Search results for " exponent"

showing 10 items of 315 documents

Numerical analysis of dynamical systems: unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimensi…

2018

In this article, on the example of the known low-order dynamical models, namely Lorenz, Rossler and Vallis systems, the difficulties of reliable numerical analysis of chaotic dynamical systems are discussed. For the Lorenz system, the problems of existence of hidden chaotic attractors and hidden transient chaotic sets and their numerical investigation are considered. The problems of the numerical characterization of a chaotic attractor by calculating finite-time time Lyapunov exponents and finite-time Lyapunov dimension along one trajectory are demonstrated using the example of computing unstable periodic orbits in the Rossler system. Using the example of the Vallis system describing the El…

Lyapunov functionHistoryMathematics::Dynamical SystemsDynamical systems theoryNumerical analysisChaoticFOS: Physical sciencesLyapunov exponentLorenz systemNonlinear Sciences - Chaotic DynamicsComputer Science ApplicationsEducationNonlinear Sciences::Chaotic Dynamicssymbols.namesakeAttractorsymbolsTrajectoryApplied mathematicsChaotic Dynamics (nlin.CD)Mathematics
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Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system

2015

The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a {hidden attractor} in the case of multistability as well as a classical {self-excited attractor}. The hidden attractor in this system can be localized by analytical-numerical methods based on the {continuation} and {perpetual points}. For numerical study of the attractors' dimension the concept of {finite-time Lyapunov dimension} is developed. A conjecture on the Lyapunov dimension of self-excited attractors and the notion of {exact Lyapunov dimension} are discussed. A comparative survey on the computation of the finite-time Lyapunov expon…

Lyapunov functionMathematics::Dynamical SystemsChaoticAerospace EngineeringFOS: Physical sciencesOcean EngineeringLyapunov exponent01 natural sciences010305 fluids & plasmasadaptive algorithmssymbols.namesakehidden attractorsDimension (vector space)0103 physical sciencesAttractorApplied mathematicsElectrical and Electronic Engineering010301 acousticsMultistabilityMathematicsAdaptive algorithmApplied MathematicsMechanical EngineeringNumerical analysisNonlinear Sciences - Chaotic DynamicsNonlinear Sciences::Chaotic DynamicsControl and Systems EngineeringLyapunov dimensionsymbolsperpetual pointsChaotic Dynamics (nlin.CD)finite-time Lyapunov exponents
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Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations

2014

Nowadays the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. It may lead to a confusion since there are at least two well-known definitions, which are used in computations: the upper bounds of the exponential growth rate of the norms of linearized system solutions (Lyapunov characteristic exponents, LCEs) and the upper bounds of the exponential growth rate of the singular values of the fundamental matrix of linearized system (Lyapunov exponents, LEs). In this work the relation between Lyapunov exponents and Lyapunov characteristic exponents is discussed. The invariance…

Lyapunov functionMathematics::Dynamical SystemsComputationFOS: Physical sciencesAerospace EngineeringOcean EngineeringDynamical Systems (math.DS)Lyapunov exponent01 natural sciencessymbols.namesakeExponential growthComputer Science::Systems and Control0103 physical sciencesFOS: MathematicsApplied mathematics0101 mathematicsElectrical and Electronic EngineeringMathematics - Dynamical Systems010301 acousticsMathematicsApplied MathematicsMechanical Engineering010102 general mathematicsNonlinear Sciences - Chaotic DynamicsNonlinear Sciences::Chaotic DynamicsSingular valueFundamental matrix (linear differential equation)Control and Systems EngineeringsymbolsDiffeomorphismChaotic Dynamics (nlin.CD)Characteristic exponent
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Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System

2015

In applied investigations, the invariance of the Lyapunov dimension under a diffeomorphism is often used. However, in the case of irregular linearization, this fact was not strictly considered in the classical works. In the present work, the invariance of the Lyapunov dimension under diffeomorphism is demonstrated in the general case. This fact is used to obtain the analytic exact upper bound of the Lyapunov dimension of an attractor of the Shimizu–Morioka system. peerReviewed

Lyapunov functionPure mathematicsMathematics::Dynamical SystemsGeneral Physics and Astronomylcsh:AstrophysicsLyapunov exponentUpper and lower boundssymbols.namesakeShimizu-Morioka systemDimension (vector space)Attractorlcsh:QB460-466Lyapunov equationLyapunov redesignlcsh:ScienceMathematicsta111Mathematical analysisShimizu–Morioka systemlcsh:QC1-999Nonlinear Sciences::Chaotic DynamicssymbolsLyapunov dimensionlcsh:QDiffeomorphismLyapunov exponentlcsh:PhysicsEntropy
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Lyapunov Functions for Second-Order Differential Inclusions: A Viability Approach

2001

AbstractIn this paper the existence of Lyapunov functions for second-order differential inclusions is analyzed by using the methodology of the Viability Theory. A necessary assumption on the initial states and sufficient conditions for the existence of local and global Lyapunov functions are obtained. An application is also provided.

Lyapunov functionsecond orderViability theoryApplied MathematicsMathematical analysisOrder (ring theory)Lyapunov exponentExponential functionsymbols.namesakeDifferential inclusiondifferential inclusionssymbolsLyapunov equationviability theoryExponential decayAnalysisMathematicsLyapunov functionsJournal of Mathematical Analysis and Applications
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Impulsive control on the synchronization for a class of chaotic Systems

2014

In this paper, the impulsive control problem on the synchronization for a class of chaotic systems is discussed. Based on Lyapunov stability theory, the new impulsive synchronization strategy is presented to realize the chaos synchronization and possesses the wider scope of application. Finally the numerical simulation examples are given to demonstrate the effectiveness of our theoretical results.

Lyapunov stabilitychaos systemClass (set theory)Computer simulationSynchronization of chaoschaos system; impulsive switching; Lyapunov stability; synchronization; Electrical and Electronic Engineering; Control and Systems EngineeringLyapunov exponentimpulsive switchingSynchronizationNonlinear Sciences::Chaotic DynamicsCHAOS (operating system)symbols.namesakeControl and Systems EngineeringControl theoryLyapunov stabilitysymbolsElectrical and Electronic EngineeringLyapunov redesignsynchronizationMathematics2014 IEEE 23rd International Symposium on Industrial Electronics (ISIE)
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H<inf>∞</inf> controller design for the synchronization of a hyper-chaotic system

2013

In this paper, the robust control on the synchronization of a hyper-chaotic system is investigated. Based on Lyapunov stability theory and linear matrix inequality techniques, the multi-dimensional and the single-dimensional robust H∞ synchronization controllers are constructed for the possible application in practical engineering. Some numerical simulations are provided to demonstrate the effectiveness of the presented controllers.

Lyapunov stabilitysymbols.namesakeControl theoryChaoticsymbolsLinear matrix inequalityControl engineeringLyapunov equationLyapunov exponentRobust controlLyapunov redesignSynchronizationMathematics2013 9th Asian Control Conference (ASCC)
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Crystal structure and magnetism of Co(HPO3)⋅H2O : A novel layered compound of Co(II)

1990

Under the terms of the Creative Commons Attribution (CC BY) license to their work.-- et al.

Magnetic PropertiesMagnetismGeneral Physics and AstronomyCrystal structureMagnetic Susceptibility:FÍSICA [UNESCO]HydratesMedium TemperatureMagnetic structureChemistryCritical ExponentsUNESCO::FÍSICASpace groupHydrogen BondsMagnetic susceptibilityCobalt Compounds ; Acid Phosphates ; Hydrates ; Layers ; Crystal Structure ; Magnetic Properties ; Lattice Parameters ; Space Groups ; Hydrogen Bonds ; Magnetic Susceptibility ; Magnetic Structure ; Critical Exponents ; Ising Model ; Medium TemperatureCrystallographySpace GroupsIsing ModelOctahedronAcid PhosphatesCrystal StructureLattice ParametersIsing modelMagnetic StructureCobalt CompoundsLayersCritical exponent
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ATXN2 trinucleotide repeat length correlates with risk of ALS

2017

We investigated a CAG trinucleotide repeat expansion in the ATXN2 gene in amyotrophic lateral sclerosis (ALS). Two new case-control studies, a British dataset of 1474 ALS cases and 567 controls, and a Dutch dataset of 1328 ALS cases and 691 controls were analyzed. In addition, to increase power, we systematically searched PubMed for case-control studies published after 1 August 2010 that investigated the association between ATXN2 intermediate repeats and ALS. We conducted a meta-analysis of the new and existing studies for the relative risks of ATXN2 intermediate repeat alleles of between 24 and 34 CAG trinucleotide repeats and ALS. There was an overall increased risk of ALS for those carry…

MaleExpansion0301 basic medicineAgingATXN2 geneSettore MED/03 - GENETICA MEDICA0302 clinical medicineTrinucleotide RepeatsGenetic Report AbstractAmyotrophic lateral sclerosisAtaxin-2GeneticsCAGGeneral NeuroscienceATXN2Triplet3. Good healthFemalePsychologyNeurovetenskaperRiskNeuroscience(all)Age of onsetClinical Neurology03 medical and health sciencesSCA2Trinucleotide repeatJournal ArticlemedicineHumansAlleleAllelesGenetic Association StudiesAmyotrophic lateral sclerosiIntermediate expansionNeuroscience (all)NeurosciencesExponential riskCase-control studyAmyotrophic lateral sclerosismedicine.diseaseClinical neurologyAgeing030104 developmental biologyCase-Control StudiesHuman medicineNeurology (clinical)ALSGeriatrics and GerontologyAge of onsetTrinucleotide Repeat ExpansionTrinucleotide repeat expansionALS; ATXN2; Age of onset; Amyotrophic lateral sclerosis; CAG; Expansion; Exponential risk; Intermediate expansion; Risk; SCA2; Trinucleotide repeat; TripletNeuroscience030217 neurology & neurosurgeryMeta-AnalysisDevelopmental BiologyNeurobiology of Aging
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Fractal analyses reveal independent complexity and predictability of gait

2017

Locomotion is a natural task that has been assessed since decades and used as a proxy to highlight impairments of various origins. Most studies adopted classical linear analyses of spatio-temporal gait parameters. Here, we use more advanced, yet not less practical, non-linear techniques to analyse gait time series of healthy subjects. We aimed at finding more sensitive indexes related to spatio-temporal gait parameters than those previously used, with the hope to better identify abnormal locomotion. We analysed large-scale stride interval time series and mean step width in 34 participants while altering walking direction (forward vs. backward walking) and with or without galvanic vestibular…

MalePhysiologyEffect of gait parameters on energetic costlcsh:MedicineWalkingMotor Neuron Diseases0302 clinical medicineElderlyMedicine and Health SciencesMastoid Processlcsh:ScienceMusculoskeletal SystemGaitMathematicsMultidisciplinary05 social sciencesNeurodegenerative DiseasesFractalsNeurologyPhysical SciencesFemale[SDV.NEU]Life Sciences [q-bio]/Neurons and Cognition [q-bio.NC]AnatomyGait AnalysisResearch ArticleAdultSTRIDEGeometryFOS: Physical sciencesSurgical and Invasive Medical ProceduresFractal dimension050105 experimental psychology03 medical and health sciencesYoung AdultFractalHumans0501 psychology and cognitive sciencesPredictabilityGalvanic vestibular stimulationSkeletonHurst exponentFunctional Electrical Stimulationbusiness.industryBiological LocomotionAmyotrophic Lateral SclerosisSkulllcsh:RBiology and Life SciencesPattern recognitionPhysics - Medical PhysicsAge GroupsGait analysis[ SDV.NEU ] Life Sciences [q-bio]/Neurons and Cognition [q-bio.NC]People and PlacesPopulation Groupingslcsh:QArtificial intelligenceMedical Physics (physics.med-ph)business030217 neurology & neurosurgeryMathematics
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