linear planar sytems in ode

Linear 2D(Planar)Systems

Piyush Patel

Introduction

Solution ofthe System

Classification

PhasePortrait

Linear 2D (Planar) Systems

Piyush Patel

Department of MathematicsFaculty of Science

The Maharaja Sayajirao University of Baroda, Vadodara.

June 17, 2010


Piyush Patel

Introduction


Classification

PhasePortrait

Linear Systems of ODE

Consider the (homogeneous) lineartwo-dimensional autonomous systems of the form

dxdt

= x = ax + by ,dydt

= y = cx + dy , (1)

where a,b, c,d are constants.

This system (1) is linear as the terms in x , y , x , and yare all linear.

System (1) can be written in the equivalent matrixform as

x = Ax, x ∈ R2 and A =

(a bc d

)(2)


Piyush Patel

Introduction


Classification

PhasePortrait

Linear Systems of ODE (contd.)

Definition

Every solution of (1) and (2), say φ(t) = (x(t), y(t)),can be represented as a curve in the plane. The solutioncurves are called trajectories or orbits.

The existence and uniqueness theorem guaranteesthat trajectories do not intersect cross.Note that there are an infinite number of trajectoriesthat would fill the plane if they were all plotted.However, the qualitative behavior can be determined byplotting just a few of the trajectories given theappropriate number of initial conditions.


Piyush Patel

Introduction


Classification

PhasePortrait

Definition

The phase portrait is a two-dimensional figureshowing how the qualitative behavior of system (1) isdetermined as x and y vary with t .

With the appropriate number of trajectories plotted, it shouldbe possible to determine where any trajectory will end upfrom any given initial condition.


Piyush Patel

Introduction


Classification

PhasePortrait

Outline

1 Solution of the System

2 Classification

3 Phase Portrait


Piyush Patel

Introduction


Classification

PhasePortrait

Homogeneous Linear Systems:Uncoupled

Two types of linear systems:1 Uncoupled linear systems2 Coupled linear systems

Uncoupled system: The most simple homogeneouslinear system is the uncoupled system for which thecoefficient matrix A is diagonal.For Example: The linear system x = Ax with

A =

(a 00 b

), where a and b are real numbers, is an

uncoupled system. The solution of this system is veryeasy to obtain and is given by

x(t) =(

eat 00 ebt

)c,

where c = x(0).


Piyush Patel

Introduction


Classification

PhasePortrait

Homogeneous Linear Systems: Coupled

Coupled system: When the system is coupled, but thecoefficient matrix A is diagonalizable, then we canobtain the solution as follows:

1 Reduce the given coupled system to an uncoupledsystem using the transformation x = Py , where P is aninvertible matrix (whose columns consists ofgeneralized eigenvectors of A) such that B = P−1AP isa diagonal matrix.

2 Obtain the vector y by solving the uncoupled systemy = By .

3 Obtain the required solution x of the given system usingthe transformation y = P−1x .

Note that the matrix B will have one of the followingforms:

B =

(λ 00 µ

), B =

(λ 10 λ

), B =

(a −bb a

),

where a,b, λ, µ are real constants.


Piyush Patel

Introduction


Classification

PhasePortrait

Exponential of Matrix

Definition

Let A ∈ M2(R). Then for t ∈ R,

eAt =∞∑

k=o

Ak tk

k !.

Now we can compute the matrix eAt for any 2× 2 matrix A.Using the some simple results of theory of Exponential ofoperators, we can compute eBt , where B as above, asfollows:

eBt =

(eλt 00 eµt

), eBt = eλt

(1 t0 1

),

eBt = eat(

cos bt − sin btsin bt cos bt

),


Piyush Patel

Introduction


Classification

PhasePortrait

Fundamental Theorem

Theorem (Fundamental Theorem for LinearSystems)

Let A ∈ M2(R). Then for a given x0 ∈ R2, the initialvalue problem

x = Ax , x(0) = x0

has a unique solution for all t ∈ R which is given by

x(t) = eAtx0.

It follows from the fundamental theorem and form ofthe matrix eBt that the solution of the initial value problemy = By , y(0) = x0 is given by y(t) = eBtx0. Thus thesolution of our original coupled system x = Ax , x(0) = x0 isnow given by

x(t) = P−1eAtPx0.


Piyush Patel

Introduction


Classification

PhasePortrait

Outline


2 Classification

3 Phase Portrait


Piyush Patel

Introduction


Classification

PhasePortrait

Classification

The critical points may be classified depending uponthe type of eigenvalues. So accordingly we list the phaseportrait that result from these solutions.

Real Distinct eigenvalues: B =

(λ 00 µ

).

If all the eigenvalues are distinct, real, and positive, thenthe critical point is called an unstable node.If all the eigenvalues are distinct, real, and negative,then the critical point is called an stable node.If one of the eigenvalue is positive and other negative,then the critical point is called a saddle point.


Piyush Patel

Introduction


Classification

PhasePortrait

Classification

Complex Eigenvalues (λ = a± ib): B =

(a −bb a

).

(The type of phase portrait depends on the values of aand b.)

If a > 0, then the critical point is called an unstablefocus.If a = 0, then the critical point is called a center.If a < 0, then the critical point is called a stable focus.If b < 0, then the trajectories spiral counter-clockwisearound the origin.If b > 0, then the trajectories spiral clockwise aroundthe origin.

If A is not in real canonical form, then the phase portraitshould look similar but may be rotated, flipped, stretched,skewed, etc.


Piyush Patel

Introduction


Classification

PhasePortrait

Outline


2 Classification

3 Phase Portrait


Piyush Patel

Introduction


Classification

PhasePortrait

Phase Portrait


Piyush Patel

Introduction


Classification

PhasePortrait

References

S. Lynch, Dynamical Systems with Applications usingMATLAB, Birkhauser Boston, 2004.

L. Perko, Differential Equations and DynamicalSystems, 3/e, Springer-verleg, New York Inc., 2001.

V. Raghvendra, Lecture Notes: Linear Systems andPhase Portraits, ATML-ODE (MSU-BARODA), June2-15, 2011.


Piyush Patel

Introduction


Classification

PhasePortrait

T HANK YOU

linear planar sytems in ode

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