
𝑐 𝑈𝑘
−𝑑𝑉𝑘 − 𝐷𝑣 𝑘 2 𝑉𝑘
𝑎 − 𝐷𝑢 𝑘 2
−𝑏
𝑈𝑘
=
𝑐
−𝑑 − 𝐷𝑣 𝑘 2 𝑉𝑘
Fourier transforms reduce PDEs to ODEs and stability analysis becomes an eigenvalue
problem in Fourier space
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For a pattern to evolve some spatial modes must be
unstable
Stability analysis of Turing model (6/7)
Fourier transforming the linearized system of differential equations yields
𝜕 𝑈𝑘
𝑎 − 𝐷𝑢 𝑘 2
=
𝜕𝑡 𝑉𝑘
𝑐
−𝑏
−𝑑 − 𝐷𝑣 𝑘 2
𝑈𝑘
𝑉𝑘
To find a criterion for unstable spatial modes find eigenvalues
𝑎 − 𝐷𝑢 𝑘 2
−𝑏
𝑑𝑒𝑡
−𝜆𝕀 =0
𝑐
−𝑑 − 𝐷𝑣 𝑘 2
⟺ 𝜆2 + 𝜆 −𝑎 + 𝑑 + 𝐷𝑢 𝑘 2 + 𝐷𝑣 𝑘 2 + 𝐷𝑢 𝑘 2 − 𝑎 𝐷𝑣 𝑘 2 + 𝑑 + 𝑏𝑐 = 0
Compare with
𝜆−𝑎 𝜆−𝑏 =0
⟺ 𝜆2 − 𝜆 𝑎 + 𝑏 + 𝑎𝑏 = 0
and
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𝑝
2
𝜆2 + 𝑝𝜆 + 𝑞 = 0 ⟺ 𝜆± = − ±
Turing Patterns by Miriam Daeubler
𝑝 2
2
−𝑞
17
Condition for instability to occur linked with typical
length scales
Stability analysis of Turing model (7/7)
To sum up, the only possibility for an instability is for the product of the roots to be negative
𝑕(𝑘 2 ) ≡ 𝐷𝑢 𝑘 2 − 𝑎 𝐷𝑣 𝑘 2 + 𝑑 + 𝑏𝑐 < 0
The Turing instability occurs as long as
𝑎
𝑑
>
and the minimum of 𝑕 𝑘 2 is
𝐷𝑢
𝐷𝑣
negative.
First requirement can be rewritten
𝐷𝑣
𝑑
>
Source:
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𝐷𝑢
𝑎
⟺ 𝑙𝑣 > 𝑙𝑢
Hoyle, Pattern Formation, Cambridge University Press, 2006, p.20
Turing Patterns by Miriam Daeubler
18
Nonlinear bifurcation theory has to be used to predict
morphologies
Results obtained from stability analysis
Stability analysis
the parameter range for which instabilities are present
the characteristic length of the resulting patterns, which are independent of the
initial conditions, therefore intrinsic to the system
initial conditions only influence the phase of the pattern, i.e. position and alignment
of pattern entities
the different possible morphologies of the patterns
the selection rules for multistable systems
Apply nonlinear bifurcation theory to approximately predict
the stability of different Turing patterns
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19
Existence of Turing Patterns and different types of
patterns will be discussed
1
Short Introduction to Turing Patterns
2
2
Linear Stability Analysis
6
3
Existence of Turing Patterns
11
4
Types of Turing Patterns in a System
20
5
Summary and Outlook
26
6
Literature
29
24.11.2010
Turing Patterns by Miriam Daeubler
20
Bifurcations describe sudden qualitative changes in the
set of solutions of a system as parameters are varied
Bifurcation theory in a nutshell (1/2)
Consider the following nonlinear system of differential equations with 𝑓 being a nonlinear
function which can be continuously differentiated a sufficient number of times
𝑦 ′ 𝑡 = 𝑓(𝑦 𝑡 )
The solutions of the system define a flow 𝜙 𝑦, 𝑡 with 𝜙 𝑦0 , 0 = 𝑦0 .
Definition bifurcation
At a bifurcation there is a sudden qualitative change in the flow 𝜙 in response to infinitesimal
changes in one or more parameters of the system of differential equations. The phase portrait,
the number and stability of fixed points or periodic orbits is usually affected.
Local vs. global bifurcation
Local bifurcation:
changes of the flow in the vicinity of a fixed point or periodic orbit
Global bifurcation: changes that affect the large scale property of the flow
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For Turing patterns local bifurcations are of interest
Bifurcation theory in a nutshell (2/2)
Local bifurcations
A local bifurcation occurs when a parameter change
causes the stability of a fixed point or periodic orbit
to change. This corresponds to the real part of an
eigenvalue of a fixed point or PO passing
through zero.
Goal of bifurcation theory
Bifurcation theory tries to produce parameter
space maps or bifurcation diagrams that divide
the parameter space into regions of topologically
equivalent systems, in our case equivalent Turing
pattern morphologies.
Source:
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Ott, Chaos in Dynamical Systems, Second Edition, Cambridge University Press, 2002, p. 46
Turing Patterns by Miriam Daeubler
22
Stripes and hexagonal spotty pattern typical for 2D
reaction-diffusion systems
Typical patterns of 2D reaction-diffusion systems
Source:
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Leppänen, The theory of Turing pattern formation, Helsinki University of Technology, 2004
Turing Patterns by Miriam Daeubler
23
Bifurcation theory able to predict stable morphologies
found in numerical simulations of 2D system
Bifurcation theory applied to Turing patterns: 2D hexagonal lattice
2D hexagonal
lattice is a
bistable
system
Source:
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Leppänen, The theory of Turing pattern formation, Helsinki University of Technology, 2004
Turing Patterns by Miriam Daeubler
24
Turing patterns discussed theoretically beforehand have
been observed experimentally
Experimentally observed patterns in chlorite-iodine-malonic acid reactions
Source:
24.11.2010
Ouyang/Swinney, Transition from a uniform state hexagonal and striped Turing patterns, Nature, Vol. 352, 610-611, 1991
Turing Patterns by Miriam Daeubler
25
Existence of Turing Patterns and different types of
patterns will be discussed
1
Short Introduction to Turing Patterns
2
2
Linear Stability Analysis
6
3
Existence of Turing Patterns
11
4
Types of Turing Patterns in a System
20
5
Summary and Outlook
26
6
Literature
29
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Turing Patterns by Miriam Daeubler
26
There is no general way to determine which state a
morphologically multistable system will choose
Results obtained from stability analysis and nonlinear bifurcation theory
Stability analysis and nonlinear bifurcation theory
the parameter range for which instabilities are present
the characteristic length of the resulting patterns, which are independent of the
initial conditions, therefore intrinsic to the system
initial conditions only influence the phase of the pattern, i.e. position and alignment
of pattern entities
the different possible morphologies of the patterns
the selection rules for multistable systems
Formal theory of self-organization based on non-equilibrium thermodynamics,
however, this theory is incomplete
There is no general way to figure out the selection rules for multistable systems
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27
Besides missing selection rules there are many more
topics concerning Turing patterns currently analyzed
Further topics concerning Turing patterns
•
•
•
•
•
Turing patterns also appear in 3D systems
Turing patterns can be generalized to allow for transport mechanisms other than diffusion
Interaction between Turing instability (constant in time) and other instabilities such as Hopf
instability (constant in space)
Spatially correlated noise can induce Turing patterns in systems that under normal
conditions would not exhibit any
…
24.11.2010
Turing Patterns by Miriam Daeubler
28
Existence of Turing Patterns and different types of
patterns will be discussed
1
Short Introduction to Turing Patterns
2
2
Linear Stability Analysis
6
3
Existence of Turing Patterns
11
4
Types of Turing Patterns in a System
20
5
Summary and Outlook
26
6
Literature
29
24.11.2010
Turing Patterns by Miriam Daeubler
29
Literature used
1) Hoyle, Pattern Formation: An Introduction to methods, Cambridge University Press, 2006
2) Ott, Chaos in Dynamical Systems, Second Edition, Cambridge University Press, 2002
3) Leppänen et al., Spatio-temporal dynamics in a Turing model, submitted to InterJournal
2004
4) Sanz-Anchelergues et al., Turing pattern formation induced by spatially correlated noise,
Phys. Rev. E, vol. 63, 056124
5) Guckenheimer, Bifurcation, doi: 10.4249/scholarpedia.1517
6) Leppänen et al., A new dimension to Turing patterns, Physica D, 168-169 (2002) 35-44
7) Leppänen et al., Dimensionality effects in Turing pattern formation, arXiv:condmat/0306121v1, 2003
8) Nakao/Mikhailov, Turing patterns in network-organized activator-inhibitor systems, doi:
10.1038/NPHYS1651
9) Gierer/Meinhardt, A Theory of Biological Pattern Formation, Kybernetik 12, 30-39 (1972)
10) Leppänen et al., Morphological transitions and bistability in Turing systems, Phys. Rev. E,
vol. 70, 066202 (2004)
11) Leppänen, The theory of Turing pattern formation, Helsinki University of Technology, 2004
24.11.2010
Turing Patterns by Miriam Daeubler
30
Literature used
12) Ouyang/Swinney, Transition from a uniform state hexagonal and striped Turing patterns,
Nature, Vol. 352, 610-611, 1991
13) Leppänen et al., Turing Systems as Models of Complex Pattern Formation, Brazil Journal of
Physics, vol. 34, No. 2A, June 2004
14) Movie: Turing-Hopf bifurcation in monostable system downloaded from
http://www.apmaths.uwo.ca/~mkarttu/turing.shtml
15) Movie: Self-organization of spherical structures under Gaussian noise in 3D downloaded
from http://www.apmaths.uwo.ca/~mkarttu/turing.shtml
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Turing Patterns by Miriam Daeubler
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Backup
24.11.2010
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32
In order to solve a system of linear differential equations
one needs to find the homogeneous solution first
General remarks on systems of linear differential equations
Definition of an explicit system of linear first order differential equations
𝑦′ 𝑡 = 𝐴 𝑡 𝑦 𝑡 + 𝑕 𝑡
with 𝐴 𝑡 ∈ ℝ
𝑛𝑥𝑛
Systems of higher order
can always be reduced to
a system of first order
and 𝑕(𝑡) ∈ ℝ𝑛 continuous functions of time 𝑡 ∈ ℝ.
General solution of an explicit system of linear first order differential equations
𝑦 𝑡 = 𝑦𝑝 𝑡 + 𝑦ℎ 𝑡
𝑦𝑝 𝑡 being a particular solution of the inhomogeneous differential equation
𝑦ℎ 𝑡 being any solution of the homogeneous differential equation
Study solutions of homogenous system of differential equations
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One needs to construct a basis of the vector space
spanned by the solutions
Solutions of a system of homogeneous linear differential equations
The solutions of the homogeneous linear system of differential equations form a subspace of
𝐶 1 ℝ, ℝ𝑛 , the real vector space of all continuously differentiable functions ℝ → ℝ𝑛 , with
finite dimension.
Receipe to construct a basis of this subspace for an initial value problem
•
•
Choose 𝑡0 ∈ ℝ and a basis 𝑣 𝑘 , 𝑘𝜖 1, . . , 𝑛 of ℝ𝑛
Solve
𝑌 ′ 𝑡 = 𝐴 𝑡 𝑌(𝑡), 𝑌 𝑡0 = (𝑣 1 , . . , 𝑣 𝑛 )
•
The matrix 𝑌(𝑡) is called a fundamental system.
Then the general solution of the homogeneous equation is
𝑦ℎ = 𝑌 𝑡 ∙ 𝑐 =
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𝑛
𝑘
𝑘=1 𝑐𝑘 𝑦 (𝑡),
𝑐 ∈ ℝ𝑛
Turing Patterns by Miriam Daeubler
34
Many methods enable one to obtain a particular
solution of the inhomogeneous system
Solutions of a system of inhomogeneous linear differential equations
•
•
First a construct a basis of the solutions of the homogeneous differential equations 𝑌(𝑡)
Use one of the abundant methods to obtain a solution for the inhomogeneous system of
differential equations for initial value problems, such as
• Variation of constants
• Green’s functions …
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