A variety of period-doubling universality classes
in multi-parameter analysis of transition to chaos
A.P. Kuznetsov, S.P. Kuznetsov and I.R. Sataev
Institute of Radio-Engineering & Electronics, Saratov, Russia
Considering period-doubling transition to chaos in multi-parameter systems one must account a possibility of non-Feigenbaum scaling behavior at some special paths in the parameter space. There are two possibilities for arising such anomaly. First, the dynamics at the onset of chaos may remain essentially one-dimensional, but the 1D map is distorted in such a way that leaves the Feigenbaum's universality class. Alternatively, a new mode may come to the threshold of instability and increase the effective dimension of the dynamics. We present here a list of 1D and 2D maps representing different classes of the period-doubling quantitative universality. The description of the Feigenbaum's behavior serves as the pattern for presentation of all other data.
* Type F ("Feigenbaum's"): 
; 
; orbit scaling factor
; parameter space scaling factor: 
.
* Type T ("tricritical"): 
; 
, 
;
; 
, 
.
* Type S ("six power"): 
; 
, 
,
; 
; 
, 
, 
.
* Type S': ; 
, 
, 
;
; , 
, 
.
* Type E ("eight power"): ; 
, 
,
; 
; 
, 
, 
.
* Type H ("Hamiltonian"): 
, 
; 
, 
;
, 
; 
, 
.
* Type B ("bicritical"): 
, 
; ,
; 
, 
; 
, 
.
* Type BT ("bi-tricritical"): , 
; 
,
; 
, 
; , 
, 
.
* Type C ("cycle"): 
, 
; 
, 
;
[for period-quadrupling]: 
, 
;
, 
.
* Type FQ: 
, 
; 
, 
;
, 
; 
,
.
Details of critical dynamics of all types are considered, possibilities of their observation and significance for understanding complex nonlinear phenomena are discussed. The work is supported by Russian Fund of Fundamental Research (grant 95-02-05818).