1. Confirmation of The Universality of The Critical Scaling behavior
Eui-Sun Lee
Department of Physics
Kangwon National University
Stability analysis
In the PFP, the stability of the periodic orbit is determined by the eigenvalues of the Linearized Map–Matrix(M) .
The eigenvalues of the M depend on the Trace and Determinant of the M
,where
about period-q orbit.
In the stability analysis of the periodic orbit,the real quantity,called the residue value(R) is more convenient than complex eigenvalues of the M.
SNB
PDB
Stable
Unstable
Unstable

2. Parameter Scaling Factor
Parameter scaling factors is obtained through the sequence of {An} at which the PDB
of period-2n orbit occurs(R=1).
Parameter Scaling Factor(γ=0.1,Ω=0.5) n
A n
δ n 1 … 2 … … 3 … … 4 … … 5 … … 6 … … 7 … … 8 …
The limit value agrees
well with that (4.669…)
for a 1D map
Accumulation point
Runge-Kutta steps=100

3. Orbital Scaling Factors Orbit(x,y) Scaling Factors(γ=0.1 ,Ω=0.5)
Orbital scaling factor is obtained through the sequence of the period –2n orbit
with a maximum distance from the nearest point for A=An in the full x-y space.
The limit values agree well with that (-2.5…) for the 1D Map
Orbit(x,y) Scaling Factors(γ=0.1 ,Ω=0.5) n
x n
α n,x 1 … 2 … … 3 … … 4 … … 5 … … 6 … … 7 … … 8 … n
y n
α n,y 1 … 2 … … 3 … … 4 … … 5 … … 6 … … 7 … … 8 …
Runge-Kutta steps=100

4. Confirmation of the Parameter and Orbital Scalings
A Sequence of Close-ups of The Bifurcation Diagram(γ=0.1 ,Ω=0.5)
1st Close-up
2nd Close-up
1st Close-up
2nd Close-up

5. Confirmation of the Chaotic Attractor Scalings
A Sequence of Close-ups of Attractors (γ=0.1, Ω=0.5)
1st Close-up
2nd Close-up

6. Parameter and Orbit Scaling Factors for γ=0.3 and Ω=0.5
The limit values agree well with that (4.66… ,-2.5… ) for 1D Map
Parameter(A) and Orbit(x,y) Scaling Factor n
A n δn 1 … 2 … … 3 … … 4 … … 5 … … 6 … … 7 … … 8 … n
x n
α n,x 1 … 2 … … 3 … … 4 … … 5 … … 6 … … 7 … … 8 … n
y n
α n,y 1 … 2 … … 3 … … 4 … … 5 … … 6 … … 7 … … 8 …
Runge-Kutta steps=100

7. Parameter and Orbit Scaling Factors for γ=0.5 and Ω=0.5
The limit values agree well with that (4.669… ,-2.5… ) for 1D Map
Parameter and Orbit(x,y) Scaling Factor n
A n
δ n 1 … 2 … … 3 … … 4 … … 5 … … 6 … 7 … n
x n
αn,x 1 … 2 … … 3 … … 4 … … 5 … … 6 … … 7 … n
y n
αn,y 1 … 2 … … 3 … … 4 … … 5 … … 6 … … 7 …
Runge-Kutta steps=100

8. Summary
1. In the PFP, the critical scaling behavior near the accumulation point of PDB’s
is the same as the 1D Map.
2.As the damping coefficient(γ) increases, the convergence rates of the parameter and
orbital scaling factors( ) of the PDB’s increase.