Strange Attractors and Competitive Modes in Dynamical Systems
Explore the localization of chaotic strange attractors in multiparameter nonlinear dynamical systems through competitive modes. Discover the fractal nature of strange attractors, chaos sensitivity, and the role of competitive modes in system behavior analysis.
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Presentation Transcript
Localizing the Chaotic Strange Attractors of Multiparameter Nonlinear Dynamical Systems using Competitive Modes A Literary Analysis 1
Table of Contents Introduction Strange Attractors and Chaos Localization through Competitive Modes Lorenz Attractor Example Conclusions 2
Dynamical Systems Introduction Strange Attractors ? = ?(? ?) ? = ? ? ? ? ? = ?? ?? Competitive Modes Conclusions 3
Strange Attractors Introduction Definition: The strange attractor A of a n-dimensional system of differential equations is an attractor that is fractal in nature. Strange Attractors Competitive Modes Conclusions 4
Chaos Introduction Definition:Chaos is the phenomenon where a dynamical system is extremely sensitive to initial conditions in some set ? ?. Strange Attractors Competitive Modes Conclusions 5
Chaos: Lorenz Introduction Strange Attractors Competitive Modes Conclusions 6
Localization Through Competitive Modes 7
Oscillators Introduction Strange Attractors ? + ?? = 0 with ? 0 Competitive Modes Conclusions 8
Oscillators Introduction Strange Attractors Competitive Modes Conclusions 9
Oscillators to Competitive Modes Introduction Strange Attractors ?1= ?1?1, ?? ??= ???1, ?? Competitive Modes Conclusions 10
Oscillators to Competitive Modes Introduction Strange Attractors ?1+ ?1?1?1, ?? = 1?2, ?? ??+ ?????1, ?? = ??1, ?? 1 Competitive Modes Conclusions 11
Competitive Mode Conjecture Introduction Conjecture: The conditions for a dynamical system to be chaotic are: there exist at least two ?? s in the system at least one ?? is a function of ? at least one ? is a function of the system variables ? so that ??? ??? and ??? ,??? > 0 for some ?? and ?? Strange Attractors Competitive Modes Conclusions 12
Competitive Modes: Lorenz Introduction Strange Attractors Competitive Modes Conclusions 13
Competitive Modes: Lorenz Introduction ??= ??: Strange Attractors ?2= 1 ?2 Competitive Modes ??= ??: ?? = ?2+ ? ? + ? ?2 ??= ??: ?? = ?? + 1 ?2 Conclusions 14
Competitive Modes: Lorenz Introduction Strange Attractors Competitive Modes Conclusions 15
Conclusions and Research Questions Introduction Localization techniques do already exist, but are not necessarily simple. Strange Attractors Competitive Modes Conclusions Is the Competitive Mode Conjecture true? Which systems fulfill the competitive mode conjecture? Does it also apply to discrete systems? 16
Introduction Questions? Strange Attractors Competitive Modes Conclusions 17
Localization through Trajectories Introduction Strange Attractors Trajectories Nambu Mechanics Competitive Modes Conclusions 18
Localization through Trajectories Introduction Self-Excited Attractors Hidden Attractors Strange Attractors Trajectories Nambu Mechanics Competitive Modes Conclusions 19
Localization of Chua Hidden Attractor Introduction ? = ?(? ? ?(?)) ? = ? ? + ? ? = ? ? ? ? Strange Attractors Trajectories Nambu Mechanics Competitive Modes ? + 1 |? 1| 2 ? ? = ?1? + (?0+ ?1) Conclusions 20
Localization of Chua Hidden Attractor Introduction Strange Attractors Trajectories Nambu Mechanics Competitive Modes Conclusions 21
Localization of Chua Hidden Attractor Introduction Strange Attractors Trajectories Nambu Mechanics Competitive Modes Conclusions 22
Nambu Systems Introduction ? =??1 ??2 ?? ??1 ??2 ?? ??1 ??2 ?? ??1 ??2 ?? ??2 ?? ??2 ?? Strange Attractors ?? ?? Trajectories ? =??1 Nambu Mechanics ?? ?? Competitive Modes ? =??1 ?? ?? Conclusions 23
Nambu Systems Introduction Then ?1= 0 and ?2= 0 Strange Attractors Trajectories Meaning = ?1? 0 ,? 0 ,? 0 = ?2? 0 ,? 0 ,? 0 Nambu Mechanics ?1? ? ,? ? ,? ? ?2? ? ,? ? ,? ? Competitive Modes Conclusions 24
Nambu Hamiltonian Localization Introduction Strange Attractors Trajectories Nambu Mechanics Competitive Modes Conclusions 25
