2 D Henon Map EuiSun Lee Department of

2 D Henon Map Eui-Sun Lee Department of Physics Kangwon National University • Purpose • 2 D Henon Map : The 2 D 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 2 D Henon Map. • Period doubling transition In the bifurcation diagram, the 2 D Henon Map exhibits the period doubling transition to chaos. Bifurcation diagram

Periodic orbits • Period-q orbit: • The Fixed Point Problem: • 2 D Newton algorithm 1 step 2 step While( )

Linear Stability Analysis • Jacobian Matrix M 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. • Eigenvalues of Jacobian Matrix M The linear Stability of The Periodic Orbit Are Determined by The Eigenvalues λ of The Jacobian matrix M. • Characteristic equation: 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.

Stability diagram in the 2 D Henon Map In the Stability diagram, the stability of the periodic orbit is confirmed directly. The stability multiplier is depend on both the trace (Tr. M) and determinant (Det. M). • Characteristic equation: PDB(λ=-1) line : Det. M= -Tr. M-1 SNB(λ =1) line : Det. M= Tr. M-1 HB (| λ|=1) line : Det. M= 1 Stability diagram of the period-2 orbit

Analysis of the Stability by Numerical Examples 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.