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Period-doubling critical behavior

T - Tricritical



1D maps
The period-doubling accumulation point in unimodal map of 4-th power. For bimodal 1D maps - the limit of period-doubling on a special curve in the parameter plane defined by the condition "extremum is mapped to extremum". T-point appears as the terminal point of the Feigenbaum curve. This type of critical behavior is known after Chang, Wortis, Wright, and Fraser and Kapral.

More general systems
T-point may appear generically only in codimension 3. In some cases the pseudo-tricritical behavior may occur, as an intermediate asymptotics.

RG equation

The fixed point

The orbital scaling factor

Critical multiplier

Relevant eigenvalues

CoDim=3 (restr. 2)

Codimension-2 example

Param. space arrangement and scaling with factors

Scaling coordinates

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Codimension-3 example
tricritical point at

Codimension-3 example in 2D invertible map
For D=0.3 the tricritical point is located at

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Saratov group
of theoretical nonlinear
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