Low-frequency climate variability: a dynamical systems a approach
|
|
- Austin Franklin
- 5 years ago
- Views:
Transcription
1 Low-frequency climate variability: a dynamical systems a approach Henk Broer Rijksuniversiteit Groningen broer 19 October 2006 Page 1 of 40
2 1. Variability at low-frequencies Making a long story short: Power spectrum of a time series related to mean solid-body rotation of atmosphere relative to Earth, from spectral atmospheric model (James & James 1989). Similar spectra typical for atmosphere/ocean data and models often at decadal-to-multidecadal scale. Page 2 of 40 Example: Atlantic Multidecadal Oscillation Can such phenomena be studied by dynamical system techniques?
3 Dynamical systems perspective Deterministic modelling plays central role in: weather prediction (Numerical Weather Prediction Models) and in climate analysis (General Circulation Models) These are intense research fields large scale computational efforts (Earth Simulator) huge financial commitments (Kyoto, etc.). Weather unpredictability and climate statistics: related to turbulence and chaos (Lorenz 1963, Ruelle-Takens 1971). Basic idea: identify regimes with stationary solutions; explain low-frequency variability and regime transitions by intermittency due to homo- and heteroclinicity Page 3 of 40
4 Dynamical systems principles Statespace with deterministic evolution (NB: Three-body problem has 3 6 = 18D state space) Stationary state point attractor Periodic state circle attractor Multiperiodic state torus Strange attractor (chaos & unpredictability, fractal structure) Search for persistent / robust phenomena, often also universal (i.e., context independent). Example: Transition upon variation of parameters like Hopf- or period doubling bifurcation (think of Feigenbaum cascade). Page 4 of 40
5 Towards low dimensional models Start with first principle PDE models: Navier-Stokes plus mass and energy conservation laws infinite dimensional dynamics in function space I. Conceptual reduction to finite dimension: invariant / inertial manifold containing attractors (Ruelle & Takens, Mallet-Paret, Temam, etc.). II. Analytical / computational approximation: Galerkin like truncations (e.g., Lorenz 1963, 1984). I and II not so easy to marry: a mathematical challenge... Search is for reduced models by projection of PDE s onto their attractors, leading to finite (low) dimensional ODE s Page 5 of 40
6 Example: Torus and Lorenz attractor Lorenz attractor (1963, from a convection model) is first strange attractor; only proven so by Tucker (1999). x2 Z Page 6 of 40 x1 3D torus admits strange attractor X
7 Example: towards reduced models Two-Layer Quasi-Geostrophic Model on circumpolar annulus domain. Ansätze: Hydrostatic atmosphere near geostrophic equilibrium, in two layers, with a Coriolis force that is linear in latitude, parametrizations o W 150 o W 150 o E 120 o W 120 o E L y 90 o W 90 o E Page 7 of o 30 N 45 o N 60 o W 22 o 30 N 60 o E 30 o W y^ ^ x 30 o E From: Lucarini, Speranza & Vitolo (2006). 0 o
8 Minimal models: Lorenz-84 Fourier expansion + Galerkin truncation + reduction to 3D manifold Lorenz-84 ODE: ẋ = ax y 2 z 2 + af, ẏ = y + xy bxz + G, ż = z + bxy + xz. E.N. Lorenz (1984,1990): simplest model for atmospheric dynamics at midlatitudes. Shil nikov, Nicolis& Nicolis (1995): comprehensive bifurcation analysis. Page 8 of 40 Pielke & Zheng (1994): low-frequency variability induced by seasonal forcing. L. van Veen (2003): derivation from two-layer quasi-geostrophic equations, usage in low-order coupled atmosphere/ocean models. Broer, Simó & Vitolo (2002): influence on dynamics of seasonal variation of (F, G), new types of strange attractors.
9 Variables and parameters in Lorenz-84 x: strength of westerly wind current y, z: sine and cosine phases of travelling waves F, G: thermal forcings Lorenz-84 is a toy -model only for detecting qualitative aspects in atmospheric baroclinic jet at mid-latitudes. Page 9 of 40
10 2. Bifurcations in Lorenz-84 Message: Lorenz 84 has rich dynamics. Codimension 2 organizing centers in (F, G) parameter plane: - Hopf-saddle-node bifo of equilibria. - Cusp of equilibria. - 1:2 strong resonance bifo of periodic orbits. - 1:1 (Bogdanov-Takens) bifo of periodic orbits. Page 10 of 40 Homoclinic bifurcations: Shil nikov tangencies. Several codimension 1 bifos of equilibria (Hopf, saddle-node) and of periodic orbits (Hopf, saddle-node, period doubling). Shil nikov, Nicolis & Nicolis (1995).
11 Page 11 of 40 Figure: Van Veen (2003).
12 Page 12 of 40 Figure: Van Veen (2003).
13 Shil nikov-like strange attractors Page 13 of 40 Parameter values are s.t. a Shil nikov bifurcation takes place in 6D ODE, similar attractors also in Lorenz-84. Blue: 4-times period-doubled periodic orbit. Red: Shil nikov homoclinic orbit. Green: strange attractor. Figure: Van Veen (2003).
14 Shil nikov bifurcation Page 14 of 40 Sketch of Shil nikov bifurcation (codimension 1) associated to chaos
15 Two broken sphere scenario s Page 15 of 40 Broer & Vegter (1984)
16 Conclusions I (see Appendices) 1. Hopf-saddle-node and Shil nikov bifurcations occur in several, very different low-order models of atmospheric circulation. 2. Rich dynamical phenonena are related to this: (a) Transversal heteroclinic orbit; (b) Chaos and strange attractors; (c) Intermittency (and regime transiton ) near both polar equilibria. 3. Traces of homo- and heteroclinic behaviour are found in reduced models of large-scale atmospheric flow (of dimensions 6D to 30D), where low-frequency variability has been observed (Charney & DeVore (1979), Branstator & Opsteegh (1989), De Swart (1989), James et al. (1994), Crommelin et al. (2004). Page 16 of 40
17 Conclusions II (see Appendices) 1. Homo- and heteroclinicity: a possible explanation for (a) low-frequency variability (b) regime transitions (heteroclinic intermittency) (c) bimodality (observed in data) 2. What is the analogue of Shil nikov in more complex, higher dimensional models such as PDE s or the General Circulation Models? 3. Can traces of homoclinic behaviour also be found in climatic data (i.e., in observations)? Page 17 of 40
18 Bibliography 1. E.N. Lorenz: Energy and numerical weather prediction, Tellus 12(4) (1960), E.N. Lorenz: Irregularity: a fundamental property of the atmosphere, Tellus 36A (1984), E.N. Lorenz: Can chaos and intransitivity lead to interannual variability? Tellus 42A (1990), E.N. Lorenz: Regimes in simple systems, J. Atmos. Sci. 63 (2006), I.N. James, P.M. James: Ultra-low-frequency variability in a simple atmospheric circulation model, Nature 342 (1989), R. Pielke, X. Zeng: Long-term variability of climate, J. Atmos. Sci. 51 (1994), Page 18 of A. Shil nikov, G. Nicolis, C. Nicolis: Bifurcation and predictability analysis of a low-order atmospheric circulation model, Int.J.Bifur.Chaos 5(6) (1995),
19 Bibliography (continued) 8. D.T. Crommelin: Homoclinic Dynamics: A Scenario for Atmospheric Ultralow-Frequency Variability, J. Atmos. Sci. 59(9) (2002), D.T. Crommelin: Regime transitions and heteroclinic connections in a barotropic atmosphere J. Atmos. Sci., 60(2) (2003), D.T. Crommelin, J.D. Opsteegh, F. Verhulst: A mechanism for atmospheric regime behaviour, J. Atmos. Sci. 61(12) (2004), L. van Veen: Baroclinic flow and the Lorenz-84 model, Int.J.Bifur.Chaos 13 (2003), L. van Veen, T. Opsteegh and F. Verhulst: Active and passive ocean regimes in a low-order climate model, Tellus 53A (2001), Page 19 of L. van Veen: Overturning and wind driven circulation in a low-order ocean-atmosphere model, Dyn.Atmos.Oceans 37 (2003),
20 Bibliography (continued) 14. H.W. Broer, C. Simó, R. Vitolo: Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing, Nonlinearity 15(4) (2002), H.W. Broer, R. Vitolo: Dynamical systems modeling of low-frequency variability in low-order atmospheric models. Submitted. 16. H. E. De Swart: Analysis of a six-component atmospheric spectral model: Chaos, predictability and vacillation. Physica D 36 (1989), Page 20 of H. Itoh, and M. Kimoto: Multiple attractors and chaotic itinerancy in a quasigeostrophic model with realistic topography: Implications for weather regimes and low-frequency variability, J. Atmos. Sci. 53 (1996),
21 Bibliography (continued) 18. V. Lucarini, A. Speranza, R. Vitolo: Physical and Mathematical Properties of a Quasi-Geostrophic Model of Intermediate Complexity of the Mid-Latitudes Atmospheric Circulation, preprint PASEF: (2006). 19. P. Glendinning: Differential equations with bifocal homoclinic orbits, IJBC 7 (1997), G. Branstator, and J. D. Opsteegh: Free solutions of the barotropic vorticity equation, J. Atmos. Sci. 46 (1989), P. M. James, K. Fraedrich, and I. N. James: Wave-zonal-flow interaction and ultra-low-frequency variability in a simplified global circulation model, Quart. J. Roy. Meteor. Soc. 120 (1994), Page 21 of M. Viana: What s new on Lorenz strange attractors, The Mathematical Intelligencer 22 (3) (2000) 6-19.
22 Appendices Summary and connection of the Appendices A, B, C and D to main text: 1. There really exists something like low frequency variability. 2. Dynamical systems concepts play a key role in current activities, which are of societal relevance (weather forecasts, climate analysis). 3. Understanding of low frequency (regimes, regime transitions) in terms of Shil nikov, homo- and heteroclinic intermittency. 4. First step: from infinite (PDE) to finite (ODE) (Lorenz-84, Charney, DeVore & De Swart) by means of filtering and spectral projection. 5. Small models: we understand something of Shil nikov, Hopf-saddlenode, etc. 6. How does all of this look when returning from small to finite and to infinite? What is the role of 3. in the PDE s of the GCM? Do we find traces of this in NATURE? Page 22 of 40
23 A. Equations of 2LQG Lorenz, 1960: t 2 ψ = J(ψ, 2 ψ + f) J(τ, 2 τ) C 2 (ψ τ), t 2 τ = J(τ, 2 ψ + f) J(ψ, 2 τ) + C 2 (ψ τ)+ t θ = J(ψ, θ) + σ 2 χ 1 h N (θ θ ). + (f χ 1 ) 2C 2 τ, Model for atmospheric dynamics at mid-latitudes (Lorenz 1960). Coordinates (x, y) [0, L x ] [0, L y ] horizontally (x periodic). Pressure p vertically: discretized at two levels! Page 23 of 40 Unknowns: ψ, τ, θ, χ 1.
24 Unknowns in 2LQG model ψ = 1 2 (Ψ 3 + Ψ 1 ) τ = 1 2 (Ψ 3 Ψ 1 ) barotropic streamfunction baroclinic streamfunction θ = 1 2 (Θ 3 + Θ 1 ) mean potential temperature σ = 1 2 (Θ 3 Θ 1 ) static stability (fixed) Ψ 1,3 : streamfunction at lower and upper layer, respectively Θ 1,3 : potential temperature at lower and upper layer, respectively Page 24 of 40 C: friction at layer interface C : friction at layer interface h N : Newtonian cooling f: Coriolis parameter: constant (f-plane approx.) θ = 1 T cos(πy/l 2 y): imposed temperature profile (halfcosine shape) T : north-south temperature gradient
25 Original variables (non-discretized model) Let v be velocity (wind) in initial 3D equations: v = v h + v v with horizontal velocity: v h = v r + v d, where v r = k Ψ divergence-free component of v h, v d = χ irrotational component of v h, v v = p t k vertical velocity. Ψ : streamfunction χ : velocity potential T : temperature Θ : potential temperature ( p cp T = Θ, where p s ) cp cv p s surface pressure, c p, c v : specific heat of dry air at constant pressure and volume Page 25 of 40
26 B. The seasonally driven Lorenz-84 model ẋ = ax y 2 z 2 + af (1 + ɛ cos(ωt)) ẏ = y + xy bxz + G(1 + ɛ cos(ωt)) (1) ż = z + bxy + xz. T = 2π/ω = 73: period of the forcing (a, b: constants) F, G, ɛ: control parameters Poincaré (time T ) map P F,G,ɛ : P F,G,ɛ : R 3 R 3 is diffeomorphism. Page 26 of 40 Problem setting: - Coherent inventory of dynamics of P F,G,ɛ depending on F, G, ɛ. H.W. Broer, C. Simó & R. Vitolo, Nonlinearity 15(4) (2002),
27 Bifurcation diagram of fixed points of P F,G Page 27 of 40 ɛ = 0.5 fixed. H 2 : Hopf, H sub 1 : subcritical Hopf, SN 0 sub: saddle-node.
28 Page 28 of 40 Magnification of box A in Fig.1. Cusp C terminates two saddle-node curves. H 1 : Hopf. A 1 1:1 resonance tongue. Strange attractors in L 1.
29 Disappearance of HSN bifurcation point Page 29 of 40 ɛ = 0.01 ɛ = 0.5 Hopf and saddle-node bifurcation curves are no longer tangent for ɛ = 0.5. Several strong resonances interrupt Hopf bifurcation curve. Codimension 3 bifurcation between ɛ = 0.01 and 0.5?
30 Quasi-periodic bifos of invariant circles z 1e-05 1e-15 1e x z -1 x e-35 1e-05 1e-15 1e-25 1e Page 30 of 40 Left: projections on (x, z) of attractors. Right: power spectra. Top: G = Bottom: G = (F = 11).
31 Quasi-periodic strange attractors e e e JJ II J I 2 4 Page 31 of S 1e-07 z x e Left: projections on (x, z) of attractors. Right: power spectra. Top: G = Bottom: G = (F = 11).
32 Hénon-like attractors Ansatz: Hénon-like strange attractor = closure of unstable manifold Page 32 of 40
33 C. Homoclinic dynamics and ultralow-frequency variability D. Crommelin: J. Atmos. Sci. 59 (2002). Ultralow-frequency: timescale beyond several months. Common physical explanations: 1. Associated to slow dynamical components: ice, oceans. 2. Seasonal variations of parameters (James & James, 1989) 3. Interaction of zonal (longitudinal) flow and baroclinic waves (James & James, 1994) Page 33 of 40 But what is mathematical structure? Connection with homoclinic dynamics? Present investigation: consider various models: 1. A General Circulation Model (GCM): NCAR CCM version 0B. 2. Empirical Orthogonal Projection (EOF) of a quasi-geostrophic model. 3. A 4D simplified model.
34 Page 34 of 40 Low-frequency variability in power spectra of EOF1 GCM (top), 30D EOF model (middle), 10D EOF (bottom)
35 Page 35 of 40 Traces of homoclinic dynamics in attractors of 4D model (2D projection) for four values of a control parameter. Last plot (h) is attracting periodic orbit: suggests bifocal homoclinic, two equilibria of saddle-focus type.
36 D. The 6D ODE model Charney and DeVore (1979), De Swart (1989). From barotropic vorticity equation (PDE), Galerkin projection ẋ 1 = γ 1 x 3 C(x 1 x 1), ẋ 2 = (α 1 x 1 β 1 )x 3 Cx 2 δ 1 x 4 x 6, ẋ 3 = (α 1 x 1 β 1 )x 2 γ 1 x 1 Cx 3 + δ 1 x 4 x 5, ẋ 4 = γ 2 x 6 C(x 4 x 4) + ε(x 2 x 6 x 3 x 5 ), ẋ 5 = (α 2 x 1 β 2 )x 6 Cx 5 δ 2 x 4 x 3, ẋ 6 = (α 2 x 1 β 2 )x 5 γ 2 x 4 Cx 6 + δ 2 x 4 x 2. Physical meaning of terms: α j : advection of waves by zonal (longitudinal) flow. β j : Coriolis force. γ j, γ j : topography. C: Newtonian damping to zonal profile (x 1, 0, 0, x 4, 0, 0). δ, ε: Fourier modes interactions due to nonlinearity. Page 36 of 40 Control parameters: x 1, γ, r (where x 4 = rx 1).
37 Bifurcations in Charney-DeVore-De Swart Crommelin, Opsteegh, Verhulst, Codimension 2 organizing centers in (F, G) parameter plane: - Hopf-saddle-node bifo of equilibria. - Cusp of equilibria. - 1:2 strong resonance bifo of periodic orbits. Homoclinic bifurcations: Shil nikov tangencies. Codimension 1 bifos of equilibria (Hopf, saddle-node) and of periodic orbits (Hopf, period doubling). Page 37 of 40 Very strong analogies with Lorenz-84!
38 Bifurcation analysis fh: Hopf-saddle-node c: cusp sn1,sn2: saddle-node pd: period doubling (of periodic orbits) Page 38 of 40
39 Homoclinic (Shil nikov) orbits Homoclinic orbits of an equilibrium eq1 occurring, from top to bottom, at different values of the parameters (x 1, r). Page 39 of 40
40 Bimodality, regimes, and heteroclinics Intermittency of saddle-node type occurs after equilibrium eq2 coalesces with another equilibrium eq3 (at sn2). Intermittent heteroclinic behaviour: orbits visit alternatively vicinity of eq1 and (formerly existing) eq2,eq3. Bimodality of probability distribution function (bottom right): near eq1 and near eq2,eq3. Page 40 of 40 Notice high speed in phase space along transitions between two regions.
A model for atmospheric circulation
Apeiron, Vol. 19, No. 3, July 2012 264 A model for atmospheric circulation B S Lakshmi JNTU College Of Engineering Hyderabad K L Vasundhara Vidya Jyothi Institute of Technology Hyderabad In this paper
More informationBaroclinic flow and the Lorenz-84 model
CHAPTER Baroclinic flow and the Lorenz-84 model To appear in the Internat. J. Bifur. Chaos. Abstract. The bifurcation diagram of a truncation to six degrees of freedom of the equations for quasi-geostrophic,
More informationBifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing
Home Search Collections Journals About Contact us My IOPscience Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing This article has been downloaded from IOPscience.
More informationElectronic Circuit Simulation of the Lorenz Model With General Circulation
International Journal of Physics, 2014, Vol. 2, No. 5, 124-128 Available online at http://pubs.sciepub.com/ijp/2/5/1 Science and Education Publishing DOI:10.12691/ijp-2-5-1 Electronic Circuit Simulation
More informationMathematical Foundations of Neuroscience - Lecture 7. Bifurcations II.
Mathematical Foundations of Neuroscience - Lecture 7. Bifurcations II. Filip Piękniewski Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland Winter 2009/2010 Filip
More informationBIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs
BIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs Yuri A. Kuznetsov August, 2010 Contents 1. Solutions and orbits: equilibria cycles connecting orbits compact invariant manifolds strange
More informationWhat is a Low Order Model?
What is a Low Order Model? t Ψ = NL(Ψ ), where NL is a nonlinear operator (quadratic nonlinearity) N Ψ (x,y,z,...,t)= Ai (t)φ i (x,y,z,...) i=-n da i = N N cijk A j A k + bij A j + f i v i j;k=-n j=-n
More informationExample of a Blue Sky Catastrophe
PUB:[SXG.TEMP]TRANS2913EL.PS 16-OCT-2001 11:08:53.21 SXG Page: 99 (1) Amer. Math. Soc. Transl. (2) Vol. 200, 2000 Example of a Blue Sky Catastrophe Nikolaĭ Gavrilov and Andrey Shilnikov To the memory of
More informationDRIVEN and COUPLED OSCILLATORS. I Parametric forcing The pendulum in 1:2 resonance Santiago de Compostela
DRIVEN and COUPLED OSCILLATORS I Parametric forcing The pendulum in 1:2 resonance Santiago de Compostela II Coupled oscillators Resonance tongues Huygens s synchronisation III Coupled cell system with
More informationTHREE DIMENSIONAL SYSTEMS. Lecture 6: The Lorenz Equations
THREE DIMENSIONAL SYSTEMS Lecture 6: The Lorenz Equations 6. The Lorenz (1963) Equations The Lorenz equations were originally derived by Saltzman (1962) as a minimalist model of thermal convection in a
More information11 Chaos in Continuous Dynamical Systems.
11 CHAOS IN CONTINUOUS DYNAMICAL SYSTEMS. 47 11 Chaos in Continuous Dynamical Systems. Let s consider a system of differential equations given by where x(t) : R R and f : R R. ẋ = f(x), The linearization
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: August 22, 2018, at 08 30 12 30 Johanneberg Jan Meibohm,
More informationPhysical and Mathematical Properties of a Quasi-Geostrophic Model of Intermediate Complexity of the Mid-Latitudes. atmospheric circulation.
Physical and Mathematical Properties of a Quasi-Geostrophic Model of Intermediate Complexity of the Mid-Latitudes Atmospheric Circulation Valerio Lucarini, Antonio Speranza, and Renato Vitolo PASEF Physics
More informationResonance and fractal geometry
Resonance and fractal geometry Henk Broer Johann Bernoulli Institute for Mathematics and Computer Science Rijksuniversiteit Groningen Summary i. resonance in parametrized systems ii. two oscillators: torus-
More informationSPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICE. Itishree Priyadarshini. Prof. Biplab Ganguli
SPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICE By Itishree Priyadarshini Under the Guidance of Prof. Biplab Ganguli Department of Physics National Institute of Technology, Rourkela CERTIFICATE This is to
More informationChaos. Lendert Gelens. KU Leuven - Vrije Universiteit Brussel Nonlinear dynamics course - VUB
Chaos Lendert Gelens KU Leuven - Vrije Universiteit Brussel www.gelenslab.org Nonlinear dynamics course - VUB Examples of chaotic systems: the double pendulum? θ 1 θ θ 2 Examples of chaotic systems: the
More informationLecture 6. Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of:
Lecture 6 Chaos Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of: Chaos, Attractors and strange attractors Transient chaos Lorenz Equations
More informationNBA Lecture 1. Simplest bifurcations in n-dimensional ODEs. Yu.A. Kuznetsov (Utrecht University, NL) March 14, 2011
NBA Lecture 1 Simplest bifurcations in n-dimensional ODEs Yu.A. Kuznetsov (Utrecht University, NL) March 14, 2011 Contents 1. Solutions and orbits: equilibria cycles connecting orbits other invariant sets
More informationEE222 - Spring 16 - Lecture 2 Notes 1
EE222 - Spring 16 - Lecture 2 Notes 1 Murat Arcak January 21 2016 1 Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Essentially Nonlinear Phenomena Continued
More informationOn the Takens-Bogdanov Bifurcation in the Chua s Equation
1722 PAPER Special Section on Nonlinear Theory and Its Applications On the Takens-Bogdanov Bifurcation in the Chua s Equation Antonio ALGABA, Emilio FREIRE, Estanislao GAMERO, and Alejandro J. RODRÍGUEZ-LUIS,
More informationSimplest Chaotic Flows with Involutional Symmetries
International Journal of Bifurcation and Chaos, Vol. 24, No. 1 (2014) 1450009 (9 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127414500096 Simplest Chaotic Flows with Involutional Symmetries
More informationBIFURCATIONS AND STRANGE ATTRACTORS IN A CLIMATE RELATED SYSTEM
dx dt DIFFERENTIAL EQUATIONS AND CONTROL PROCESSES N 1, 25 Electronic Journal, reg. N P23275 at 7.3.97 http://www.neva.ru/journal e-mail: diff@osipenko.stu.neva.ru Ordinary differential equations BIFURCATIONS
More informationTWO DIMENSIONAL FLOWS. Lecture 5: Limit Cycles and Bifurcations
TWO DIMENSIONAL FLOWS Lecture 5: Limit Cycles and Bifurcations 5. Limit cycles A limit cycle is an isolated closed trajectory [ isolated means that neighbouring trajectories are not closed] Fig. 5.1.1
More informationSurvey of strong normal-internal k : l resonances in quasi-periodically driven oscillators for l = 1, 2, 3.
June, : WSPC - Proceedings Trim Size: in x in SPT-broer Survey of strong normal-internal k : l resonances in quasi-periodically driven oscillators for l =,,. H.W. BROER and R. VAN DIJK Institute for mathematics
More informationScenarios for the transition to chaos
Scenarios for the transition to chaos Alessandro Torcini alessandro.torcini@cnr.it Istituto dei Sistemi Complessi - CNR - Firenze Istituto Nazionale di Fisica Nucleare - Sezione di Firenze Centro interdipartimentale
More informationModeling the atmosphere of Jupiter
Modeling the atmosphere of Jupiter Bruce Turkington UMass Amherst Collaborators: Richard S. Ellis (UMass Professor) Andrew Majda (NYU Professor) Mark DiBattista (NYU Postdoc) Kyle Haven (UMass PhD Student)
More informationChapter 4. Transition towards chaos. 4.1 One-dimensional maps
Chapter 4 Transition towards chaos In this chapter we will study how successive bifurcations can lead to chaos when a parameter is tuned. It is not an extensive review : there exists a lot of different
More informationPredictability of large-scale atmospheric motions: Lyapunov exponents and error dynamics
Accepted in Chaos Predictability of large-scale atmospheric motions: Lyapunov exponents and error dynamics Stéphane Vannitsem Royal Meteorological Institute of Belgium Meteorological and Climatological
More informationChapitre 4. Transition to chaos. 4.1 One-dimensional maps
Chapitre 4 Transition to chaos In this chapter we will study how successive bifurcations can lead to chaos when a parameter is tuned. It is not an extensive review : there exists a lot of different manners
More information6.2 Brief review of fundamental concepts about chaotic systems
6.2 Brief review of fundamental concepts about chaotic systems Lorenz (1963) introduced a 3-variable model that is a prototypical example of chaos theory. These equations were derived as a simplification
More informationDynamics of the Extratropical Response to Tropical Heating
Regional and Local Climate Modeling and Analysis Research Group R e L o C l i m Dynamics of the Extratropical Response to Tropical Heating (1) Wegener Center for Climate and Global Change (WegCenter) and
More informationTorus Maps from Weak Coupling of Strong Resonances
Torus Maps from Weak Coupling of Strong Resonances John Guckenheimer Alexander I. Khibnik October 5, 1999 Abstract This paper investigates a family of diffeomorphisms of the two dimensional torus derived
More informationIntroduction to Applied Nonlinear Dynamical Systems and Chaos
Stephen Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos Second Edition With 250 Figures 4jj Springer I Series Preface v L I Preface to the Second Edition vii Introduction 1 1 Equilibrium
More informationDiscussion of the Lorenz Equations
Discussion of the Lorenz Equations Leibniz Universität Hannover Proseminar Theoretische Physik SS/2015 July 22, 2015 (LUH) Lorenz Equations July 22, 2015 1 / 42 Outline 1 2 3 4 5 6 7 8 (LUH) Lorenz Equations
More informationIrregularity and Predictability of ENSO
Irregularity and Predictability of ENSO Richard Kleeman Courant Institute of Mathematical Sciences New York Main Reference R. Kleeman. Stochastic theories for the irregularity of ENSO. Phil. Trans. Roy.
More informationHomoclinic bifurcations in Chua s circuit
Physica A 262 (1999) 144 152 Homoclinic bifurcations in Chua s circuit Sandra Kahan, Anibal C. Sicardi-Schino Instituto de Fsica, Universidad de la Republica, C.C. 30, C.P. 11 000, Montevideo, Uruguay
More informationTowards a Global Theory of Singularly Perturbed Dynamical Systems John Guckenheimer Cornell University
Towards a Global Theory of Singularly Perturbed Dynamical Systems John Guckenheimer Cornell University Dynamical systems with multiple time scales arise naturally in many domains. Models of neural systems
More informationA Novel Three Dimension Autonomous Chaotic System with a Quadratic Exponential Nonlinear Term
ETASR - Engineering, Technology & Applied Science Research Vol., o.,, 9-5 9 A Novel Three Dimension Autonomous Chaotic System with a Quadratic Exponential Nonlinear Term Fei Yu College of Information Science
More informationNonlinear Balance on an Equatorial Beta Plane
Nonlinear Balance on an Equatorial Beta Plane David J. Raymond Physics Department and Geophysical Research Center New Mexico Tech Socorro, NM 87801 April 26, 2009 Summary Extension of the nonlinear balance
More informationResonance and fractal geometry
Resonance and fractal geometry Henk Broer Johann Bernoulli Institute for Mathematics and Computer Science Rijksuniversiteit Groningen Summary i. resonance in parametrized systems ii. two oscillators: torus-
More informationRecent new examples of hidden attractors
Eur. Phys. J. Special Topics 224, 1469 1476 (2015) EDP Sciences, Springer-Verlag 2015 DOI: 10.1140/epjst/e2015-02472-1 THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS Review Recent new examples of hidden
More informationThe influence of noise on two- and three-frequency quasi-periodicity in a simple model system
arxiv:1712.06011v1 [nlin.cd] 16 Dec 2017 The influence of noise on two- and three-frequency quasi-periodicity in a simple model system A.P. Kuznetsov, S.P. Kuznetsov and Yu.V. Sedova December 19, 2017
More informationBy STEVEN B. FELDSTEINI and WALTER A. ROBINSON* University of Colorado, USA 2University of Illinois at Urbana-Champaign, USA. (Received 27 July 1993)
Q. J. R. Meteorol. SOC. (1994), 12, pp. 739-745 551.513.1 Comments on Spatial structure of ultra-low frequency variability of the flow in a simple atmospheric circulation model by I. N. James and P. M.
More informationShilnikov bifurcations in the Hopf-zero singularity
Shilnikov bifurcations in the Hopf-zero singularity Geometry and Dynamics in interaction Inma Baldomá, Oriol Castejón, Santiago Ibáñez, Tere M-Seara Observatoire de Paris, 15-17 December 2017, Paris Tere
More informationChapter 23. Predicting Chaos The Shift Map and Symbolic Dynamics
Chapter 23 Predicting Chaos We have discussed methods for diagnosing chaos, but what about predicting the existence of chaos in a dynamical system. This is a much harder problem, and it seems that the
More informationDifference Resonances in a controlled van der Pol-Duffing oscillator involving time. delay
Difference Resonances in a controlled van der Pol-Duffing oscillator involving time delay This paper was published in the journal Chaos, Solitions & Fractals, vol.4, no., pp.975-98, Oct 9 J.C. Ji, N. Zhang,
More informationThe application of the theory of dynamical systems in conceptual models of environmental physics The thesis points of the PhD thesis
The application of the theory of dynamical systems in conceptual models of environmental physics The thesis points of the PhD thesis Gábor Drótos Supervisor: Tamás Tél PhD School of Physics (leader: László
More informationObservational Zonal Mean Flow Anomalies: Vacillation or Poleward
ATMOSPHERIC AND OCEANIC SCIENCE LETTERS, 2013, VOL. 6, NO. 1, 1 7 Observational Zonal Mean Flow Anomalies: Vacillation or Poleward Propagation? SONG Jie The State Key Laboratory of Numerical Modeling for
More informationno eddies eddies Figure 3. Simulated surface winds. Surface winds no eddies u, v m/s φ0 =12 φ0 =0
References Held, Isaac M., and Hou, A. Y., 1980: Nonlinear axially symmetric circulations in a nearly inviscid atmosphere. J. Atmos. Sci. 37, 515-533. Held, Isaac M., and Suarez, M. J., 1994: A proposal
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: January 14, 2019, at 08 30 12 30 Johanneberg Kristian
More informationA Truncated Model for Finite Amplitude Baroclinic Waves in a Channel
A Truncated Model for Finite Amplitude Baroclinic Waves in a Channel Zhiming Kuang 1 Introduction To date, studies of finite amplitude baroclinic waves have been mostly numerical. The numerical models,
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 5. Global Bifurcations, Homoclinic chaos, Melnikov s method Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Motivation 1.1 The problem 1.2 A
More informationMultistability in the Lorenz System: A Broken Butterfly
International Journal of Bifurcation and Chaos, Vol. 24, No. 10 (2014) 1450131 (7 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127414501314 Multistability in the Lorenz System: A Broken
More informationCOMPARISON OF THE INFLUENCES OF INITIAL ERRORS AND MODEL PARAMETER ERRORS ON PREDICTABILITY OF NUMERICAL FORECAST
CHINESE JOURNAL OF GEOPHYSICS Vol.51, No.3, 2008, pp: 718 724 COMPARISON OF THE INFLUENCES OF INITIAL ERRORS AND MODEL PARAMETER ERRORS ON PREDICTABILITY OF NUMERICAL FORECAST DING Rui-Qiang, LI Jian-Ping
More informationElements of Applied Bifurcation Theory
Yuri A. Kuznetsov Elements of Applied Bifurcation Theory Third Edition With 251 Illustrations Springer Introduction to Dynamical Systems 1 1.1 Definition of a dynamical system 1 1.1.1 State space 1 1.1.2
More informationEdward Lorenz. Professor of Meteorology at the Massachusetts Institute of Technology
The Lorenz system Edward Lorenz Professor of Meteorology at the Massachusetts Institute of Technology In 1963 derived a three dimensional system in efforts to model long range predictions for the weather
More informationLecture #2 Planetary Wave Models. Charles McLandress (Banff Summer School 7-13 May 2005)
Lecture #2 Planetary Wave Models Charles McLandress (Banff Summer School 7-13 May 2005) 1 Outline of Lecture 1. Observational motivation 2. Forced planetary waves in the stratosphere 3. Traveling planetary
More informationEdward Lorenz: Predictability
Edward Lorenz: Predictability Master Literature Seminar, speaker: Josef Schröttle Edward Lorenz in 1994, Northern Hemisphere, Lorenz Attractor I) Lorenz, E.N.: Deterministic Nonperiodic Flow (JAS, 1963)
More information7 Two-dimensional bifurcations
7 Two-dimensional bifurcations As in one-dimensional systems: fixed points may be created, destroyed, or change stability as parameters are varied (change of topological equivalence ). In addition closed
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: January 08, 2018, at 08 30 12 30 Johanneberg Kristian
More informationRotating stratified turbulence in the Earth s atmosphere
Rotating stratified turbulence in the Earth s atmosphere Peter Haynes, Centre for Atmospheric Science, DAMTP, University of Cambridge. Outline 1. Introduction 2. Momentum transport in the atmosphere 3.
More informationModeling Large-Scale Atmospheric and Oceanic Flows 2
Modeling Large-Scale Atmospheric and Oceanic Flows V. Zeitlin known s of the Laboratoire de Météorologie Dynamique, Univ. P. and M. Curie, Paris Mathematics of the Oceans, Fields Institute, Toronto, 3
More informationGoverning Equations and Scaling in the Tropics
Governing Equations and Scaling in the Tropics M 1 ( ) e R ε er Tropical v Midlatitude Meteorology Why is the general circulation and synoptic weather systems in the tropics different to the those in the
More informationThe Big, Big Picture (Bifurcations II)
The Big, Big Picture (Bifurcations II) Reading for this lecture: NDAC, Chapter 8 and Sec. 10.0-10.4. 1 Beyond fixed points: Bifurcation: Qualitative change in behavior as a control parameter is (slowly)
More informationContents. Parti Fundamentals. 1. Introduction. 2. The Coriolis Force. Preface Preface of the First Edition
Foreword Preface Preface of the First Edition xiii xv xvii Parti Fundamentals 1. Introduction 1.1 Objective 3 1.2 Importance of Geophysical Fluid Dynamics 4 1.3 Distinguishing Attributes of Geophysical
More informationAuthor: Francine Schevenhoven Utrecht University Supervisors: Dr. ir. Frank Selten (KNMI) Prof. dr. ir. Jason Frank (UU)
Author: Francine Schevenhoven Utrecht University 3697355 Supervisors: Dr. ir. Frank Selten (KNMI) Prof. dr. ir. Jason Frank (UU) June 3, 26 Abstract Historically, weather and climate forecasting has always
More informationThermosolutal Convection at Infinite Prandtl Number with or without rotation: Bifurcation and Stability in Physical Space
1/29 Thermosolutal Convection at Infinite Prandtl Number with or without rotation: Bifurcation and Stability in Physical Space Jungho Park Department of Mathematics New York Institute of Technology SIAM
More informationOn low speed travelling waves of the Kuramoto-Sivashinsky equation.
On low speed travelling waves of the Kuramoto-Sivashinsky equation. Jeroen S.W. Lamb Joint with Jürgen Knobloch (Ilmenau, Germany) Marco-Antonio Teixeira (Campinas, Brazil) Kevin Webster (Imperial College
More informationLecture 1: A Preliminary to Nonlinear Dynamics and Chaos
Lecture 1: A Preliminary to Nonlinear Dynamics and Chaos Autonomous Systems A set of coupled autonomous 1st-order ODEs. Here "autonomous" means that the right hand side of the equations does not explicitly
More informationElements of Applied Bifurcation Theory
Yuri A. Kuznetsov Elements of Applied Bifurcation Theory Second Edition With 251 Illustrations Springer Preface to the Second Edition Preface to the First Edition vii ix 1 Introduction to Dynamical Systems
More informationNWP Equations (Adapted from UCAR/COMET Online Modules)
NWP Equations (Adapted from UCAR/COMET Online Modules) Certain physical laws of motion and conservation of energy (for example, Newton's Second Law of Motion and the First Law of Thermodynamics) govern
More informationPHYSFLU - Physics of Fluids
Coordinating unit: 230 - ETSETB - Barcelona School of Telecommunications Engineering Teaching unit: 748 - FIS - Department of Physics Academic year: Degree: 2018 BACHELOR'S DEGREE IN ENGINEERING PHYSICS
More informationTopological Bifurcations of Knotted Tori in Quasiperiodically Driven Oscillators
Topological Bifurcations of Knotted Tori in Quasiperiodically Driven Oscillators Brian Spears with Andrew Szeri and Michael Hutchings University of California at Berkeley Caltech CDS Seminar October 24,
More informationSolutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10)
Solutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10) Mason A. Porter 15/05/2010 1 Question 1 i. (6 points) Define a saddle-node bifurcation and show that the first order system dx dt = r x e x
More informationPredictive Understanding of the Oceans' Wind-Driven Circulation on Interdecadal Time Scales
CCPP Mtg. Seattle, Oct. 18 20, 2004 Predictive Understanding of the Oceans' Wind-Driven Circulation on Interdecadal Time Scales Michael Ghil Atmospheric & Oceanic Sciences Dept. and IGPP, UCLA Roger Temam
More informationModel equations for planetary and synoptic scale atmospheric motions associated with different background stratification
Model equations for planetary and synoptic scale atmospheric motions associated with different background stratification Stamen Dolaptchiev & Rupert Klein Potsdam Institute for Climate Impact Research
More informationObservation Impact Assessment for Dynamic. Data-Driven Coupled Chaotic System
Applied Mathematical Sciences, Vol. 10, 2016, no. 45, 2239-2248 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.65170 Observation Impact Assessment for Dynamic Data-Driven Coupled Chaotic
More informationUniversity of Groningen. Atmospheric variability and the Atlantic multidecadal oscillation Sterk, Alef
University of Groningen Atmospheric variability and the Atlantic multidecadal oscillation Sterk, Alef IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to
More informationA Parameter Sweep Experiment on. Quasi-Periodic Variations of a Polar Vortex. due to Wave-Wave Interaction. in a Spherical Barotropic Model
A Parameter Sweep Experiment on Quasi-Periodic Variations of a Polar Vortex due to Wave-Wave Interaction in a Spherical Barotropic Model Yasuko Hio and Shigeo Yoden 1 Department of Geophysics, Kyoto University,
More informationHopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance bubble
Physica D www.elsevier.com/locate/physd Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance bubble Henk Broer a, Carles Simó b, Renato Vitolo c, a Department of
More informationLFV in a mid-latitude coupled model:
LFV in a mid-latitude coupled model: Gulf Stream influence on the NAO M. Ghil 1,2, E. Simonnet 3, Y. Feliks 1,4 1 Dept. of Atmospheric and Oceanic Sciences and IGPP, UCLA, USA. 2 Geosciences Dept. and
More informationarxiv: v1 [math.ds] 15 Jun 2010
CLIMATE DYNAMICS AND FLUID MECHANICS: NATURAL VARIABILITY AND RELATED UNCERTAINTIES arxiv:1006.2864v1 [math.ds] 15 Jun 2010 Michael Ghil Département Terre-Atmosphère-Océan, Laboratoire de Météorologie
More informationContents Dynamical Systems Stability of Dynamical Systems: Linear Approach
Contents 1 Dynamical Systems... 1 1.1 Introduction... 1 1.2 DynamicalSystems andmathematical Models... 1 1.3 Kinematic Interpretation of a System of Differential Equations... 3 1.4 Definition of a Dynamical
More informationThe Martian Climate Revisited
Peter L. Read and Stephen R. Lewis The Martian Climate Revisited Atmosphere and Environment of a Desert Planet Springer Published in association with Praxis Publishing Chichester, UK Contents Preface Abbreviations
More informationAn Introduction to Coupled Models of the Atmosphere Ocean System
An Introduction to Coupled Models of the Atmosphere Ocean System Jonathon S. Wright jswright@tsinghua.edu.cn Atmosphere Ocean Coupling 1. Important to climate on a wide range of time scales Diurnal to
More informationATMOSPHERIC AND OCEANIC FLUID DYNAMICS
ATMOSPHERIC AND OCEANIC FLUID DYNAMICS Fundamentals and Large-scale Circulation G E O F F R E Y K. V A L L I S Princeton University, New Jersey CAMBRIDGE UNIVERSITY PRESS An asterisk indicates more advanced
More informationFinite aspect ratio Taylor Couette flow: Shil nikov dynamics of 2-tori
Physica D 211 (2005) 168 191 Finite aspect ratio Taylor Couette flow: Shil nikov dynamics of 2-tori Juan M. Lopez a,, Francisco Marques b a Department of Mathematics and Statistics, Arizona State University,
More informationFROM EQUILIBRIUM TO CHAOS
FROM EQUILIBRIUM TO CHAOS Practica! Bifurcation and Stability Analysis RÜDIGER SEYDEL Institut für Angewandte Mathematik und Statistik University of Würzburg Würzburg, Federal Republic of Germany ELSEVIER
More informationIntroduction to Dynamical Systems Basic Concepts of Dynamics
Introduction to Dynamical Systems Basic Concepts of Dynamics A dynamical system: Has a notion of state, which contains all the information upon which the dynamical system acts. A simple set of deterministic
More information8.1 Bifurcations of Equilibria
1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integro-differential equations
More informationSecular and oscillatory motions in dynamical systems. Henk Broer Johann Bernoulli Instituut voor Wiskunde en Informatica Rijksuniversiteit Groningen
Secular and oscillatory motions in dynamical systems Henk Broer Johann Bernoulli Instituut voor Wiskunde en Informatica Rijksuniversiteit Groningen Contents 1. Toroidal symmetry 2. Secular (slow) versus
More informationLectures on Dynamical Systems. Anatoly Neishtadt
Lectures on Dynamical Systems Anatoly Neishtadt Lectures for Mathematics Access Grid Instruction and Collaboration (MAGIC) consortium, Loughborough University, 2007 Part 3 LECTURE 14 NORMAL FORMS Resonances
More informationWIDELY SEPARATED FREQUENCIES IN COUPLED OSCILLATORS WITH ENERGY-PRESERVING QUADRATIC NONLINEARITY
WIDELY SEPARATED FREQUENCIES IN COUPLED OSCILLATORS WITH ENERGY-PRESERVING QUADRATIC NONLINEARITY J.M. TUWANKOTTA Abstract. In this paper we present an analysis of a system of coupled oscillators suggested
More informationClimate Change and Variability in the Southern Hemisphere: An Atmospheric Dynamics Perspective
Climate Change and Variability in the Southern Hemisphere: An Atmospheric Dynamics Perspective Edwin P. Gerber Center for Atmosphere Ocean Science Courant Institute of Mathematical Sciences New York University
More informationNOTES AND CORRESPONDENCE. On the Seasonality of the Hadley Cell
1522 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 NOTES AND CORRESPONDENCE On the Seasonality of the Hadley Cell IOANA M. DIMA AND JOHN M. WALLACE Department of Atmospheric Sciences, University of Washington,
More informationWind Gyres. curl[τ s τ b ]. (1) We choose the simple, linear bottom stress law derived by linear Ekman theory with constant κ v, viz.
Wind Gyres Here we derive the simplest (and oldest; Stommel, 1948) theory to explain western boundary currents like the Gulf Stream, and then discuss the relation of the theory to more realistic gyres.
More informationIn two-dimensional barotropic flow, there is an exact relationship between mass
19. Baroclinic Instability In two-dimensional barotropic flow, there is an exact relationship between mass streamfunction ψ and the conserved quantity, vorticity (η) given by η = 2 ψ.the evolution of the
More informationProblem Set Number 2, j/2.036j MIT (Fall 2014)
Problem Set Number 2, 18.385j/2.036j MIT (Fall 2014) Rodolfo R. Rosales (MIT, Math. Dept.,Cambridge, MA 02139) Due Mon., September 29, 2014. 1 Inverse function problem #01. Statement: Inverse function
More informationBIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs
BIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs Yuri A. Kuznetsov August, 2010 Contents 1. Solutions and orbits. 2. Equilibria. 3. Periodic orbits and limit cycles. 4. Homoclinic orbits.
More information= F ( x; µ) (1) where x is a 2-dimensional vector, µ is a parameter, and F :
1 Bifurcations Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 A bifurcation is a qualitative change
More information