scholarly journals Polynomial diffeomorphisms of C2: V. critical points and Lyapunov exponents

1998 ◽  
Vol 8 (3) ◽  
pp. 349-383 ◽  
Author(s):  
Eric Bedford ◽  
John Smillie
Keyword(s):  
Lyapunov Exponents ◽  
Critical Points ◽  
Polynomial Diffeomorphisms

scholarly journals Invariant measures for interval maps without Lyapunov exponents

2021 ◽  
pp. 1-29
Author(s):  
JORGE OLIVARES-VINALES
Keyword(s):  
Lyapunov Exponent ◽  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Critical Points ◽  
Invariant Measures ◽  
Full Measure ◽  
Interval Maps ◽  
Interval Map

Abstract We construct an invariant measure for a piecewise analytic interval map whose Lyapunov exponent is not defined. Moreover, for a set of full measure, the pointwise Lyapunov exponent is not defined. This map has a Lorenz-like singularity and non-flat critical points.

scholarly journals Experimentally Viable Techniques for Accessing Coexisting Attractors Correlated with Lyapunov Exponents

2021 ◽  
Vol 11 (21) ◽  
pp. 9905
Author(s):  
Joshua Ray Hall ◽  
Erikk Kenneth Tilus Burton ◽  
Dylan Michael Chapman ◽  
Donna Kay Bandy
Keyword(s):  
Lyapunov Exponents ◽  
Critical Points ◽  
Control Parameter ◽  
Experimental System ◽  
Coexisting Attractors ◽  
Shift Method ◽  
Large Domain ◽  
Lag Effects ◽  
Power Shift ◽  
Small Domain

Universal, predictive attractor patterns configured by Lyapunov exponents (LEs) as a function of the control parameter are shown to characterize periodic windows in chaos just as in attractors, using a coherent model of the laser with injected signal. One such predictive pattern, the symmetric-like bubble, foretells of an imminent bifurcation. With a slight decrease in the gain parameter, we find the symmetric-like bubble changes to a curved trajectory of two equal LEs in one attractor, while an increase in the gain reverses this process in another attractor. We generalize the power-shift method for accessing coexisting attractors or periodic windows by augmenting the technique with an interim parameter shift that optimizes attractor retrieval. We choose the gain as our parameter to interim shift. When interim gain-shift results are compared with LE patterns for a specific gain, we find critical points on the LE spectra where the attractor is unlikely to survive the gain shift. Noise and lag effects obscure the power shift minimally for large domain attractors. Small domain attractors are less accessible. The power-shift method in conjunction with the interim parameter shift is attractive because it can be experimentally applied without significant or long-lasting modifications to the experimental system.

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.

Lyapunov Exponents

2016 ◽  
Author(s):  
Arkady Pikovsky ◽  
Antonio Politi
Keyword(s):  
Lyapunov Exponents
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