Study of chaos theory and different maps(Logistic,Lorenz,Circle,Arnold cat maps) and determination of stability by Lypunov exponent.

1. Chaos theory and Logistic maps
Summer Project Report
Narendra Kumar
Supervisor: Prof. Srikanth Sastry

2. Future Plan:
Topological Entropy : Quantify chaos
➢ Calculation of Topological entropy for higher dimensional map
➢ Calculation of topological entropy in spin-glass

3. Q.Why and how we study non-linear dynamics?
Ans. Most of the systems are non-linear in nature. A theoretical study of such a
system leads to the solution of nonlinear differential equation. we can extract
a lot of informations about solution from the equation themselves. If the
dynamical systems be autonomous systems then it can be decomposed Into
coupled first order differential equations and then can be solved computationally.
Chaos:
Its main features are :
➢ extreme sensitivity to initial conditions
➢ Non-linearity
➢ strangeness
“Edward Lorenz” described chaos as “when the present determines the future, but
the approximate present doesn't approximately determines the future”.

4. Lyapunov Exponent: Sensitivity to initial conditions and parameters of a
non-linear system is determined by the value of Lyapunov exponent.
n
λ = lim (1/n) ∑ |df(xi
)/dx|
n→∞ i=0
If λ (<>0) then orbit is stable, unstable(show chaos).
Arnold Transform: a n×n pixel size 2D image and transform each point
(x,y) by transformation equation Γ : (x,y) → (x+y,x+2y) mod n after a
finite no. of iteration(say k) same image is restored. But there is no any
specific relation between k & n.

5. RESULTS: (1) Logistic function map:

6. (2) Lorenz map:

8. (3)Arnold cat map:

10. (4) Circle map:
θ(t+1) = θ(t)+Ω+(K/2π) sin(2πθ(t)) mod1 0<θ(t)<=1
Lyapunov constant graph

11. Applications:
(1)weather prediction
(2)population growth
(3)cardiotocography
(4) signal analysis
(5) random number generator
(6) Information theory
(7) optics
References:
(1)Periodic entrainment of chaotic logistic map dynamics:E. Atlee Jackson, Alfred Hübler
(2)Lyapunov graph for two-parameters map: Application to the circle map by Figueiredo
(3)Deterministic Non-periodic flow: Edward N. Lorenz
(4)Nonlinear dynamics and chaos (Book) : Strogatz
(5) Arnold cat map : Gabriel Peterson

12. THANK YOU