Stability of critical points under small perturbations part II: Analytic theory

1973 ◽  
Vol 8 (1) ◽  
pp. 69-92 ◽  
Author(s):  
Michael Reeken
Keyword(s):  
Critical Points ◽  
Analytic Theory ◽  
Small Perturbations

ANALYTIC CENTER OF NILPOTENT CRITICAL POINTS

2012 ◽  
Vol 22 (08) ◽  
pp. 1250198 ◽  
Author(s):  
TAO LIU ◽  
LIANGGANG WU ◽  
FENG LI
Keyword(s):  
Critical Points ◽  
Necessary Conditions ◽  
Computation Method ◽  
Analytic Center ◽  
Center Problem ◽  
Third Order ◽  
Small Perturbations ◽  
Integrating Factor ◽  
Sufficient And Necessary Conditions ◽  
Lyapunov Constants

For third-order nilpotent critical points of a planar dynamical system, the analytic center problem is completely solved in this article by using the integrating factor method. The associated quasi-Lyapunov constants are defined and their computation method is given. For a class of cubic-order systems under small perturbations, sufficient and necessary conditions for an analytic center are obtained.

scholarly journals Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators

2002 ◽  
Vol 165 ◽  
pp. 123-158 ◽  
Author(s):  
Alano Ancona
Keyword(s):  
Harmonic Function ◽  
Elliptic Operator ◽  
Critical Points ◽  
Harmonic Functions ◽  
Elliptic Operators ◽  
Second Order ◽  
Small Perturbations ◽  
Order Elliptic Operator ◽  
Symmetry Assumption ◽  
Second Order Elliptic Operators

Let M be a manifold and let L be a sufficiently smooth second order elliptic operator in M such that (M, L) is a transient pair. It is first shown that if L is symmetric with respect to some density in M, there exists a positive L-harmonic function in M which dominates L-Green’s function at infinity. Other classes of elliptic operators are investigated and examples are constructed showing that this property may fail if the symmetry assumption is removed. Another part of the paper deals with the existence of critical points for certain L-harmonic functions with periodicity properties. A class of small perturbations of second order elliptic operators is also described.

ON THIRD-ORDER NILPOTENT CRITICAL POINTS: INTEGRAL FACTOR METHOD

2011 ◽  
Vol 21 (05) ◽  
pp. 1293-1309 ◽  
Author(s):  
YIRONG LIU ◽  
JIBIN LI
Keyword(s):  
Dynamical System ◽  
Critical Points ◽  
Small Amplitude ◽  
Computation Method ◽  
Third Order ◽  
Small Perturbations ◽  
Factor Method ◽  
Cubic Systems ◽  
Nilpotent Critical Point ◽  
Lyapunov Constants

For third-order nilpotent critical points of a planar dynamical system, the problem of characterizing its center and focus is completely solved in this article by using the integral factor method. The associated quasi-Lyapunov constants are defined and their computation method is given. For a class of cubic systems under small perturbations, it is proved that there exist eight small-amplitude limit cycles created from a nilpotent critical point.

The relationship of the scleractinian corals to the rugose corals

Paleobiology ◽  
1980 ◽  
Vol 6 (02) ◽  
pp. 146-160 ◽  
Author(s):  
William A. Oliver
Keyword(s):  
Critical Points ◽  
Scleractinian Corals ◽  
Convincing Evidence ◽  
Early Triassic ◽  
Rugose Corals ◽  
History Of ◽  
The Subject ◽  
Relationship Of ◽  
The Relationship ◽  
Biologic Factors

The Mesozoic-Cenozoic coral Order Scleractinia has been suggested to have originated or evolved (1) by direct descent from the Paleozoic Order Rugosa or (2) by the development of a skeleton in members of one of the anemone groups that probably have existed throughout Phanerozoic time. In spite of much work on the subject, advocates of the direct descent hypothesis have failed to find convincing evidence of this relationship. Critical points are:(1) Rugosan septal insertion is serial; Scleractinian insertion is cyclic; no intermediate stages have been demonstrated. Apparent intermediates are Scleractinia having bilateral cyclic insertion or teratological Rugosa.(2) There is convincing evidence that the skeletons of many Rugosa were calcitic and none are known to be or to have been aragonitic. In contrast, the skeletons of all living Scleractinia are aragonitic and there is evidence that fossil Scleractinia were aragonitic also. The mineralogic difference is almost certainly due to intrinsic biologic factors.(3) No early Triassic corals of either group are known. This fact is not compelling (by itself) but is important in connection with points 1 and 2, because, given direct descent, both changes took place during this only stage in the history of the two groups in which there are no known corals.

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