Chapter 2 Equivalent Systems

In this chapter, we describe how highly erratic dynamic behavior can arise from a nonlinear logistic map, and how this apparently random behavior is governed by a surprising order. With this lesson in mind, we should not be overly surprised that highly erratic and random appearing observed data might also be generated by parsimonious deterministic dynamic systems. At a minimum, we contend that researchers should apply NLTS to test for this possibility. We also introduced tools to analyze dynamic behavior that form the foundation for NLTS. In particular, we have stressed the quite unexpected capability to achieve some form of predictability even with only one trajectory at hand. In subsequent chapters, we treat known nonlinear dynamical systems as unknown, and investigate how NLTS methods rely on a single solution (or multiple solutions) generated by them to reconstruct equivalent systems. This is a conventional approach in the literature for seeing how NLTS methods work since we know what needs to be reconstructed.

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Partial aperture imaging system based on sparse point spread holograms and nonlinear cross-correlations

Scientific Reports ◽

10.1038/s41598-020-77912-3 ◽

2020 ◽

Vol 10 (1) ◽

Author(s):

Angika Bulbul ◽

Joseph Rosen

Keyword(s):

Structural Changes ◽

Imaging System ◽

Current System ◽

Three Dimensional ◽

Phase Mask ◽

Coded Aperture ◽

Space Telescopes ◽

Reconstruction Process ◽

Cross Correlations ◽

Equivalent Systems

AbstractPartial aperture imaging system (PAIS) is a recently developed concept in which the traditional disc-shaped aperture is replaced by an aperture with a much smaller area and yet its imaging capabilities are comparable to the full aperture systems. Recently PAIS was demonstrated as an indirect incoherent digital three-dimensional imaging technique. Later it was successfully implemented in the study of the synthetic marginal aperture with revolving telescopes (SMART) to provide superresolution with subaperture area that was less than one percent of the area of the full synthetic disc-shaped aperture. In the study of SMART, the concept of PAIS was tested by placing eight coded phase reflectors along the boundary of the full synthetic aperture. In the current study, various improvements of PAIS are tested and its performance is compared with the other equivalent systems. Among the structural changes, we test ring-shaped eight coded phase subapertures with the same area as of the previous circular subapertures, distributed along the boundary of the full disc-shaped aperture. Another change in the current system is the use of coded phase mask with a point response of a sparse dot pattern. The third change is in the reconstruction process in which a nonlinear correlation with optimal parameters is implemented. With the improved image quality, the modified-PAIS can save weight and cost of imaging devices in general and of space telescopes in particular. Experimental results with reflective objects show that the concept of coded aperture extends the limits of classical imaging.

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On the equivalence of systems of different sizes, with applications to system comparisons

Advances in Applied Probability ◽

10.1017/apr.2016.3 ◽

2016 ◽

Vol 48 (2) ◽

pp. 332-348 ◽

Cited By ~ 10

Author(s):

Bo H. Lindqvist ◽

Francisco J. Samaniego ◽

Arne B. Huseby

Keyword(s):

Optimization Problem ◽

Coherent System ◽

Sufficient Condition ◽

Equivalent System ◽

Coherent Systems ◽

Two Systems ◽

Equivalent Systems ◽

Engineered Systems ◽

Signature Vector ◽

Mixed Systems

Abstract The signature of a coherent system is a useful tool in the study and comparison of lifetimes of engineered systems. In order to compare two systems of different sizes with respect to their signatures, the smaller system needs to be represented by an equivalent system of the same size as the larger system. In the paper we show how to construct equivalent systems by adding irrelevant components to the smaller system. This leads to simpler proofs of some current key results, and throws new light on the interpretation of mixed systems. We also present a sufficient condition for equivalence of systems of different sizes when restricting to coherent systems. In cases where for a given system there is no equivalent system of smaller size, we characterize the class of lower-sized systems with a signature vector which stochastically dominates the signature of the larger system. This setup is applied to an optimization problem in reliability economics.

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