Cryptography – Key management – Having particular key generator
Statutory Invention Registration
1998-12-04
2001-03-06
Pihulic, Daniel T. (Department: 3662)
Cryptography
Key management
Having particular key generator
C703S007000
Statutory Invention Registration
active
H0001945
ABSTRACT:
FIELD OF THE INVENTION
In general, the present invention relates to a method and apparatus for sustaining chaos in a system. In particular, the present invention relates to a method and apparatus for preserving chaos in non-chaotic parameter regions by using the natural dynamics of the subject system to redirect flow towards the chaotic region along unstable manifolds of basin boundary saddles, by utilizing small, infrequent parameter perturbations.
BACKGROUND OF THE INVENTION
In many nonlinear systems, such as chaotic vibrations in structures or lasers, there exist regions in which much of the energy is present within a small range of frequencies. Typically, this occurs when the system is operating at or near resonance. Systems, such as aluminum wings or combustion engines, which operate at resonance, may fail due to repetitive stress caused by driving such a system near resonance.
Although numerous areas in science are now known to exhibit chaos as a natural occurrence, many situations would benefit from the inducement of chaos. In biology, the disappearance of chaos may signal pathological phenomena. In mechanics, chaos could be induced in order to prevent resonance, such as with the aluminum wings or combustion engines noted above. For example, in a system of coupled pendulums, one can excite chaotic motion of several modes to spread the energy over a wide frequency range. In optics, material damage is caused by lasers having a peak intensity at a given temporal frequency, so chaos is desirable since it has broadband spectra. It has also been suggested that chaos occur for normal machine tool cutting, making chaos preservation a desired control for deeper than normal cutting.
A conventional method maintains chaos in a regime where only chaotic transients exist, based on accurate analytical knowledge of the dynamical system, and requiring not only a priori phase space knowledge of escape regions from chaos, but also of preiterates of these regions. Such conventional methods maintain chaos using time series which require monitoring and adjusting the system prior to entering an escape region of the attractor by several iterates, an overly burdensome process.
Moreover, for the flow considered herein, preimages of sets in the escape regions cover much of the chaotic transient region, so monitoring all preimages, as done conventionally, would be tedious.
SUMMARY OF THE INVENTION
It is therefore an object of the present invention to provide a method and apparatus for controlling and optimizing the sustain of chaotic transients merely by examining behavior near a saddle.
According to a preferred embodiment of the present invention, such a method for sustaining chaos includes monitoring output parameters of a system, defining a saddle region containing a basin saddle for the system, and redirecting the output parameters toward a chaotic region when the output parameters enter the saddle region, by applying changes to input parameters of the system based on information describing the chaotic region.
According to another aspect of the present invention, a method for maintaining a stable state of a system includes monitoring an adjustable parameter of a system, obtaining information describing an unstable state of the system, and applying parameter perturbations to the system based on the obtained information describing the unstable state. The information describing the unstable state of the system may be the location of the unstable state. Obtaining the information describing the unstable state of the system may include finding a branch connecting a current, stable state of the system to the unstable state, and estimating the location of the unstable state based on the connecting branch. Applying parameter perturbations to the system based on the obtained information describing the unstable state may include implementing a target procedure to direct the system from the stable state toward the unstable state. Implementing a targeting procedure may include varying the adjustable parameter. The method may further include using the location of the unstable state as a reference for control of the system by the application of parameter perturbations. The method may also include determining the parameter perturbations based on the location of the unstable state. Preferably, the stable state is observed, and the unstable state is unobservable.
According to a further aspect of the invention, an apparatus for sustaining chaos includes means for monitoring output parameters of a system, means for defining a saddle region containing a basin saddle for the system, and means for redirecting the output parameters toward a chaotic region when the output parameters enter the saddle region. The latter means includes means for applying changes to input parameters of the system based on received information describing the chaotic region.
According to yet another aspect of the invention, an apparatus for maintaining a stable state of a system includes means for monitoring an adjustable parameter of a system, means for obtaining information describing an unstable state of the system, and means for applying perturbations to the adjustable parameter based on the obtained information describing the unstable state. The information describing the unstable state of the system may be the location of the unstable state. The means for obtaining the information describing the unstable state of the system may include means for finding a branch connecting a current, stable state of the system to the unstable state, and means for estimating the location of the unstable state based on the connecting branch. The means for applying perturbations to the adjustable parameter based on the obtained information describing the unstable state may include means for implementing a target procedure to direct the system from the stable state toward the unstable state. The apparatus may also include means for using the location of the unstable state as a reference for control of the system by the application of parameter perturbations. The apparatus may further include means for determining the parameter perturbations based on the location of the unstable state. The means for implementing a targeting procedure may include means for varying the adjustable parameter. Preferably, the stable state is observed, and the unstable state is unobservable.
These and other features and advantages of the invention will be better understood upon consideration of the following detailed description of the preferred embodiments, as illustrated by the accompanying drawings.
REFERENCES:
patent: 5402334 (1995-03-01), Pecora et al.
patent: 5442510 (1995-08-01), Schwartz et al.
patent: 5447520 (1995-09-01), Spano et al.
patent: 5486280 (1996-01-01), Bullock, IV et al.
patent: 5522863 (1996-06-01), Spano et al.
patent: 5706192 (1998-01-01), Schwartz et al.
patent: 5729388 (1998-03-01), Gershenfeld
Ira B. Schwartz and Ioana Triandaf, “Sustaining Chaos by Using Basin Boundary Saddles,” Physical Review Letters, vol. 77, No. 23, pp 4740-4743, Dec. 2, 1996.
Edelberg Barry A.
Miles Edward F.
Pihulic Daniel T.
The United States of America as represented by the Secretary of
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