Slide 1Slide 2Slide 3Slide 4Mackey-Glass delay differential equationsSlide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Concrete problems of chaotic and clusteringtime-series analysisA. Bershadskii, ICAR, Jerusalem, IsraelCHAOTIC TIME-SERIES ANALYSIS1. Spectral discrimination between chaotic and stochastic (turbulent) time series 1.1 Exponential and power-law broad-band spectra 1.2 Time-series related to low dimensional dynamic systems 1.3 Time-series related to infinite dimensional dynamic system 1.4 Singularities at complex times and the exponential spectra2. Hidden periods of chaotic time series 2.1 Chaotic Sun 2.2 Global temperature anomaly at millennial timescales 2.3 Chaotic dynamics of atmospheric CO2 and glaciation cycles 2.4 Chaotic-chaotic climate response Part I :Scaling: no fixed scale Periodic: fixed scale (period) Chaotic: fixed scale Te STOCHASTIC DETERMINISTICO.E. Rössler formulated a very simple continuous dynamical system generating chaotic solutions. dx/dt = -(y + z) dy/dt = x + a y dz/dt = b + x z - c z where a, b and c are parameters-----------------------------------------------------Rössler, O. E. An equation for continuous chaos. Phys. Lett. A 35, 397 (1976).standard values a=0.15, b=0.20, c=10.0Rössler attractorMackey-Glass delay differential equations0.00 0.04 0.08 0.10tf=200, a=0.2, b=0.1B. Mensour and A. Longtin, Power spectra and dynamical invariants for delay-differential and difference equations, Physica D, 113, 1 (1998)A generic property, first observed by J.D. Farmer (1982)Nature of the exponential decay of the power spectra of the chaotic systems is still an unsolved mathematical problem. A progress in solution of this problem has been achieved by the use of the analytical continuation of the equations in the complex domain: U. Frisch and R. Morf, Intermittency in non-linear dynamicsand singularities at complex times, Phys. Rev. 23, 2673 (1981).In this approach the exponential decay of chaotic spectrumis related to a singularity in the plane of complex time, which lies nearest to the real axis. Distance between this singularity and the real axis determines the rate of the exponential decay. For many interesting cases chaotic solutions are analytic in a finite strip around the real time axis. This takes place, in particular, for attractors bounded in the real domain (the Lorentz attractor, for instance). In this case the radius of convergence of the Taylor series is also bounded (uniformly) at any real time.Fourier transformThe theorem of residues: the sum runs over all poles located in the relevant half plane, Rj being their residue and xj + iyj their location.-T/2T/2Asymptotic behavior of E() at large ymin is the imaginary part of the location ofthe pole which lies nearest to the real axis.Singularities at complex times and the exponential spectraThe 176y period is the third doubling of the period 22y. The 22y period corresponds to the Sun’s magnetic poles polarity switching.A delay chaotic system ?A. Bershadskii, Europhys. Lett., 85, 49002 (2009).Parametric modulation with period TeIf parameters of the dynamical system fluctuate periodically around their mean values with period Te, then the restriction of the Taylor series convergence (at certain conditions) is determined by Te, and the width of the analytic strip aroundreal time axis equals Te/2High frequency tail of the spectrum of a reconstruction of Northern Hemisphere temperature anomaly for the past 2,000 yearsarXiv:0903.2795 Chaotic climate response to periodic forcingA quasi-linear response to a weak periodic forcingProblem of glacial cycles26 kyr41 kyr100 kyr ?The inclination of the Earth's orbit has a 100,000 year cycle relative to the invariable plane (the invariable plane is the plane that represents the angular momentum of the solar system).The multi-millennium timescale changes in orientation change the amount of solar radiation reaching the Earth in different latitudes. In high latitudes the annual mean insolation (incident solar radiation) decreases with axial tilt, while it increases in lower latitudes. Axial tilt forcing effect is maximum at the poles and comparatively small in the tropics.100,000-year problem of glacial cyclesObservations show that glacial changes from -1.5 to -2.5 Myr (early Pleistocene) were dominated by 41 kyr cycle, whereas the period from -0.8 Myr to present (late Pleistocene) is characterized by approximately 100 kyr glacial cycles. While the 41 kyr cycle of early Pleistocene glaciation is readily related to the 41 kyrperiod of Earth’s axial tilt oscillations the 100 kyr periodof the glacial cycles in last 0.8 Myr presents a serious problem.It was speculated in literature that influence of the axial tilt variations on global climate started amplifying around - 2.5 Myr, and became nonlinear from -0.8 Myr to present. Long term decrease in atmospheric CO2, which could result in a change in the internal response of the global carbon cycle to the axial tilt oscillations forcing, has been mentioned as one of the principal reasons for this phenomenon.A reconstruction of atmospheric CO2 based on deep sea proxies, for the past 650kyr. The data were taken from W.H. Berger, Database for reconstruction of atmospheric CO2, IGBP PAGES/World Data Center-A for Paleoclimatology Data Contribution Series # 96-031. NOAA/NGDC Paleoclimatology Program, Boulder CO, USA100,000-year problem of the glacial cyclesSpectrum of the atmospheric CO2 fluctuationsarXiv:0903.2795Thus, the axial tilt oscillations period of 41 kyr is still a dominating factor in the chaotic CO2 fluctuations, although it is hidden for linear interpretation of the power spectrumA quasi-linear response to a weak periodic forcingChaotic response to strong periodic forcing by axial tilt Te = 1/fe = 41,000 yrEarth’s orbit variations with about 100 kyr periodChaotic-chaotic climate responsearXiv:0903.2795Spectrum of temperature fluctuation in the semi-logarithmical scales (the reconstructed data for the last 10000 years – the interglacial warm period). Data at ftp://ftp.ncdc.noaa.gov/pub/data/paleo/ icecore/antarctica/epica_domec/edc3deuttemp2007.txtSSN TemperaturePART IICLUSTER ANALYSIS of TURBULENT (STOCHASTIC) TIME-SERIES1.Telegraph approximation of turbulent signals2. Clustering and scaling: