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

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

3. Linear Stability Analysis 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, 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. Characteristic equation:

4. Stability diagram in the 2D Henon Map PDB(λ=-1) line : DetM= -TrM-1 SNB(λ =1) line : DetM= TrM-1 HB (| 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).

5. 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