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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