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Analysis of ODE models
Analysis of ODE models with MATLAB
Shan He
School for Computational ScienceUniversity of Birmingham
Module 06-23836: Computational Modelling with MATLAB
Analysis of ODE models
Outline
Outline of Topics
Some Concepts
Eigenvalue stability method
Phase Plane Analysis
Classification of ODE using phase-plane
Analysis of ODE models
Outline
What will we learn from this lecture
I Aim of analysis.
I Some concpets
I Eigenvalue Stability.
I Phase Plane Analysis.
Analysis of ODE models
Outline
Aim of analysis:
Given a large and complex system of ODEs describing thedynamics of a complex biological system, we need to answer:
I Where will it go?I Stable and damped?I Or unstable and undamped, e.g., constant fluctuation?I Or unstable with increasing fluctuation?
I What will it do?I Exponential decay?I Exponential increase?I Stable oscillation?I Chaotic solution?
Analysis of ODE models
Outline
Why such analysis matters?
I The MATLAB ODE solver only give solutions for one initialconditions.
I But we usually want to know how, given a set of initialconditions, the system will evolve with time?
I Provides insights into how biological systems behavior changeswhen stimuli and rate constants are modified.
I Biomedical applications: could provide information formechanism-based drug discovery.
Analysis of ODE models
Some Concepts
Equilibrium of a dynamic system
I Definitions: Equilibrium, or fixed point, is a state of asystem which does not change.
I In ODE, we can calculate equilibria by setting all derivativesto zero because nothing is changing with respect to time.
Analysis of ODE models
Some Concepts
Example: Equilibria of the Lotka-Volterra equation
I The Lotka-Volterra equation{dxdt = x(α− βy)dydt = −y(γ − δx)
I Set all derivatives to zero:{x(α− βy) = 0
−y(γ − δx) = 0
I When solved for x and y the above system of equations yieldstwo equilibria: {x = 0, y = 0} and {x = γ
δ , y = αβ }
Analysis of ODE models
Some Concepts
Stable and unstable equilibrium
I Stable equilibrium: a system return to its equilibrium andremain there after disturbances.
I Unstable equilibrium: a system moves away from theequilibrium after disturbances
I Stability analysis is important:I Everything should be stable to be observableI Crucial for understanding complex systems
I The eigenvalues of a system linearized around a equilibriumcan determine the stability behavior of a system around thefixed point.
Analysis of ODE models
Some Concepts
Eigenvector and Eigenvalue
Definitions:
I Eigenvector: A vector that maintains its direction afterundergoing a linear transformation. Also called characteristicvectors.
I Eigenvalue: The scalar value that the eigenvector wasmultiplied by during the linear transformation. Also calledcharacteristic values.
Analysis of ODE models
Some Concepts
Eigenvalue and Eigenvector with Mona Lisa
I In the left picture, two vectors were drawn on the Mona Lisa.
I The picture is then linear transformed (sheared) and shown onthe right.
I The red arrow changes direction but the blue arrow does not.
I The blue arrow is an eigenvector, with eigenvalue 1 (lengthunchanged).
Analysis of ODE models
Some Concepts
Fundamental Equation
Av = λv
where A is a square matrix, v is the Eigenvector and λ is theEigenvalue.
Analysis of ODE models
Some Concepts
Eigenvalue and Eigenvector in MATLAB
[V,D] = eig(A) produces matrices of eigenvalues D andeigenvectors V of matrix A
Analysis of ODE models
Eigenvalue stability method
Eigenvalue stability method
Steps:
I Step 1: Determining the equilibria
I Step 2: Determine the eigenvalue of the equilibria
I Step 3: Determine the stability based on the sign of theeigenvalue
Analysis of ODE models
Eigenvalue stability method
Example: eigenvalue stability methodLet’s consider a very simple linear system:{
dxdt = ydydt = 2x + y
I Step 1: Determining the equilibria{y = 0
2x + y = 0
. We have one equilibrium (0, 0)I Step 2: Determine the eigenvalue of the equilibrium
A =
(0 12 1
)Using D = eig(A) we have: λ1 = −1 and λ2 = 2.
Analysis of ODE models
Eigenvalue stability method
Example: eigenvalue stability method
Eigenvalue Type Stability OscillatoryBehavior
Notation
All Real and + Unstable None Nodal sourceAll Real and - Stable None Nodal sink
Mixed Real Unstable None Saddle point+a + bi Unstable Undamped Spiral source-a + bi Stable Damped Spiral sink0 + bi Unstable Undamped Centre
I Step 3: Determine the stability based on the sign of theeigenvalue.λ1 = −1 and λ2 = 2: Unstable saddle point
Analysis of ODE models
Eigenvalue stability method
Pros/Cons of eigenvalue stability
Advantages:
I Very accurate for linear systems.
I Can be applied to a variety of processes.
I Can be used for systems with undefined parameters.
Disadvantages:
I Only applicable for linear models.
Analysis of ODE models
Phase Plane Analysis
Phase Plane Analysis
I Eigenvalue stability method is only applicable for linearmodels. How about non-linear system?
I Usually there is no analytical solution for a nonlinear systemmodel.
I If the systems are second order (two-dimensional) systems, wecan use Phase Plane Analysis.
I Phase Plane Analysis is a very useful technique fordetermining the qualitative behaviour of solutions of lowdimensional nonlinear systems.
Analysis of ODE models
Phase Plane Analysis
Our first Phase Plane
x ’ = y y ’ = 2 x + y
−2 −1 0 1 2 3 4
−4
−3
−2
−1
0
1
2
x
yNullclines
Analysis of ODE models
Phase Plane Analysis
Phase Plane Explained:
I A phase-plane plot usually consists of curves of one dimension(state variable) versus the other dimension (x1(t) vs. x2(t)).
I Each curve is based on a different initial condition.
I The original PPlan plot has vector fields or slope fields ordirection fields.
I Each vector in vector field is a point (x,y) to a unit vectorwith slope x ′
1, x′2.
Analysis of ODE models
Phase Plane Analysis
Nullclines
Nullclines:
I For an ODE system, nullclines are the geometric shape for
whichdxj
dt = 0 for any j .
I Sometimes called zero-growth isoclines.
I They are boundaries for determining the direction of themotion along the trajectories.
I They split the phase plane into regions of similar flow.
I The intersection point of all the nullclines is an equilibriumpoint of the system.
Analysis of ODE models
Phase Plane Analysis
Phase Plane Analysis tool: PPlane
I A MATLAB tool useful for Phase Plane Analysis.
I Download from [here]
I Can run in MATLAB or as a JAVA Applet.
I Plot solution curves in the phase plane by simple clicking onthem.
I A number of advanced features, including finding equilibriumpoints, eigenvalues and nullclines.
Analysis of ODE models
Phase Plane Analysis
How to use Pplane
Analysis of ODE models
Classification of ODE using phase-plane
Classification of 2D ODE
I We have seen how to use eigenvalue to determine stable andunstable equilibrium.
I We can also use phase-plane to analyse ODE, but only for 1stand 2nd order ODE.
I We will use a few examples in MATLAB Pplane to illustratehow to do this.
Analysis of ODE models
Classification of ODE using phase-plane
Saddle
I Example:dy
dx=
(1 42 −1
)y
−2
−1
0
1
2−3
−2−1
01
23
−1
−0.5
0
0.5
1
yx
t
Analysis of ODE models
Classification of ODE using phase-plane
Nodal Source
I Example:dy
dx=
(3 11 3
)y
−10−5
05
10 −10−5
05
100
0.2
0.4
0.6
0.8
1
1.2
1.4
yx
t
Analysis of ODE models
Classification of ODE using phase-plane
Nodal Sink
I Example:dy
dx=
(−3 −1−1 −3
)y
−2
−1
0
1
2 −2
−1
0
1
20
0.5
1
1.5
2
yx
t
Analysis of ODE models
Classification of ODE using phase-plane
Centre
I Example:dy
dx=
(4 −102 −4
)y
−20
−10
0
10
20 −10−5
05
10
0
0.5
1
1.5
2
yx
t
Analysis of ODE models
Classification of ODE using phase-plane
Spiral Source
I Example:dy
dx=
(0.2 1−1 0.2
)y
−4−2
02
4 −4−2
02
4
0
0.5
1
1.5
2
yx
t
Analysis of ODE models
Classification of ODE using phase-plane
Spiral Sink
I Example:dy
dx=
(−0.2 1−1 −0.2
)y
−2
−1
01
2 −2
−1
0
1
20
2
4
6
8
10
yx
t
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