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Finding Periodic Orbits and Their Stability Analysis

Eui-Sun Lee Department of Physics Kangwon National University. Finding Periodic Orbits and Their Stability Analysis. Poincar é Map(T). Period Doubling Transition To Chaos. The Poincar é Map of The PFP is obtained by The integration of the ordinary differential equation.

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Finding Periodic Orbits and Their Stability Analysis

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  1. Eui-Sun Lee Department of Physics Kangwon National University Finding Periodic Orbits and Their Stability Analysis • Poincaré Map(T) • Period Doubling Transition To Chaos The Poincaré Map of The PFP is obtained by The integration of the ordinary differential equation. The Poincare Map of The PFP Exhibits Period-doubling Transition To Chaos. Bifurcation diagram

  2. Linearized Map of The PFP • Linearized Differential Equations • The linear stability of fixed point is determined from the linearized-map matrix M of T. • M is obtained by the integration of the linearized differential equations over the q-period. • Perturbation: • The Two Variables Taylor expansion.

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

  4. Linear Stability Analysis • Linearized Map Matrix M The stability of the periodic orbit is determined by eigenvalues of M . • Characteristic equation of Matrix M: The Linear Stability of The Periodic Orbit Are Determined by The Eigenvalues λ of M. The Eigenvalues, and of M is Called Floquet stability multipliers. • 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.

  5. 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 Multiplier is pass through λ= -1 on the real axis. Stability of The Period-2

  6. Summary 1. With The Constant Jacobian Determinant , The Poincare Map of PFP is like the 2D-Henon Map. 2. The Periodic orbit of the PFP is Obtain by The 2D Newton Method. 3. The Period Doubling Bifurcation (PDB) occur when the Stability Multiplier is pass through λ= -1 on the real axis.

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