Sinai–Ruelle–Bowen measure for normal form map of grazing bifurcations of impact oscillators

Grazing bifurcations in impact oscillators characterize the transition in asymptotic dynamics between impacting and nonimpacting motions. Several different grazing bifurcation scenarios under variations of a single system parameter have been previously documented in the literature. In the present paper, the transition between two characteristically different co-dimension-one grazing bifurcation scenarios is found to be associated with the presence of certain co-dimension-two grazing bifurcation points and their unfolding in parameter space. The analysis investigates the distribution of such degenerate bifurcation points along the grazing bifurcation manifold in examples of single-degree-of-freedom oscillators. Unfoldings obtained with the discontinuity-mapping technique are used to explore the possible influence on the global dynamics of the smooth co-dimension-one bifurcations of the impacting dynamics that emanate from such co-dimension-two points. It is shown that attracting impacting motion may result from parameter variations through a co-dimension-two grazing bifurcation of an initially unstable limit cycle in a nonlinear micro-electro-mechanical systems (MEMS) oscillator.

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Control of Chaotic Impacts

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127497000765 ◽

1997 ◽

Vol 07 (04) ◽

pp. 951-955 ◽

Cited By ~ 16

Author(s):

Fernando Casas ◽

Celso Grebogi

Keyword(s):

Periodic Orbit ◽

Unstable Periodic Orbit ◽

Controlling Chaos ◽

Impact Oscillators ◽

Universal Properties ◽

Grazing Bifurcations

We apply controlling chaos techniques to select the desired sequence of impacts in a map that captures universal properties of impact oscillators near grazing. For instance, we can choose the period and then stabilize an unstable periodic orbit with, say, one impact per period involved in the grazing bifurcations that take place in the system.

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Closing the GAP—A 5Å Scanning Microscope

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100061938 ◽

1969 ◽

Vol 27 ◽

pp. 6-7

Author(s):

A. V. Crewe

Keyword(s):

Normal Form ◽

The Other ◽

Resolving Power ◽

Scanning Microscope ◽

Conventional Microscope ◽

Other Hand ◽

Closing The Gap ◽

Thermal Sources ◽

Better Than ◽

Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

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