141 research outputs found
Operator-Based Detecting, Learning, and Stabilizing Unstable Periodic Orbits of Chaotic Attractors
This paper examines the use of operator-theoretic approaches to the analysis
of chaotic systems through the lens of their unstable periodic orbits (UPOs).
Our approach involves three data-driven steps for detecting, identifying, and
stabilizing UPOs. We demonstrate the use of kernel integral operators within
delay coordinates as an innovative method for UPO detection. For identifying
the dynamic behavior associated with each individual UPO, we utilize the
Koopman operator to present the dynamics as linear equations in the space of
Koopman eigenfunctions. This allows for characterizing the chaotic attractor by
investigating its principal dynamical modes across varying UPOs. We extend this
methodology into an interpretable machine learning framework aimed at
stabilizing strange attractors on their UPOs. To illustrate the efficacy of our
approach, we apply it to the Lorenz attractor as a case study.Comment: arXiv admin note: text overlap with arXiv:2304.0783
Stabilization of heterodimensional cycles
We consider diffeomorphisms with heteroclinic cycles associated to
saddles and of different indices. We say that a cycle of this type can
be stabilized if there are diffeomorphisms close to with a robust cycle
associated to hyperbolic sets containing the continuations of and . We
focus on the case where the indices of these two saddles differ by one. We
prove that, excluding one particular case (so-called twisted cycles that
additionally satisfy some geometrical restrictions), all such cycles can be
stabilized.Comment: 31 pages, 9 figure
Chaos Criminology: A Critical Inquiry
There has been a push since the early 1980âs for a paradigm shift in criminology from a Newtonian-based ontology to one of quantum physics. Primarily this effort has taken the form of integrating Chaos Theory into Criminology into what this thesis calls âChaos Criminologyâ. However, with the melding of any two fields, terms and concepts need to be translated properly, which has yet to be done. In addition to proving a translation between fields, this thesis also uses a set of criteria to evaluate the effectiveness of the current use of Chaos Theory in Criminology. While the results of the theory evaluation reveal that the current Chaos Criminology work is severely lacking and in need of development, there is some promise in the development of Marxâs dialectical materialism with Chaos Theory
Dynamics of Patterns
Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction
Algebraic Aspects of (Bio) Nano-chemical Reaction Networks and Bifurcations in Various Dynamical Systems
The dynamics of (bio) chemical reaction networks have been studied by different methods. Among these methods, the chemical reaction network theory has been proven to successfully predicate important qualitative properties, such as the existence of the steady state and the asymptotic behavior of the steady state. However, a constructive approach to the steady state locus has not been presented. In this thesis, with the help of toric geometry, we propose a generic strategy towards this question. This theory is applied to (bio)nano particle configurations. We also investigate Hopf bifurcation surfaces of various dynamical systems
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