1. Limit Cycles of Algebraic Systems
Shouvik Bhattacharya
Minnesota State University Moorhead
April 16, 2013
Shouvik Bhattacharya Limit Cycles of Algebraic Systems

2. Outline
Cycles
Limit Cycles
Van der Pol Equation
Shouvik Bhattacharya Limit Cycles of Algebraic Systems

3. Cycles
Definition
A cycle is a periodic solution of a Differential Equation.
If the independent variable t does not appear explicitly in the
differential equation, then it is called an Autonomous Differential
Equation.
Definition
An algebraic system is a differential system whose equations are
polynomials.
Shouvik Bhattacharya Limit Cycles of Algebraic Systems

4. An Example of a Cycle
Problem 
 dx(t)
 = y · (1 − x 2 )
dt (1)
 dy (t)
 = −x · (1 − y 2 )
dt
How do you solve this problem?
Shouvik Bhattacharya Limit Cycles of Algebraic Systems

5. An Example of a Cycle Contd.
Steps
1. You rewrite the equations in the form of dy /dx.
2. Then you apply separation of variables techniques.
3. The solution has the form:
(1 − y 2 ) · (1 − x 2 ) = A (2)
Shouvik Bhattacharya Limit Cycles of Algebraic Systems

6. An Example of a Cycle Contd.
These are some solutions of the system
Shouvik Bhattacharya Limit Cycles of Algebraic Systems

7. Hamiltonian
Theorem
The system

 dx(t)
 = X (x, y )
dt (3)
 dy (t)
 = Y (x, y )
dt
∂X ∂Y
is Hamiltonian, iff + = 0.
∂x ∂y
Shouvik Bhattacharya Limit Cycles of Algebraic Systems

8. Limit Cycles
Definition
A limit cycle represents a periodic and isolated solution of the
Differential Equation.
Shouvik Bhattacharya Limit Cycles of Algebraic Systems

9. An Example of a limit cycle
Problem

 dx(t)
 = −y + x · (1 − x 2 + y 2 )
dt (4)
 dy (t)
 = x + y · (1 − x 2 + y 2 )
dt
Shouvik Bhattacharya Limit Cycles of Algebraic Systems

10. An Example of a Limit Cycle Contd.
Steps
1. You transform the coordinates.
2. Carefully compute the derivatives.
3. The solution has the form:
θ(t) = t + C (5)
dr /dt = r · (1 − r ) (6)
When r is not changing the solution curve is the equation of a
circle, centered at (0,0).
Shouvik Bhattacharya Limit Cycles of Algebraic Systems

11. Van der Pol Equation
Theorem
If a polynomial system of Lienard equation

 dx(t)
 =y
dt (7)
 dy (t)
 = −f (x) · y − g (x)
dt
satifies (i) (f, g) = 0, (ii)deg (f ) ≥ deg (g ), and(g /f ) = const,
then it has no algebraic solution(Odani 1992).
Shouvik Bhattacharya Limit Cycles of Algebraic Systems

12. Van der Pol Equation Contd.
Corollary
The system of the Van der Pol System

 dx(t)
 =y
dt (8)
 dy (t)
 = µ · (x 2 − 1) · y − x, µ = 0,
dt
has no algebraic solution curves. In particular the limit cycle of it is
not algebraic (Odani 1992).
Shouvik Bhattacharya Limit Cycles of Algebraic Systems

13. Van der Pol Equation Contd.
A triode may be modeled by the Van der Pol equation
x + (x 2 − 1)x + x = 0
¨ ˙
Problem
For some values of , Van der Pol system has a limit cycle.
Shouvik Bhattacharya Limit Cycles of Algebraic Systems

14. Acknowledgements
I would like to thank Dr. Damiano Fulghesu for his patience and
support. I would also like to thank Dr. Linda Winkler for always
encouraging me to appreciate mathematics problems.
Shouvik Bhattacharya Limit Cycles of Algebraic Systems