Tricritical Behavior of Period Doubling in Unidirectionally Coupled Maps

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 zdzu. 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*=1.189 249 …, C*=0.699 339 … Parameter scaling factors 1=7.284 68 (obtained from the sequence of doubly superstable points) 2=2.867 (obtained from the bistability point) Orbital scaling factors =-1.690 30

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*=1.189 429 …, C*=0.699 339 … Critical stability multiplier: Parameter scaling factors: 1=7.284 685, 2=2.857 124 Orbital scaling factors: =-1.690 304

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.