1. Tricritical Behavior of Period Doubling in Unidirectionally Coupled Maps
W. Lim and S.-Y. Kim
Department of Physics
Kangwon National University
 Unidirectionally-coupled 1D maps:
 Basic bifurcation structure
Two PDB lines D8 along with the
SNB lines S8 emanating from a cusp
exists inside the PDB line D4.
Two superstable lines cross twice
at the doubly superstable and
bistability points.

2. Periodic Orbits on The Left Superstable line
Orbits on the superstable line 8d have the critical orbit point zd=(-0.014…,0),
where the Jacobian determinant DT becomes zero.
Four-times iteration maps: 1D map behavior in the second subsystem
Quadratic maximum
Quartic maximum
Quadratic maximum
Four-times iteration map has the quartic maximum at the doubly superstable point.

3. Periodic Orbits on The Right Superstable line
Orbits on the superstable line 8u have the critical orbit point zu=(0.883…,0),
where the Jacobian determinant DT becomes zero.
Eight-times iteration map has the quartic maximum at the doubly superstable point.

4. Doubly Superstable Points
Orbits on the doubly superstable point contains both zd and zu.
Binary tree of the doubly superstable point
Types of doubly superstable periodic orbits
are defined in terms of the numbers of
iterations between two critical orbit points.
r-times iterations
s-times iterations
zd zu.
Doubly superstable period-q (=r+s) of type (r,s)
Squences of the doubly superstable points
with […(L,)] or […(R,)]
 Tricritical point

5. Tricritical Scaling Behavior
Tricritical point lies near an edge of the complicated parts of the boundary
of chaos, and at an end of a Feigenbaum’s critical line.
 Tricritical scaling behavior associated with
the sequence of [(L,)]
Tricritical point
B*= …, C*= …
Parameter scaling factors
1= (obtained from the sequence
of doubly superstable points)
2=2.867 (obtained from the bistability point)
Orbital scaling factors =

6. Eigenvalue-matching Renormalization-group (RG) Analysis
Recurrence relation of level n and n+1:
Linearizing abouat fixed point (B*,C*)
of the RG equation
Eigenvalues of n, 1,n and 2,n:
Parameter scaling factors along the
eigendirections 1 and 2
 Tricritical scaling behaviors obtained through the eigenvalue-matching RG analysis
Tricritical point: B*= …, C*= …
Critical stability multiplier:
Parameter scaling factors: 1= , 2=
Orbital scaling factors: =

7. Self-similar Topography near The Tricritical Point
Relation between the physical coordinate (B, C)=(B-B*,C-C*)
and the scaling coordinate (C1,C2)
Normalized eigenvectors along the eigendirections

8. Summary
 The second response subsystem in the unidirectionally-coupled 1D map
exhibits the tricritical behavior.
By using the eigenvalue-matching RG analysis, we analysis the scaling
behaviors near the tricritical point.
These RG results agree well with those obtained by a direct numerical method.