2. A graphical zoo of strange and peculiar attractors

Holden, Arun V.; Muhamad, Mudiana

doi:10.1515/9781400858156.15

Cited by 20 publications

(11 citation statements)

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“…The above class includes many systems displaying rich nonlinear dynamics (period-doubling cascade and chaotic behavior) such as the single mode CO 2 laser [9] and the Brusselator system [10], which are not contained in the class of Lur'e systems recently investigated in [8]. Note that system (20.2.1) reduces to a Lur'e system if f(x) = Ax + Bn(Cx), where n : R → R is a scalar function.…”

Section: Problem Formulation and Preliminary Resultsmentioning

confidence: 99%

“…This chapter considers the same problem for a much larger class of sinusoidally forced nonlinear systems, containing meaningful examples which exhibit rich nonlinear dynamics, such as the single-mode CO 2 laser [9] or the Brusselator system [10], and do not fit into the previously considered Lur'e class. A synthesis technique is proposed by mixing results concerning absolute stability of nonlinear systems and robustness of uncertain linear systems.…”

Section: Introductionmentioning

confidence: 99%

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Controller Synthesis for Periodically Forced Chaotic Systems

Basso¹,

Genesio²,

Giovanardi

2002

World Scientific Series on Nonlinear Science Series B

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Delayed feedback controllers are an appealing tool for stabilization of periodic orbits in chaotic systems. Despite their conceptual simplicity, specific and reliable design procedures are difficult to obtain, partly also because of their inherent infinite-dimensional structure. This chapter considers the use of finite dimensional linear time invariant controllers for stabilization of periodic solutions in a general class of sinusoidally forced nonlinear systems. For such controllers -which can be interpreted as rational approximations of the delayed ones -we provide a computationally attractive synthesis technique based on Linear Matrix Inequalities (LMIs), by mixing results concerning absolute stability of nonlinear systems and robustness of uncertain linear systems. The resulting controllers prove to be effective for chaos suppression in electronic circuits and systems, as shown by two different application examples.

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Section: Problem Formulation and Preliminary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Controller Synthesis for Periodically Forced Chaotic Systems

Basso¹,

Genesio²,

Giovanardi

2002

World Scientific Series on Nonlinear Science Series B

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“…In addition, Hayek and Wilson (2013, Table 3) showed that the species affected by the turnover differed between the peaks in ATI, such that one would not expect to find identical faunas in all glacials or all interglacials. This implies that glacial and interglacial conditions act as attractors comparable to a butterfly-shaped representation of the Lorenz attractor of Chaos Theory (Figure 1), in which minor differences in the initial conditions of any two or more systems (such as at, say, the onset of glacials or interglacials) prevent the perfect convergence of those systems onto a point representing a single state (Schaffer 1985, Holden and Muhamad 1986, Chang 2013. Thus, the details of the abyssal foraminiferal assemblage in one glacial cannot be predicted from those of the assemblage in the preceding glacial.…”

Section: Ontological Decisions and The Use Of Paleontological Data Fomentioning

confidence: 99%

Ontology Confounds Reproducibility in Ecology and Climate Science

Wilson¹,

Hayek²

2014

Life: The Excitement of Biology

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Abstract:The ability to reproduce ecological or climate change experiments or quantitative results is a hallmark of science, but not the only one. In addition, a scientist who subsequently examines original research must be able to reproduce the interpretation of the observations. Confidence in both ecology and climate science research is undermined when an a priori interpretation cannot be confirmed. This will happen when an a priori (before investigation) assumption was introduced by the original author(s), even inadvertently, regarding the nature of the universe or a subset of it. Such an assumption is here termed an ontological assumption when it then is treated as invariant; that is, it is not changed as unsupportive evidence, whether positive or negative, is accumulated. It thus conditions the interpretation of observed scientific phenomena. The results of the scientific investigation are therefore interpreted within the framework of this assumption, even though this framework is at variance with actual results obtained.Recent decades have seen increasing use of paleoclimatic studies as predictors of what will happen as current global climate change unfolds. Here we consider how the results from impact studies of paleoclimatic change on biotic communities have been used as a basis for neoclimatic and ecological research and prediction. We show that in some cases at least the observed results of these paleoclimatic studies were not in accord with the original interpretations, the paleoclimatic workers having held onto unwarranted ontological assumptions in the face of observations to the contrary. This practice is a threat to validity and undermines the applicability of these original paleonvironmental studies to later neoenvironmental work. For example, contrary to expectations, Pleistocene glacial-interglacial cycles did not always impact on community diversity, whether measured by species richness or the information function. Likewise, the quantitative assemblage turnover index (ATI) shows that in the greenhouse world of the Eocene, communities actually were little impacted by lithological changes associated with quantitatively-determined climatic perturbations. We demonstrate that the application of simple statistical measures can draw attention to unwarranted paleoecological ontological assumptions and avoid their being applied inappropriately to neoecological work.

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“…In fact, one can imagine a catalogue of relevant chaos modules in S-system form, similar to the one collated for monotonic and oscillatory responses (Morrison 1992). Such a catalogue would present modules classified by mathematical properties, such as dissipation, conservation, and structure of their maps, as well as phenomenological aspects, as pictured in Holden's and Muhamad's "graphical zoo of strange and peculiar attractors" (Holden and Muhamad 1986). Equipped with such a catalogue, the investigator could select an input module that would best mimic the general type of fluctuations observed in the environmental system of interest and then customize it to satisfy quantitative requirements such as its frequency, amplitude, center, and other statistical properties.…”

Section: Deterministic Chaos Offers a Valuable Modelling Toolmentioning

confidence: 99%

S‐system modelling of complex systems with chaotic input

Voit

1993

Environmetrics

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Environmental systems are characterized by large numbers of constituents and processes at hierarchical levels of organization. These levels range from elemental chemical and physical phenomena to multi-faceted ecosystems that are subject to natural and anthropogenic influences. The analysis of environmental systems is complicated because the governing processes are usually complex and ill defined. In addition to the structural complexity of the investigated phenomena themselves, environmental systems are difficult to analyze because they are constantly exposed to inputs that appear to fluctuate in a chaotic fashion. S-system models have the potential to address this situation. They are sets of nonlinear ordinary differential equations that were developed as representations for organizationally complex models, primarily in biology and biochemistry. They are characterized by a mathematical structure that allows efficient symbolic and numerical analysis of key features such as steady states, stabilities, sensitivities, and gains. At the same time, S-systems are structurally rich enough to capture virtually all relevant continuous nonlinearities. This paper begins with a brief review of two key features of S-systems: modelling and simulation based on powerlaw approximation; and the transformation method of recasting which allows differential equations to be formulated exactly as S-systems. The paper then discusses how chaotically fluctuating input can be simulated with a recast S-system based on deterministic chaos and how this recast S-system can be used as an input module for environmental phenomena that are represented as S-system models via power-law approximation.

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2. A graphical zoo of strange and peculiar attractors

Cited by 20 publications

References 0 publications

Controller Synthesis for Periodically Forced Chaotic Systems

Controller Synthesis for Periodically Forced Chaotic Systems

Ontology Confounds Reproducibility in Ecology and Climate Science

S‐system modelling of complex systems with chaotic input

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