Order within chaos:
An attractor is the set approached asymptotically by solutions to a system of ordinary differential equations, after initial transients have relaxed. In chaotic (strange) attractors instabilities cause trajectories to appear noisy, with the statistical properties normally associated with random func...
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| Format: | Abschlussarbeit Buch |
| Sprache: | English |
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| Zusammenfassung: | An attractor is the set approached asymptotically by solutions to a system of ordinary differential equations, after initial transients have relaxed. In chaotic (strange) attractors instabilities cause trajectories to appear noisy, with the statistical properties normally associated with random functions. These chaotic trajectories are thus both random and deterministic; their proper description requires a synthesis of random signal analysis and the theory of differential equations. In this dissertation, I present several methods of quantifying the orderly properties of chaotic attractors, and use these methods to investigate several examples where order and chaos coexist. This dissertation is divided into four chapters. The first two discuss spatial order in the phase space. Chapter I discusses the microscopic structure present in the probability distributions of chaotic attractors, and introduces the information dimension as a means of quantifying this structure. In Chapter II a detailed study is made of the attractors of an infinite dimensional system (a differential delay equation). Computations of the spectrum of Lyapunov exponents demonstrate that as the delay parameter is increased, the dimension of these chaotic attractors increases but nevertheless remains finite. Chapters III and IV concern temporal order. This temporal order is manifested as sharp peaks in a power spectrum, shown in Chapter III to arise from coherence in the phase of solutions on the attractor. In Chapter IV the rate of phase diffusion c is introduced as a means of quantifying this temporal order. For period doubling sequences c exhibits universal scaling properties. |
| Beschreibung: | Power spectra and mixing properties of strange attractors / Doyne Farmer [and others]: leaves l63-182 |
| Beschreibung: | IX, 196 Blätter Illustrationen 29 cm |
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| 100 | 1 | |a Farmer, J. Doyne |d 1952- |e Verfasser |0 (DE-588)171854373 |4 aut | |
| 245 | 1 | 0 | |a Order within chaos |c by James Doyne Farmer, Jr. |
| 246 | 1 | 3 | |a Power spectra and mixing properties of strange attractors |
| 300 | |a IX, 196 Blätter |b Illustrationen |c 29 cm | ||
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| 500 | |a Power spectra and mixing properties of strange attractors / Doyne Farmer [and others]: leaves l63-182 | ||
| 502 | |b Dissertation |c University of California, Santa Cruz |d 1981 | ||
| 520 | 3 | |a An attractor is the set approached asymptotically by solutions to a system of ordinary differential equations, after initial transients have relaxed. In chaotic (strange) attractors instabilities cause trajectories to appear noisy, with the statistical properties normally associated with random functions. These chaotic trajectories are thus both random and deterministic; their proper description requires a synthesis of random signal analysis and the theory of differential equations. In this dissertation, I present several methods of quantifying the orderly properties of chaotic attractors, and use these methods to investigate several examples where order and chaos coexist. This dissertation is divided into four chapters. The first two discuss spatial order in the phase space. Chapter I discusses the microscopic structure present in the probability distributions of chaotic attractors, and introduces the information dimension as a means of quantifying this structure. In Chapter II a detailed study is made of the attractors of an infinite dimensional system (a differential delay equation). Computations of the spectrum of Lyapunov exponents demonstrate that as the delay parameter is increased, the dimension of these chaotic attractors increases but nevertheless remains finite. Chapters III and IV concern temporal order. This temporal order is manifested as sharp peaks in a power spectrum, shown in Chapter III to arise from coherence in the phase of solutions on the attractor. In Chapter IV the rate of phase diffusion c is introduced as a means of quantifying this temporal order. For period doubling sequences c exhibits universal scaling properties. | |
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Datensatz im Suchindex
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| index_date | 2024-07-03T21:59:11Z |
| indexdate | 2024-07-10T09:51:12Z |
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| spelling | Farmer, J. Doyne 1952- Verfasser (DE-588)171854373 aut Order within chaos by James Doyne Farmer, Jr. Power spectra and mixing properties of strange attractors IX, 196 Blätter Illustrationen 29 cm txt rdacontent n rdamedia nc rdacarrier Power spectra and mixing properties of strange attractors / Doyne Farmer [and others]: leaves l63-182 Dissertation University of California, Santa Cruz 1981 An attractor is the set approached asymptotically by solutions to a system of ordinary differential equations, after initial transients have relaxed. In chaotic (strange) attractors instabilities cause trajectories to appear noisy, with the statistical properties normally associated with random functions. These chaotic trajectories are thus both random and deterministic; their proper description requires a synthesis of random signal analysis and the theory of differential equations. In this dissertation, I present several methods of quantifying the orderly properties of chaotic attractors, and use these methods to investigate several examples where order and chaos coexist. This dissertation is divided into four chapters. The first two discuss spatial order in the phase space. Chapter I discusses the microscopic structure present in the probability distributions of chaotic attractors, and introduces the information dimension as a means of quantifying this structure. In Chapter II a detailed study is made of the attractors of an infinite dimensional system (a differential delay equation). Computations of the spectrum of Lyapunov exponents demonstrate that as the delay parameter is increased, the dimension of these chaotic attractors increases but nevertheless remains finite. Chapters III and IV concern temporal order. This temporal order is manifested as sharp peaks in a power spectrum, shown in Chapter III to arise from coherence in the phase of solutions on the attractor. In Chapter IV the rate of phase diffusion c is introduced as a means of quantifying this temporal order. For period doubling sequences c exhibits universal scaling properties. Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Mathematical physics (DE-588)4113937-9 Hochschulschrift gnd-content Mathematische Physik (DE-588)4037952-8 s DE-604 |
| spellingShingle | Farmer, J. Doyne 1952- Order within chaos Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4037952-8 (DE-588)4113937-9 |
| title | Order within chaos |
| title_alt | Power spectra and mixing properties of strange attractors |
| title_auth | Order within chaos |
| title_exact_search | Order within chaos |
| title_exact_search_txtP | Order within chaos |
| title_full | Order within chaos by James Doyne Farmer, Jr. |
| title_fullStr | Order within chaos by James Doyne Farmer, Jr. |
| title_full_unstemmed | Order within chaos by James Doyne Farmer, Jr. |
| title_short | Order within chaos |
| title_sort | order within chaos |
| topic | Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Mathematische Physik Hochschulschrift |
| work_keys_str_mv | AT farmerjdoyne orderwithinchaos AT farmerjdoyne powerspectraandmixingpropertiesofstrangeattractors |