1. Tricritical Behavior of Period Doublings in Unidirectionally Coupled Maps
W. Lim and S.-Y. Kim
Department of Physics
Kangwon National University
A=1.3 (drive system: period-4 behavior)
 Unidirectionally-coupled 1D maps:
 Basic Bifurcation Structure
Two superstable lines, 8d and 8u
with critical orbit points (zd, 0) and (zu,0), cross twice at the doubly superstable point and bistability point

2. Four-Times Iterated Maps on The Superstable lines
A Quartic Maximum at the doubly superstable point
 Four-times iterated maps on the left superstable line 8d
Quadratic maximum
Quartic maximum
Quadratic maximum
 Four-times iterated maps on the right superstable line 8u
Quartic maximum through interaction
of two quadratic maxima
Quadratic maximum
Quadratic maximum 2

3. Tricritical Scaling Behavior
Following a sequence of doubly superstable points or bistability points, one can arrive at a Tricritical Point lying at an end of a Feigenbaum’s critical line (associated with period-doubling bifurcations in the usual quadratic-maximum case) and near an edge of the complicated parts (multistability associated with saddle-node bifurcations) of the boundary of chaos.
 Tricritical scaling behavior
Tricritical point:
B*= …, C*= …
Parameter scaling factors:
1= (obtained from the sequence
of doubly superstable points)
2=2.867 (obtained from the sequence of
bistability points)
Orbital scaling factor: =

4. Eigenvalue-matching Renormalization-group (RG) Analysis
Recurrence relations for the successive three
levels n, n+1, and n+2: ’ ’ ’ ’
Fixed points of the RG Eq.
 Tricritical Point
Linearizing the RG Eq. about the fixed point,
we have ’ ’
Eigenvalues of n, 1,n and 2,n:
Parameter scaling factors along the
eigendirections 1 and 2
 Scaling fractos obtained through the eigenvalue-matching RG analysis
(They agree well with those obtained by a direct numerical method.)
Tricritical point: B*= …, C*= …
Parameter scaling factors: 1= , 2=
Orbital scaling factors: =

5. 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

6. Summary
 The second response subsystem in the unidirectionally-coupled 1D map
exhibits the tricritical behavior.
By using the eigenvalue-matching RG method, we analyze the scaling
behaviors near the tricritical point that has two relevant eigenvalues
1 (= ) and 2 (= ).
These RG results agree well with those obtained by a direct numerical method.

7. 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)
Sequences of the doubly superstable points
with […(L,)] or […(R,)]
 Tricritical point