Classification of bifurcation diagrams for elliptic equations with exponential growth in a ball

2014 ◽  
Vol 194 (4) ◽  
pp. 931-952 ◽  
Author(s):  
Yasuhito Miyamoto
Keyword(s):  
Exponential Growth ◽  
Elliptic Equations ◽  
Bifurcation Diagrams

ELLIPTIC EQUATIONS IN ℝ2 WITH ONE-SIDED EXPONENTIAL GROWTH

2004 ◽  
Vol 06 (06) ◽  
pp. 947-971 ◽  
Author(s):  
ZHITAO ZHANG ◽  
MARTA CALANCHI ◽  
BERNHARD RUF
Keyword(s):  
Exponential Growth ◽  
Elliptic Equations ◽  
Morse Index ◽  
Linear Growth ◽  
Critical Growth ◽  
The Other ◽  
Critical Groups ◽  
Bounded Domains ◽  
Eigen Value

We consider elliptic equations in bounded domains Ω⊂ℝ2 with nonlinearities which have exponential growth at +∞ (subcritical and critical growth, respectively) and linear growth λ at -∞, with λ>λ1, the first eigen value of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms; one solution is negative, the other one is sign-changing. Some critical groups and Morse index of these solutions are given. Also the case λ<λ1 is considered.

Classification of periodic and chaotic passive limit cycles for a compass-gait biped with gait asymmetries

Robotica ◽  
2011 ◽  
Vol 29 (7) ◽  
pp. 967-974 ◽  
Author(s):  
Jae-Sung Moon ◽  
Mark W. Spong
Keyword(s):  
Limit Cycles ◽  
Period Doubling ◽  
Mapping Method ◽  
Stable Limit ◽  
Bifurcation Diagrams ◽  
Cell Mapping ◽  
Cyclic Patterns ◽  
Stable Limit Cycles ◽  
Qualitative Changes

In this paper we study the problem of passive walking for a compass-gait biped with gait asymmetries. In particular, we identify and classify bifurcations leading to chaos caused by the gait asymmetries because of unequal leg masses. We present bifurcation diagrams showing step period versus the ratio of leg masses at various walking slopes. The cell mapping method is used to find stable limit cycles as the parameters are varied. It is found that a variety of bifurcation diagrams can be grouped into six stages that consist of three expanding and three contracting stages. The analysis of each stage shows that marginally stable limit cycles exhibit period-doubling, period-remerging, and saddle-node bifurcations. We also show qualitative changes regarding chaos, i.e., generation and extinction of chaos follow cyclic patterns in passive dynamic walking.

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