Differential Equations ,A Qualitative Approach 2

1
Differential Equations ,A Qualitative Approach (2)

• Baktash Babadi
• Fall 2004,
• IPM, SCS, Tehran, Iran
• baktash_at_ipm.ir

2
Two Dimensional Systems
3
Two Dimensional Linear Systems
4
Example Simple Harmonic Oscillator
m
k
x
5
Simple Harmonic Oscillator Phase Plane (1)
Phase Portrait
Trajectory
Fixed point
Closed orbit
6
Simple Harmonic Oscillator Phase Plane (2)
(a)
(b)
(b)
(a)
(c)
(c)
(d)
(d)
X0
7
Example Uncoupled System
8
Symmetric node (Star)

• a -1

9
Stable node

• alt-1

-1ltalt0
10
Line of fixed points

• a 0

11
Saddle Node

• a gt 0

Stable Manifold
Unstable Manifold
12
General Analysis of Linear Systems (1)

• In every case the trajectories that start on x or
  y axes remain on the same axis for ever. The
  fixed points are the intersection of these axes.

13
General Analysis of Linear Systems (2)

• Straight line trajectories in general

14
General Analysis of Linear Systems (2)
15
General Analysis of Linear Systems (3)
16
Example
Eigenvectors
17
Example
y
Fast eigendirection
Slow eigendirection
x
18
Example
y
x
19
Complex Eigenvalues(1)
20
Complex Eigenvalues (2)
Center
21
Complex Eigenvalues (3)
Stable Spiral
22
Complex Eigenvalues (3)
Unstabke Spiral
23
Classification of fixed points
24
Nonlinear Systems

• A typical phase portrait of a nonlinear 2D system
  

25
Linear Stability Analysis

• For nonlinear systems, we study the qualitative
  behavior near the fixed points
• 1) Finding the fixed points
• 2) Linearizing the system near the fixed points
• 3) Classifying the fixed points

26
Finding the fixed points

• NullClines

• The intersections of nullclines are the fixed
  points

27
Example Lotka-Volterra competition (1)

• Rabbits x(t) vs. Sheep y(t)

• Rabbits reproduce more
• Sheep reproduce less

• Sheep are stronger
• Rabbits are weaker

28
Example Lotka-Volterra competition (2)

• Numerical example

29
Linearization (1)

• Equations
• Fixed point
• Intersection of nullclines
• Perturbation
• Stability

30
Linearization (2)
31
Linearization (3)
32
Example Lotka-Volterra competition (1)
33
Example Lotka-Volterra competition (2)
Unstable node
34
Example Lotka-Volterra competition (3)
Stable Node
35
Example Lotka-Volterra competition (4)
Stable Node
36
Example Lotka-Volterra competition (3)
Saddle Point
37
Phase Portrait of the Lotka-Volterra system
Stable Manifold Basin Boundary Separatrix
Sheep
Basin of (3,0)
Rabbits
38
Interpretation of Lotka-Volterra Competition

• Bistability
• Principle of competitive exclusion

39
Question

• Analyze the phase portrait of the pendulum

L
?