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

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

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*=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 sequence of bistability points) Orbital scaling factor: =-1.690 30

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

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

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 (=7.284 685) and 2 (=2.857 124). These RG results agree well with those obtained by a direct numerical method.

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