2D Henon Map

The 2D Henon Map is similar to a real model of the forced nonlinear oscillator.The Purpose is The Investigation of The Period Doubling Transition to Chaos in The 2D Henon Map.

•Purpose•2D Henon Map : nnn yAxx

21 1

Eui-Sun LeeDepartment of PhysicsKangwon National University

nn xby 1

•Period doubling transition In the bifurcation diagram, the 2D Henon Map exhibits the period doubling transition to chaos.

3.0b

Bifurcation diagram

Periodic orbits

• Period-q orbit

.1

),(),(

)(,2

bxyAx

yxgyxf

zTyx

z

:)...)((...()( ,,, nbAbAnqbA zTTzT

.)( 21010 zzzzzz q

•The Fixed Point Problem: .*)(* , zTz qbA

• 2D Newton algorithm

).()( 11 kkkk zFzFzz

While( )Errorzz kk || 1

}

1 step2 step

)(1)(),()( ,, zTzFzTzzF qbA

qbA

1

0

1

0)()(

)()(

1)(1q

i

q

i

yzf

xzf

yzf

xzf

zT

.012

1001

)(1

0

q

i

kk b

zAzF

.0{ k

.1kk

•Period-q orbit:

Linear Stability Analysis

,1 nn M

., 211222112211 MMMMDetMMMTrM

• Eigenvalues of Jacobian Matrix M

• Jacobian Matrix M

The linear Stability of The Periodic Orbit Are Determined by The Eigenvalues λ of The Jacobian matrix M.

Where,

• Stability Analysis

If | λ |<1 , the periodic orbit is linearly stable.

If | λ |>1 or |λ|<1,|λ|>1, the periodic orbit is linearly unstable.

The Henon Map is linearized to Jacobian Matrix M at the period-q orbit( ) .The Determinant of The Jacobian Matrix Determine The Convergence of The Trajectories of Perturbation.

.012

2212

12111

0

*

MMMM

bzA

Mq

i

i

*z

.02 DetMTrM • Characteristic equation:

.21122211 MMMMDetM

,0))(( .}4{2/1 2 DetMTrMTrM

Stability diagram in the 2D Henon Map

02 DetMTrM

PDB(λ=-1) line : DetM= -TrM-1SNB(λ =1) line : DetM= TrM-1HB (| λ|=1) line : DetM= 1

In the Stability diagram, the stability of the periodic orbit is confirmed directly.

Stability diagram of the period-2 orbit

• Characteristic equation:

The stability multiplier is depend on both the trace (TrM) and determinant (DetM).

qbDetM

When the stability multiplier are complex number, they lie on the circle with radius inside the unit circle.

The period doubling bifurcation (PDB) occur when the stability value is pass through λ=-1 on the real axis.

Analysis of the Stability by Numerical Examples

DetM