Microscopic Models under a Macroscopic Perspective
|
|
- Geoffrey Benson
- 5 years ago
- Views:
Transcription
1 Mathematical Models of Traffic Flow, Luminy, France, October 27 Microscopic Models under a Macroscopic Perspective Ingenuin Gasser Department of Mathematics University of Hamburg, Germany Tilman Seidel, Gabriele Sirito, Bodo Werner
2 Outline: General Dynamics of the microscopic model (homogeneous case) Dynamics of the microscopic model (non-homogeneous case and roadworks) Macroscopic view Fundamental diagrams Future Basic concept: Take a very simple microscopic model (Bando), study the full dynamics, take a macroscopic view on the results.
3 Microscopic Bando model on a circular road (scaled) N cars on a circular road of lenght L: Behaviour: x j position of the j-th car ẍ j (t) = { V ( x j+ (t) x j (t) ) ẋ j (t) }, j =,..., N, x N+ = x +L V = V (x) optimal velocity function: V () =, V strictly monoton increasing, lim x V (x) = V max
4 System for the headways: y j = x j+ x j ẏ j = z j ż j = { V (y j+ ) V (y j ) ż j }, j =,..., N, yn+ = y Additional condition: N j= y j = L quasistationary solutions: y s;j = L N, z s;j =, j =,..., N. Linear stability-analysis around this solution gives for the Eigenvalues λ: (λ 2 + λ + β) N β N =, β = V ( L N ) Result (Huijberts ( 2)): For For +cos 2π N +cos 2π N > β max = max x V (x) asymptotic stability = V ( N L ) loss of stability
5 What kind of loss of stability? (I.G., G.Sirito, B. Werner 4): Eigenvalues as functions of β = V ( L N ) k=4 k=3 k=2 B k= Imaginary axis M C A A O Beta 2 3 k= k=2 k=3 B k= Real Axis L/N Bifurcation analysis gives a Hopfbifurcation. Therefore we have locally periodic solutions. Are these solutions stable? (i.e. is the bifurcation sub- or supercritical?)
6 Criterion: Sign of the first Ljapunov-coefficient l Theorem: ( ( ) l = c 2 V L V ( L N )) 2 N V ( N L ) Conclusion: For the mostly used (Bando et al (95)) V (x) = V maxtanh(a(x )) + tanh a + tanh a the bifurction is supercritical (i.e. stable periodic orbits). But: Similar functions V give also subcritical bifurcations.
7 Problem: It seems to be very sensitive with respect to V Global bifurcation analysis: numerical tool (AUTO2) Norm of solution H 3 H Norm of solution Norm of solution H 2 H H L L L Conclusion: Globally similar functions V give similar behaviour. The bifurcation is macroscopically subcritical Conclusion for the application: the critical parameters from the linear analysis are not relevant
8 Example Example 2 Example Norm of solution H Norm of solution H Norm of solution H L L L Relative Velocity Relative Velocity.5.5 Relative Velocity Headway Headway Headway
9 More bifurcations: Eigenvalues as functions of β = V ( L N ) k=4 k=3 k=2 B k= Imaginary axis M C A A O 2 3 k= k=2 k=3 B k= Real Axis Conclusion: There are many other (weakly unstable) periodic solutions
10 (J.Greenberg 4, 7) Solutions with many oscillations finally tend to a solution with one oscillation (G. Oroz, R.E. Wilson, B.Krauskopf 4, 5) Qualitatively the same global bifurcation diagram for the model with delay
11 Extension to standart microscopic model Every driver is aggressive with weight α ẍ j (t) = α { ( V xj+ (t) x j (t) ) ẋ j (t) } +α { ẋ j+ (t) ẋ j (t) }, τ j =,..., N, x N+ = x + L optimal velocity-function: loss of stability similar Aggressive drivers stabilize the fraffic flow! unfortunately also the number of accidents increases! (Olmos & Munos, Condensed matter 24)
12
13 V max ǫ Road works Symmetry breaking, the above theory is not easily applicable A solution is called ponies on a Merry-Go-Round solution (short POM), if there is a T R, such that (i) x i (t + T) = x i (t) + L (i =,..., N) (ii) x i (t) = x i ( t + T N ) (i =,..., N) hold (Aronson, Golubitsky, Mallet-Paret 9). We call T rotation number and N T the phase (phase shift).
14 Theorem:The above model has POM solutions for small ǫ > ẋ x Velocity of the quasistationary solution (no roadwork) versus roadwork solution (The red line indicates maximum velocity).
15 Technique: Poincare maps Π(η) = Φ T(η) (η) Λ, where Φ is the induced flow and Λ reduces the spacial components by L. Π(ξ) Λ ξ x x + Λ Σ Φ T(ǫ,ξ) Σ + Λ T Study fixed points of the corresponding Poincare and reduced Poincare maps. Roadworks are (regular) perturbations. x ǫ
16 Bifurcation diagram in (L, ǫ)-plane: I ε II L A curve of Neimark-Sacker bifurcations in the (L, ǫ)-plane for N = 5.
17 Four different attractors.: ǫ I II ǫ = trivial POM x Hopf periodic solution ǫ > POM x ǫ quasi-pom i.e. here POM s are typically perturbed quasistationary solutions quasi-pom s are perturbed (Hopf) periodic solutions
18 Invariant curves:.8.8 velocity v(ρ).6.4 velocity v(ρ) headway /ρ headway /ρ Two closed invariant curves (ǫ = and ǫ > ) of the reduced Poincaré map π. On the left also the optimal velocity function V is given in gray.
19 The 4 different scenarios: v.5 v x mod L L x L v.5 v x mod L L x L above: no roadworks, below: with roadworks
20 Macroscopic view of the 4 different scenarios: t t x mod L 2 6 x mod L t t x mod L x mod L 8 2 above: no roadworks, below: with roadworks
21 Macroscopic view of density and velocity: t t x mod L x mod L strong road work influence (ǫ =.32) 25
22 Fundamental diagrams: A real world point of view on the reduced Poincaré map π for N =, ǫ =.
23 Fundamental diagrams I: ρ v(ρ) ρ ρ v(ρ) ρ velocity v(ρ) headway /ρ Overlapped fundamental diagrams for N =, L = 5,..., 4 measuring at a fixed point.
24 Fundamental diagrams II: ρ v(ρ) ρ v(ρ) N/L N/L Fundamental diagram of time-averaged flow versus average density for N =, L =
25 Fundamental diagrams III (with roadworks): traffic flow ρ i v i density ρ i Overlapped fundamental diagrams for N =, L = 5,...,4, ǫ =. measuring at a fixed point.
26 Current and future work: Is this dynamics contained in macroscopic models? Which marcoscopic model has the same (rich) dynamics than the basic Bando model Micro-macro link (Aw, Klar, Materne, Rascle 22)
Dynamical Phenomena induced by Bottleneck
March 23, Sophia Antipolis, Workshop TRAM2 Dynamical Phenomena induced by Bottleneck Ingenuin Gasser Department of Mathematics University of Hamburg, Germany Tilman Seidel, Gabriele Sirito, Bodo Werner
More informationCar-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation R. Eddie Wilson, University of Bristol EPSRC Advanced Research Fellowship EP/E055567/1 http://www.enm.bris.ac.uk/staff/rew
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 informationDynamical system theory and bifurcation analysis for microscopic traffic models
Dynamical system theory and bifurcation analysis for microscopic traffic models Bodo Werner University of Hamburg L Aquila April 2010 2010 MathMods IP mailto:werner@math.uni-hamburg.de April 30, 2010 Abstract
More informationTwo-dimensional macroscopic models for traffic flow on highways
Two-dimensional macroscopic models for traffic flow on highways Giuseppe Visconti Institut für Geometrie und Praktische Mathematik RWTH Aachen University (Germany) XVII Italian Meeting on Hyperbolic Equations
More information2 Discrete growth models, logistic map (Murray, Chapter 2)
2 Discrete growth models, logistic map (Murray, Chapter 2) As argued in Lecture 1 the population of non-overlapping generations can be modelled as a discrete dynamical system. This is an example of an
More informationTHE EXACTLY SOLVABLE SIMPLEST MODEL FOR QUEUE DYNAMICS
DPNU-96-31 June 1996 THE EXACTLY SOLVABLE SIMPLEST MODEL FOR QUEUE DYNAMICS arxiv:patt-sol/9606001v1 7 Jun 1996 Yūki Sugiyama Division of Mathematical Science City College of Mie, Tsu, Mie 514-01 Hiroyasu
More information1.7. Stability and attractors. Consider the autonomous differential equation. (7.1) ẋ = f(x),
1.7. Stability and attractors. Consider the autonomous differential equation (7.1) ẋ = f(x), where f C r (lr d, lr d ), r 1. For notation, for any x lr d, c lr, we let B(x, c) = { ξ lr d : ξ x < c }. Suppose
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 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 information(8.51) ẋ = A(λ)x + F(x, λ), where λ lr, the matrix A(λ) and function F(x, λ) are C k -functions with k 1,
2.8.7. Poincaré-Andronov-Hopf Bifurcation. In the previous section, we have given a rather detailed method for determining the periodic orbits of a two dimensional system which is the perturbation of a
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 4. Bifurcations Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Local bifurcations for vector fields 1.1 The problem 1.2 The extended centre
More information1. (i) Determine how many periodic orbits and equilibria can be born at the bifurcations of the zero equilibrium of the following system:
1. (i) Determine how many periodic orbits and equilibria can be born at the bifurcations of the zero equilibrium of the following system: ẋ = y x 2, ẏ = z + xy, ż = y z + x 2 xy + y 2 + z 2 x 4. (ii) Determine
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 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 informationStability and Hopf bifurcation analysis of the Mackey-Glass and Lasota equations
Stability and Hopf bifurcation analysis of the Mackey-Glass and Lasota equations Sreelakshmi Manjunath Department of Electrical Engineering Indian Institute of Technology Madras (IITM), India JTG Summer
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 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 informationTorus Doubling Cascade in Problems with Symmetries
Proceedings of Institute of Mathematics of NAS of Ukraine 4, Vol., Part 3, 11 17 Torus Doubling Cascade in Problems with Symmetries Faridon AMDJADI School of Computing and Mathematical Sciences, Glasgow
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 informationCylindrical Manifolds and Tube Dynamics in the Restricted Three-Body Problem
C C Dynamical A L T E C S H Cylindrical Manifolds and Tube Dynamics in the Restricted Three-Body Problem Shane D. Ross Control and Dynamical Systems, Caltech www.cds.caltech.edu/ shane/pub/thesis/ April
More informationOn a Codimension Three Bifurcation Arising in a Simple Dynamo Model
On a Codimension Three Bifurcation Arising in a Simple Dynamo Model Anne C. Skeldon a,1 and Irene M. Moroz b a Department of Mathematics, City University, Northampton Square, London EC1V 0HB, England b
More informationString and robust stability of connected vehicle systems with delayed feedback
String and robust stability of connected vehicle systems with delayed feedback Gopal Krishna Kamath, Krishna Jagannathan and Gaurav Raina Department of Electrical Engineering Indian Institute of Technology
More information5.2.2 Planar Andronov-Hopf bifurcation
138 CHAPTER 5. LOCAL BIFURCATION THEORY 5.. Planar Andronov-Hopf bifurcation What happens if a planar system has an equilibrium x = x 0 at some parameter value α = α 0 with eigenvalues λ 1, = ±iω 0, ω
More informationFeedback-mediated oscillatory coexistence in the chemostat
Feedback-mediated oscillatory coexistence in the chemostat Patrick De Leenheer and Sergei S. Pilyugin Department of Mathematics, University of Florida deleenhe,pilyugin@math.ufl.edu 1 Introduction We study
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 informationBifurcations and multiple traffic jams in a car-following model with reaction-time delay
Physica D 211 (2005) 277 293 Bifurcations and multiple traffic jams in a car-following model with reaction-time delay Gábor Orosz, Bernd Krauskopf, R. Eddie Wilson Bristol Centre for Applied Nonlinear
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 informationLecture 5. Outline: Limit Cycles. Definition and examples How to rule out limit cycles. Poincare-Bendixson theorem Hopf bifurcations Poincare maps
Lecture 5 Outline: Limit Cycles Definition and examples How to rule out limit cycles Gradient systems Liapunov functions Dulacs criterion Poincare-Bendixson theorem Hopf bifurcations Poincare maps Limit
More informationLecture 3 : Bifurcation Analysis
Lecture 3 : Bifurcation Analysis D. Sumpter & S.C. Nicolis October - December 2008 D. Sumpter & S.C. Nicolis General settings 4 basic bifurcations (as long as there is only one unstable mode!) steady state
More informationPart II. Dynamical Systems. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 34 Paper 1, Section II 30A Consider the dynamical system where β > 1 is a constant. ẋ = x + x 3 + βxy 2, ẏ = y + βx 2
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 informationGeometry of Resonance Tongues
Geometry of Resonance Tongues Henk Broer with Martin Golubitsky, Sijbo Holtman, Mark Levi, Carles Simó & Gert Vegter Instituut voor Wiskunde en Informatica Rijksuniversiteit Groningen Resonance p.1/36
More informationHopf Bifurcations in Problems with O(2) Symmetry: Canonical Coordinates Transformation
Proceedings of Institute of Mathematics of NAS of Ukraine 2002, Vol. 43, Part 1, 65 71 Hopf Bifurcations in Problems with O(2) Symmetry: Canonical Coordinates Transformation Faridon AMDJADI Department
More informationModelling in Biology
Modelling in Biology Dr Guy-Bart Stan Department of Bioengineering 17th October 2017 Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October 2017 1 / 77 1 Introduction 2 Linear models of
More informationDYNAMICS OF THREE COUPLED VAN DER POL OSCILLATORS WITH APPLICATION TO CIRCADIAN RHYTHMS
Proceedings of IDETC/CIE 2005 ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference September 24-28, 2005, Long Beach, California USA DETC2005-84017
More informationMaster Thesis. Equivariant Pyragas control. Isabelle Schneider Freie Universität Berlin Fachbereich Mathematik und Informatik Berlin
Master Thesis Equivariant Pyragas control Isabelle Schneider Freie Universität Berlin Fachbereich Mathematik und Informatik 14195 Berlin February 3, 2014 Abstract In this thesis we show how the concept
More informationDelayed feedback control of three diusively coupled Stuart-Landau oscillators: a case study in equivariant Hopf bifurcation
Delayed feedback control of three diusively coupled Stuart-Landau oscillators: a case study in equivariant Hopf bifurcation Isabelle Schneider Freie Universität Berlin, Fachbereich Mathematik und Informatik,
More informationPoincaré Map, Floquet Theory, and Stability of Periodic Orbits
Poincaré Map, Floquet Theory, and Stability of Periodic Orbits CDS140A Lecturer: W.S. Koon Fall, 2006 1 Poincaré Maps Definition (Poincaré Map): Consider ẋ = f(x) with periodic solution x(t). Construct
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 informationPersistent Chaos in High-Dimensional Neural Networks
Persistent Chaos in High-Dimensional Neural Networks D. J. Albers with J. C. Sprott and James P. Crutchfield February 20, 2005 1 Outline: Introduction and motivation Mathematical versus computational dynamics
More informationDelay-induced chaos with multifractal attractor in a traffic flow model
EUROPHYSICS LETTERS 15 January 2001 Europhys. Lett., 57 (2), pp. 151 157 (2002) Delay-induced chaos with multifractal attractor in a traffic flow model L. A. Safonov 1,2, E. Tomer 1,V.V.Strygin 2, Y. Ashkenazy
More informationThe Existence of Chaos in the Lorenz System
The Existence of Chaos in the Lorenz System Sheldon E. Newhouse Mathematics Department Michigan State University E. Lansing, MI 48864 joint with M. Berz, K. Makino, A. Wittig Physics, MSU Y. Zou, Math,
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 informationTypicality paradigm in Quantum Statistical Thermodynamics Barbara Fresch, Giorgio Moro Dipartimento Scienze Chimiche Università di Padova
Typicality paradigm in Quantum Statistical Thermodynamics Barbara Fresch, Giorgio Moro Dipartimento Scienze Chimiche Università di Padova Outline 1) The framework: microcanonical statistics versus the
More informationHamad Talibi Alaoui and Radouane Yafia. 1. Generalities and the Reduced System
FACTA UNIVERSITATIS (NIŠ) Ser. Math. Inform. Vol. 22, No. 1 (27), pp. 21 32 SUPERCRITICAL HOPF BIFURCATION IN DELAY DIFFERENTIAL EQUATIONS AN ELEMENTARY PROOF OF EXCHANGE OF STABILITY Hamad Talibi Alaoui
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering Department of Mathematics & Statistics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MS08 SEMESTER: Spring 0 MODULE TITLE: Dynamical
More informationCHAPTER 6 HOPF-BIFURCATION IN A TWO DIMENSIONAL NONLINEAR DIFFERENTIAL EQUATION
CHAPTER 6 HOPF-BIFURCATION IN A TWO DIMENSIONAL NONLINEAR DIFFERENTIAL EQUATION [Discussion on this chapter is based on our paper entitled Hopf-Bifurcation Ina Two Dimensional Nonlinear Differential Equation,
More informationChapter 3. Gumowski-Mira Map. 3.1 Introduction
Chapter 3 Gumowski-Mira Map 3.1 Introduction Non linear recurrence relations model many real world systems and help in analysing their possible asymptotic behaviour as the parameters are varied [17]. Here
More informationDynamical systems tutorial. Gregor Schöner, INI, RUB
Dynamical systems tutorial Gregor Schöner, INI, RUB Dynamical systems: Tutorial the word dynamics time-varying measures range of a quantity forces causing/accounting for movement => dynamical systems dynamical
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 information2 Lecture 2: Amplitude equations and Hopf bifurcations
Lecture : Amplitude equations and Hopf bifurcations This lecture completes the brief discussion of steady-state bifurcations by discussing vector fields that describe the dynamics near a bifurcation. From
More informationMacroscopic Simulation of Open Systems and Micro-Macro Link
Macroscopic Simulation of Open Systems and Micro-Macro Link Ansgar Hennecke 1 and Martin Treiber 1 and Dirk Helbing 1 II Institute for Theoretical Physics, University Stuttgart, Pfaffenwaldring 57, D-7756
More informationas Hopf Bifurcations in Time-Delay Systems with Band-limited Feedback
as Hopf Bifurcations in Time-Delay Systems with Band-limited Feedback Lucas Illing and Daniel J. Gauthier Department of Physics Center for Nonlinear and Complex Systems Duke University, North Carolina
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 informationHopf Bifurcation Analysis of a Dynamical Heart Model with Time Delay
Applied Mathematical Sciences, Vol 11, 2017, no 22, 1089-1095 HIKARI Ltd, wwwm-hikaricom https://doiorg/1012988/ams20177271 Hopf Bifurcation Analysis of a Dynamical Heart Model with Time Delay Luca Guerrini
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 informationThe projects listed on the following pages are suitable for MSc/MSci or PhD students. An MSc/MSci project normally requires a review of the
The projects listed on the following pages are suitable for MSc/MSci or PhD students. An MSc/MSci project normally requires a review of the literature and finding recent related results in the existing
More informationStationary radial spots in a planar threecomponent reaction-diffusion system
Stationary radial spots in a planar threecomponent reaction-diffusion system Peter van Heijster http://www.dam.brown.edu/people/heijster SIAM Conference on Nonlinear Waves and Coherent Structures MS: Recent
More informationA lattice traffic model with consideration of preceding mixture traffic information
Chin. Phys. B Vol. 0, No. 8 011) 088901 A lattice traffic model with consideration of preceding mixture traffic information Li Zhi-Peng ) a), Liu Fu-Qiang ) a), Sun Jian ) b) a) School of Electronics and
More informationarxiv: v1 [physics.class-ph] 5 Jan 2012
Damped bead on a rotating circular hoop - a bifurcation zoo Shovan Dutta Department of Electronics and Telecommunication Engineering, Jadavpur University, Calcutta 700 032, India. Subhankar Ray Department
More informationFrom experimemts to Modeling
Traffic Flow: From experimemts to Modeling TU Dresden 1 1 Overview Empirics: Stylized facts Microscopic and macroscopic models: typical examples: Linear stability: Which concepts are relevant for describing
More informationProf. Krstic Nonlinear Systems MAE281A Homework set 1 Linearization & phase portrait
Prof. Krstic Nonlinear Systems MAE28A Homework set Linearization & phase portrait. For each of the following systems, find all equilibrium points and determine the type of each isolated equilibrium. Use
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 informationDYNAMICAL MODEL AND NUMERICAL SIMULATION OF TRAFFIC CONGESTION
DYNAMICAL MODEL AND NUMERICAL SIMULATION OF TRAFFIC CONGESTION A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree
More informationHopf Bifurcation in a Scalar Reaction Diffusion Equation
journal of differential equations 140, 209222 (1997) article no. DE973307 Hopf Bifurcation in a Scalar Reaction Diffusion Equation Patrick Guidotti and Sandro Merino Mathematisches Institut, Universita
More informationIntroduction to fractal analysis of orbits of dynamical systems. ZAGREB DYNAMICAL SYSTEMS WORKSHOP 2018 Zagreb, October 22-26, 2018
Vesna Županović Introduction to fractal analysis of orbits of dynamical systems University of Zagreb, Croatia Faculty of Electrical Engineering and Computing Centre for Nonlinear Dynamics, Zagreb ZAGREB
More informationQuasiperiodic phenomena in the Van der Pol - Mathieu equation
Quasiperiodic penomena in te Van der Pol - Matieu equation F. Veerman and F. Verulst Department of Matematics, Utrect University P.O. Box 80.010, 3508 TA Utrect Te Neterlands April 8, 009 Abstract Te Van
More informationNumerical techniques: Deterministic Dynamical Systems
Numerical techniques: Deterministic Dynamical Systems Henk Dijkstra Institute for Marine and Atmospheric research Utrecht, Department of Physics and Astronomy, Utrecht, The Netherlands Transition behavior
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 informationarxiv: v1 [physics.soc-ph] 17 Oct 2016
Local stability conditions and calibrating procedure for new car-following models used in driving simulators arxiv:1610.05257v1 [physics.soc-ph] 17 Oct 2016 Valentina Kurc and Igor Anufriev Abstract The
More informationSieber, J., Kowalczyk, P. S., Hogan, S. J., & di Bernardo, M. (2007). Dynamics of symmetric dynamical systems with delayed switching.
Sieber, J., Kowalczyk, P. S., Hogan, S. J., & di Bernardo, M. (2007). Dynamics of symmetric dynamical systems with delayed switching. Early version, also known as pre-print Link to publication record in
More informationAn extended microscopic traffic flow model based on the spring-mass system theory
Modern Physics Letters B Vol. 31, No. 9 (2017) 1750090 (9 pages) c World Scientific Publishing Company DOI: 10.1142/S0217984917500907 An extended microscopic traffic flow model based on the spring-mass
More informationLecture 5. Numerical continuation of connecting orbits of iterated maps and ODEs. Yu.A. Kuznetsov (Utrecht University, NL)
Lecture 5 Numerical continuation of connecting orbits of iterated maps and ODEs Yu.A. Kuznetsov (Utrecht University, NL) May 26, 2009 1 Contents 1. Point-to-point connections. 2. Continuation of homoclinic
More informationNonlinear Dynamics and Chaos
Ian Eisenman eisenman@fas.harvard.edu Geological Museum 101, 6-6352 Nonlinear Dynamics and Chaos Review of some of the topics covered in homework problems, based on section notes. December, 2005 Contents
More informationThe LWR model on a network
Mathematical Models of Traffic Flow (October 28 November 1, 2007) Mauro Garavello Benedetto Piccoli DiSTA I.A.C. Università del Piemonte Orientale C.N.R. Via Bellini, 25/G Viale del Policlinico, 137 15100
More informationMathematical Models of Traffic Flow: Macroscopic and Microscopic Aspects Zurich, March 2011
Mathematical Models of Traffic Flow: Macroscopic and Microscopic Aspects Zurich, March 2011 Michel Rascle Laboratoire JA Dieudonné, Université de Nice Sophia-Antipolis Parc Valrose 06108 Nice Cedex 02,
More informationResearch Article Hopf Bifurcation Analysis and Anticontrol of Hopf Circles of the Rössler-Like System
Abstract and Applied Analysis Volume, Article ID 3487, 6 pages doi:.55//3487 Research Article Hopf Bifurcation Analysis and Anticontrol of Hopf Circles of the Rössler-Like System Ranchao Wu and Xiang Li
More informationDynamics of Two Coupled van der Pol Oscillators with Delay Coupling Revisited
Dynamics of Two Coupled van der Pol Oscillators with Delay Coupling Revisited arxiv:1705.03100v1 [math.ds] 8 May 017 Mark Gluzman Center for Applied Mathematics Cornell University and Richard Rand Dept.
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 informationNumerical qualitative analysis of a large-scale model for measles spread
Numerical qualitative analysis of a large-scale model for measles spread Hossein Zivari-Piran Department of Mathematics and Statistics York University (joint work with Jane Heffernan) p./9 Outline Periodic
More informationPhase Synchronization
Phase Synchronization Lecture by: Zhibin Guo Notes by: Xiang Fan May 10, 2016 1 Introduction For any mode or fluctuation, we always have where S(x, t) is phase. If a mode amplitude satisfies ϕ k = ϕ k
More informationSHANGHAI JIAO TONG UNIVERSITY LECTURE
Lecture 4 SHANGHAI JIAO TONG UNIVERSITY LECTURE 4 017 Anthony J. Leggett Department of Physics University of Illinois at Urbana-Champaign, USA and Director, Center for Complex Physics Shanghai Jiao Tong
More informationCalculating Fractal Dimension of Attracting Sets of the Lorenz System
Dynamics at the Horsetooth Volume 6, 2014. Calculating Fractal Dimension of Attracting Sets of the Lorenz System Jamie Department of Mathematics Colorado State University Report submitted to Prof. P. Shipman
More informationCanards at Folded Nodes
Canards at Folded Nodes John Guckenheimer and Radu Haiduc Mathematics Department, Ithaca, NY 14853 For Yulij Il yashenko with admiration and affection on the occasion of his 60th birthday March 18, 2003
More informationSummary of topics relevant for the final. p. 1
Summary of topics relevant for the final p. 1 Outline Scalar difference equations General theory of ODEs Linear ODEs Linear maps Analysis near fixed points (linearization) Bifurcations How to analyze a
More informationBifurcation of Fixed Points
Bifurcation of Fixed Points CDS140B Lecturer: Wang Sang Koon Winter, 2005 1 Introduction ẏ = g(y, λ). where y R n, λ R p. Suppose it has a fixed point at (y 0, λ 0 ), i.e., g(y 0, λ 0 ) = 0. Two Questions:
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 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 informationExistence, stability, and mitigation of gridlock in beltway networks
Existence, stability, and mitigation of gridlock in beltway networks Wen-Long Jin a, a Department of Civil and Environmental Engineering, 4000 Anteater Instruction and Research Bldg, University of California,
More informationConnecting orbits and invariant manifolds in the spatial three-body problem
C C Dynamical A L T E C S H Connecting orbits and invariant manifolds in the spatial three-body problem Shane D. Ross Control and Dynamical Systems, Caltech Work with G. Gómez, W. Koon, M. Lo, J. Marsden,
More informationStability and bifurcation in network traffic flow: A Poincaré map approach
Stability and bifurcation in network traffic flow: A Poincaré map approach arxiv:1307.7671v1 [math.ds] 29 Jul 2013 Wen-Long Jin July 30, 2013 Abstract Previous studies have shown that, in a diverge-merge
More informationPositive Solutions of Operator Equations
M. A. KRASNOSEL'SKII Positive Solutions of Operator Equations Translated from the Russian by RICHARD E. FLAHERTY Edited by LEO F. BORON P. NOORDHOFF LTD GRONINGEN THE NETHERLANDS CONTENTS Foreword 15 Chapter
More informationBoulder School for Condensed Matter and Materials Physics. Laurette Tuckerman PMMH-ESPCI-CNRS
Boulder School for Condensed Matter and Materials Physics Laurette Tuckerman PMMH-ESPCI-CNRS laurette@pmmh.espci.fr Dynamical Systems: A Basic Primer 1 1 Basic bifurcations 1.1 Fixed points and linear
More informationAPPPHYS217 Tuesday 25 May 2010
APPPHYS7 Tuesday 5 May Our aim today is to take a brief tour of some topics in nonlinear dynamics. Some good references include: [Perko] Lawrence Perko Differential Equations and Dynamical Systems (Springer-Verlag
More informationEffect of the Trace Anomaly on the Cosmological Constant. Jurjen F. Koksma
Effect of the Trace Anomaly on the Cosmological Constant Jurjen F. Koksma Invisible Universe Spinoza Institute Institute for Theoretical Physics Utrecht University 2nd of July 2009 J.F. Koksma T. Prokopec
More informationNonlinear Analysis of a New Car-Following Model Based on Internet-Connected Vehicles
Nonlinear Analysis of a New Car-Following Model Based on Internet-Connected Vehicles Lei Yu1*, Bingchang Zhou, Zhongke Shi1 1 College School of Automation, Northwestern Polytechnical University, Xi'an,
More informationSingularities in cosmologies with interacting fluids
Georgia Kittou gkittou@icsd.aegean.gr co-author Spiros Cotsakis University of the Aegean, Research Group GeoDySyC MG13 2012 Stockholm Parallel Session GT2: Cosmological Singularities and Asymptotics 02/07/2012
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 informationThe Forced van der Pol Equation II: Canards in the Reduced System
SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 2, No. 4, pp. 57 68 c 23 Society for Industrial and Applied Mathematics The Forced van der Pol Equation II: Canards in the Reduced System Katherine Bold, Chantal
More information