Periodic windows within windows within windows
|
|
- Emerald Cox
- 5 years ago
- Views:
Transcription
1 Periodic windows within windows within windows Madhura Joglekar Applied Mathematics & Statistics, and Scientific Computation University of Maryland College Park March 29, 2014 References C. Grebogi, S. McDonald, E. Ott and J. A. Yorke, Phys. Let. A 110, (1985), 1-4 B. R. Hunt and E. Ott, J. Phys. A 30 (1997), J. D. Farmer, Phys. Rev. Lett. 55, (1985),
2 Basins of attraction of the Forced Damped Pendulum Equation: X + 0.2X + sin X = ρ cos T ρ = 2.45 gives a chaotic attractor ρ = 2.55 gives periodic attractors. Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 2/17
3 What is an ɛ-uncertain point? ɛ-uncertainty: A point (x, C) lying in the basin of a periodic attractor is ɛ-uncertain if there is a point within ɛ-distance that can result in chaos Can be defined in state space as well as in parameter space Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 3/17
4 Questions What fraction of the space consists of ɛ-uncertain points? Where are ɛ-uncertain points most likely to lie? In higher dimensional systems, difficult to predict asymptotic behavior given initial state Study the 1-dim quad map x n+1 = C x n 2 (start from x = 0) Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 4/17
5 Questions What fraction of the space consists of ɛ-uncertain points? Where are ɛ-uncertain points most likely to lie? In higher dimensional systems, difficult to predict asymptotic behavior given initial state Study the 1-dim quad map x n+1 = C x n 2 (start from x = 0) Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 4/17
6 Periodic Windows in the 1 Dim Quadratic Map x n+1 = C x n 2 where C [ 0.25, 2] Infinitely many windows Dense in parameter space, fractal structure Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 5/17
7 ɛ-uncertain C values in x n+1 = C x n 2 Given C results in a periodic attractor C is within ɛ of chaos Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 6/17
8 ɛ-uncertain C values in x n+1 = C x n 2 The fraction of ɛ-uncertain C values in a window depends on its width Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 7/17
9 Randomly choosing an ɛ-uncertain value of C C. Grebogi, S. W. McDonald, E. Ott and J. A. Yorke, Exterior dimension of fat fractals" Phys. Let. A 110, 1-4, 1985 fraction of ɛ-uncertain C values ɛ 0.41 Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 8/17
10 Study distribution of primary-window widths Using kneading theory, determine sequence of all windows Compute C width = C crisis C saddlenode Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 9/17
11 For x n+1 = C x n 2, what is distribution of C-widths? N(ɛ): No. of primary windows with C-width > ɛ Cluster computation in quadruple precision Computed windows of periods 25 : (No. of windows with C-width ɛ) vs ɛ Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 10/17
12 For x n+1 = C x n 2, what is distribution of C-widths? N(ɛ): No. of primary windows with C-width > ɛ Cluster computation in quadruple precision Computed windows of periods 25 : (No. of windows with C-width ɛ) vs ɛ Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 10/17
13 For x n+1 = C x n 2, what is distribution of C-widths? N(ɛ) = 0.133ɛ.51 Scaling exponent α 0.51 : (No. of windows with C-width ɛ) vs ɛ How does this relate to the fraction of ɛ-uncertain C values? Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 11/17
14 For x n+1 = C x n 2, what is distribution of C-widths? N(ɛ) = 0.133ɛ.51 Scaling exponent α 0.51 : (No. of windows with C-width ɛ) vs ɛ How does this relate to the fraction of ɛ-uncertain C values? Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 11/17
15 Relation between N(ɛ) and f P (ɛ) f P (ɛ): fraction of ɛ-uncertain C values in primary windows lim ɛ 0 log f P (ɛ) log ɛ 1 α Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 12/17
16 Relation between N(ɛ) and f P (ɛ) f P (ɛ): fraction of ɛ-uncertain C values in primary windows lim ɛ 0 log f P (ɛ) log ɛ 1 α As ɛ 0, most of the ɛ-uncertain C values lie in higher order windows! Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 13/17
17 Relation between N(ɛ) and f P (ɛ) f P (ɛ): fraction of ɛ-uncertain C values in primary windows lim ɛ 0 log f P (ɛ) log ɛ 1 α As ɛ 0, most of the ɛ-uncertain C values lie in higher order windows! Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 13/17
18 Where does a randomly chosen ɛ-uncertain value of C lie? Primary window width scaling N 1 (ɛ) = 0.133ɛ.51 : (No. of windows with C-width ɛ) vs ɛ Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 14/17
19 Where does a randomly chosen ɛ-uncertain value of C lie? Primary window width scaling N 1 (ɛ) = 0.133ɛ.51 Assume exact self-similarity of periodic windows N k (ɛ): No. of k th order windows with width > ɛ Derive a generalized formula for scaling of higher order windows, i.e., N k (ɛ) for all positive integers k Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 15/17
20 Where does a randomly chosen ɛ-uncertain value of C lie? Primary window width scaling N 1 (ɛ) = 0.133ɛ.51 Assume exact self-similarity of periodic windows N k (ɛ): No. of k th order windows with width > ɛ Derive a generalized formula for scaling of higher order windows, i.e., N k (ɛ) for all positive integers k Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 15/17
21 Where does a randomly chosen ɛ-uncertain value of C lie? Theorem: Choose an ɛ-uncertain point randomly. Say this point lies in a window of order r. As per the theorem, for all positive integers n, lim Probability(r > n) = 1 ɛ 0 As ɛ takes small values, Most ɛ-uncertain points lie in a window within a window within a window... N th order window for large N. Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 16/17
22 Where does a randomly chosen ɛ-uncertain value of C lie? Theorem: Choose an ɛ-uncertain point randomly. Say this point lies in a window of order r. As per the theorem, for all positive integers n, lim Probability(r > n) = 1 ɛ 0 As ɛ takes small values, Most ɛ-uncertain points lie in a window within a window within a window... N th order window for large N. Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 16/17
23 Where does a randomly chosen ɛ-uncertain value of C lie? Theorem: Choose an ɛ-uncertain point randomly. Say this point lies in a window of order r. As per the theorem, for all positive integers n, lim Probability(r > n) = 1 ɛ 0 As ɛ takes small values, Most ɛ-uncertain points lie in a window within a window within a window... N th order window for large N. Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 16/17
24 And thus, As ɛ takes small values, Most ɛ-uncertain points lie in a window within a window within a window... N th order window for large N. Acknowledgements: I would like to thank my advisor Jim Yorke, and Ed Ott for their suggestions. Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 17/17
Rotational Number Approach to a Damped Pendulum under Parametric Forcing
Journal of the Korean Physical Society, Vol. 44, No. 3, March 2004, pp. 518 522 Rotational Number Approach to a Damped Pendulum under Parametric Forcing Eun-Ah Kim and K.-C. Lee Department of Physics,
More informationABOUT UNIVERSAL BASINS OF ATTRACTION IN HIGH-DIMENSIONAL SYSTEMS
International Journal of Bifurcation and Chaos, Vol. 23, No. 12 (2013) 1350197 (7 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127413501976 ABOUT UNIVERSAL BASINS OF ATTRACTION IN HIGH-DIMENSIONAL
More informationMechanisms of Chaos: Stable Instability
Mechanisms of Chaos: Stable Instability Reading for this lecture: NDAC, Sec. 2.-2.3, 9.3, and.5. Unpredictability: Orbit complicated: difficult to follow Repeatedly convergent and divergent Net amplification
More informationABSTRACT. dynamics, particularly in the case of simple flows in which the base laminar flow
ABSTRACT Title of dissertation: Dissertation directed by: ROBUSTNESS OF ATTRACTING ORBITS Madhura Joglekar, Doctor of Philosophy, 2014 Research Professor James A. Yorke Coadvisors: Professors Ed. Ott and
More informationChaotic motion. Phys 750 Lecture 9
Chaotic motion Phys 750 Lecture 9 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t =0to
More informationInvariant manifolds of the Bonhoeffer-van der Pol oscillator
Invariant manifolds of the Bonhoeffer-van der Pol oscillator R. Benítez 1, V. J. Bolós 2 1 Dpto. Matemáticas, Centro Universitario de Plasencia, Universidad de Extremadura. Avda. Virgen del Puerto 2. 10600,
More informationCharacterization of the natural measure by unstable periodic orbits in nonhyperbolic chaotic systems
PHYSICAL REVIEW E VOLUME 56, NUMBER 6 DECEMBER 1997 Characterization of the natural measure by unstable periodic orbits in nonhyperbolic chaotic systems Ying-Cheng Lai * Department of Physics and Astronomy
More informationarxiv: v1 [nlin.cd] 20 Jul 2010
Invariant manifolds of the Bonhoeffer-van der Pol oscillator arxiv:1007.3375v1 [nlin.cd] 20 Jul 2010 R. Benítez 1, V. J. Bolós 2 1 Departamento de Matemáticas, Centro Universitario de Plasencia, Universidad
More informationA Two-dimensional Discrete Mapping with C Multifold Chaotic Attractors
EJTP 5, No. 17 (2008) 111 124 Electronic Journal of Theoretical Physics A Two-dimensional Discrete Mapping with C Multifold Chaotic Attractors Zeraoulia Elhadj a, J. C. Sprott b a Department of Mathematics,
More informationOscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. Oscillatory solution
Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic angular frequency
More informationOscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. small angle approximation. Oscillatory solution
Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ small angle approximation θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic
More informationChaotic motion. Phys 420/580 Lecture 10
Chaotic motion Phys 420/580 Lecture 10 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t
More informationChaos suppression of uncertain gyros in a given finite time
Chin. Phys. B Vol. 1, No. 11 1 1155 Chaos suppression of uncertain gyros in a given finite time Mohammad Pourmahmood Aghababa a and Hasan Pourmahmood Aghababa bc a Electrical Engineering Department, Urmia
More informationChaos. Dr. Dylan McNamara people.uncw.edu/mcnamarad
Chaos Dr. Dylan McNamara people.uncw.edu/mcnamarad Discovery of chaos Discovered in early 1960 s by Edward N. Lorenz (in a 3-D continuous-time model) Popularized in 1976 by Sir Robert M. May as an example
More information3. Controlling the time delay hyper chaotic Lorenz system via back stepping control
ISSN 1746-7659, England, UK Journal of Information and Computing Science Vol 10, No 2, 2015, pp 148-153 Chaos control of hyper chaotic delay Lorenz system via back stepping method Hanping Chen 1 Xuerong
More informationChaos synchronization of nonlinear Bloch equations
Chaos, Solitons and Fractal7 (26) 357 361 www.elsevier.com/locate/chaos Chaos synchronization of nonlinear Bloch equations Ju H. Park * Robust Control and Nonlinear Dynamics Laboratory, Department of Electrical
More informationFrom Last Time. Gravitational forces are apparent at a wide range of scales. Obeys
From Last Time Gravitational forces are apparent at a wide range of scales. Obeys F gravity (Mass of object 1) (Mass of object 2) square of distance between them F = 6.7 10-11 m 1 m 2 d 2 Gravitational
More informationRELAXATION AND TRANSIENTS IN A TIME-DEPENDENT LOGISTIC MAP
International Journal of Bifurcation and Chaos, Vol. 12, No. 7 (2002) 1667 1674 c World Scientific Publishing Company RELAATION AND TRANSIENTS IN A TIME-DEPENDENT LOGISTIC MAP EDSON D. LEONEL, J. KAMPHORST
More informationSimple approach to the creation of a strange nonchaotic attractor in any chaotic system
PHYSICAL REVIEW E VOLUME 59, NUMBER 5 MAY 1999 Simple approach to the creation of a strange nonchaotic attractor in any chaotic system J. W. Shuai 1, * and K. W. Wong 2, 1 Department of Biomedical Engineering,
More informationEffect of various periodic forces on Duffing oscillator
PRAMANA c Indian Academy of Sciences Vol. 67, No. 2 journal of August 2006 physics pp. 351 356 Effect of various periodic forces on Duffing oscillator V RAVICHANDRAN 1, V CHINNATHAMBI 1, and S RAJASEKAR
More informationControl and synchronization of Julia sets of the complex dissipative standard system
Nonlinear Analysis: Modelling and Control, Vol. 21, No. 4, 465 476 ISSN 1392-5113 http://dx.doi.org/10.15388/na.2016.4.3 Control and synchronization of Julia sets of the complex dissipative standard system
More informationLYAPUNOV GRAPH FOR TWO-PARAMETERS MAP: APPLICATION TO THE CIRCLE MAP
International Journal of Bifurcation and Chaos, Vol. 8, No. 2 (1998) 281 293 c World Scientific Publishing Company LYAPUNOV GRAPH FOR TWO-PARAMETERS MAP: APPLICATION TO THE CIRCLE MAP J. C. BASTOS DE FIGUEIREDO
More informationThe nonsmooth pitchfork bifurcation. Glendinning, Paul. MIMS EPrint: Manchester Institute for Mathematical Sciences School of Mathematics
The nonsmooth pitchfork bifurcation Glendinning, Paul 2004 MIMS EPrint: 2006.89 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from:
More informationMore Details Fixed point of mapping is point that maps into itself, i.e., x n+1 = x n.
More Details Fixed point of mapping is point that maps into itself, i.e., x n+1 = x n. If there are points which, after many iterations of map then fixed point called an attractor. fixed point, If λ
More informationUSING THE RANDOM ITERATION ALGORITHM TO CREATE FRACTALS
USING THE RANDOM ITERATION ALGORITHM TO CREATE FRACTALS UNIVERSITY OF MARYLAND DIRECTED READING PROGRAM FALL 205 BY ADAM ANDERSON THE SIERPINSKI GASKET 2 Stage 0: A 0 = 2 22 A 0 = Stage : A = 2 = 4 A
More informationHowever, in actual topology a distance function is not used to define open sets.
Chapter 10 Dimension Theory We are used to the notion of dimension from vector spaces: dimension of a vector space V is the minimum number of independent bases vectors needed to span V. Therefore, a point
More informationVideo-Captured Dynamics of a Double Pendulum
WJP, PHY81 (010) Wabash Journal of Physics v.0, p.1 Video-Captured Dynamics of a Double Pendulum Scott Pond, Thomas Warn, M. J. Madsen, and J. rown Department of Physics, Wabash College, Crawfordsville,
More informationNon-normal parameter blowout bifurcation: An example in a truncated mean-field dynamo model
PHYSICAL REVIEW E VOLUME 56, NUMBER 6 DECEMBER 1997 Non-normal parameter blowout bifurcation: An example in a truncated mean-field dynamo model Eurico Covas, 1, * Peter Ashwin, 2, and Reza Tavakol 1, 1
More informationDiscrete Time Coupled Logistic Equations with Symmetric Dispersal
Discrete Time Coupled Logistic Equations with Symmetric Dispersal Tasia Raymer Department of Mathematics araymer@math.ucdavis.edu Abstract: A simple two patch logistic model with symmetric dispersal between
More informationMATH115. Sequences and Infinite Series. Paolo Lorenzo Bautista. June 29, De La Salle University. PLBautista (DLSU) MATH115 June 29, / 16
MATH115 Sequences and Infinite Series Paolo Lorenzo Bautista De La Salle University June 29, 2014 PLBautista (DLSU) MATH115 June 29, 2014 1 / 16 Definition A sequence function is a function whose domain
More informationIs the Hénon map chaotic
Is the Hénon map chaotic Zbigniew Galias Department of Electrical Engineering AGH University of Science and Technology, Poland, galias@agh.edu.pl International Workshop on Complex Networks and Applications
More informationStrange dynamics of bilinear oscillator close to grazing
Strange dynamics of bilinear oscillator close to grazing Ekaterina Pavlovskaia, James Ing, Soumitro Banerjee and Marian Wiercigroch Centre for Applied Dynamics Research, School of Engineering, King s College,
More informationGLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS BY ACTIVE NONLINEAR CONTROL
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS BY ACTIVE NONLINEAR CONTROL Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical
More informationHandling of Chaos in Two Dimensional Discrete Maps
Handling of Chaos in Two Dimensional Discrete Maps Anil Kumar Jain Assistant Professor, Department of Mathematics Barama College, Barama, Assam, Pincode-781346, India (Email: jainanil965@gmail.com) Abstract:
More informationMultistability and Self-Similarity in the Parameter-Space of a Vibro-Impact System
Universidade de São Paulo Biblioteca Digital da Produção Intelectual - BDPI Departamento de Física Aplicada - IF/FAP Artigos e Materiais de Revistas Científicas - IF/FAP 29 Multistability and Self-Similarity
More informationChaos and Cryptography
Chaos and Cryptography Vishaal Kapoor December 4, 2003 In his paper on chaos and cryptography, Baptista says It is possible to encrypt a message (a text composed by some alphabet) using the ergodic property
More informationQuantitative Description of Robot-Environment Interaction Using Chaos Theory 1
Quantitative Description of Robot-Environment Interaction Using Chaos Theory 1 Ulrich Nehmzow Keith Walker Dept. of Computer Science Department of Physics University of Essex Point Loma Nazarene University
More informationPHY411 Lecture notes Part 5
PHY411 Lecture notes Part 5 Alice Quillen January 27, 2016 Contents 0.1 Introduction.................................... 1 1 Symbolic Dynamics 2 1.1 The Shift map.................................. 3 1.2
More informationOn a conjecture about monomial Hénon mappings
Int. J. Open Problems Compt. Math., Vol. 6, No. 3, September, 2013 ISSN 1998-6262; Copyright c ICSRS Publication, 2013 www.i-csrs.orgr On a conjecture about monomial Hénon mappings Zeraoulia Elhadj, J.
More informationCHAOS -SOME BASIC CONCEPTS
CHAOS -SOME BASIC CONCEPTS Anders Ekberg INTRODUCTION This report is my exam of the "Chaos-part" of the course STOCHASTIC VIBRATIONS. I m by no means any expert in the area and may well have misunderstood
More informationExample Chaotic Maps (that you can analyze)
Example Chaotic Maps (that you can analyze) Reading for this lecture: NDAC, Sections.5-.7. Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January
More informationCreating and Analyzing Chaotic Attractors Using Mathematica Presented at the 2013 MAA MathFest
Creating and Analyzing Chaotic Attractors Using Mathematica Presented at the 2013 MAA MathFest Ulrich Hoensch 1 Rocky Mountain College Billings, Montana hoenschu@rocky.edu Saturday, August 3, 2013 1 Travel
More informationAre numerical studies of long term dynamics conclusive: the case of the Hénon map
Journal of Physics: Conference Series PAPER OPEN ACCESS Are numerical studies of long term dynamics conclusive: the case of the Hénon map To cite this article: Zbigniew Galias 2016 J. Phys.: Conf. Ser.
More informationDynamical behaviour of a controlled vibro-impact system
Vol 17 No 7, July 2008 c 2008 Chin. Phys. Soc. 1674-1056/2008/17(07)/2446-05 Chinese Physics B and IOP Publishing Ltd Dynamical behaviour of a controlled vibro-impact system Wang Liang( ), Xu Wei( ), and
More informationADAPTIVE CHAOS SYNCHRONIZATION OF UNCERTAIN HYPERCHAOTIC LORENZ AND HYPERCHAOTIC LÜ SYSTEMS
ADAPTIVE CHAOS SYNCHRONIZATION OF UNCERTAIN HYPERCHAOTIC LORENZ AND HYPERCHAOTIC LÜ SYSTEMS Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
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 informationApproximating Chaotic Saddles for Delay Differential Equations
Approximating Chaotic Saddles for Delay Differential Equations S. Richard Taylor Thompson Rivers University Sue Ann Campbell University of Waterloo (Dated: 27 February 2007) Chaotic saddles are unstable
More informationComplete Synchronization, Anti-synchronization and Hybrid Synchronization Between Two Different 4D Nonlinear Dynamical Systems
Mathematics Letters 2016; 2(5): 36-41 http://www.sciencepublishinggroup.com/j/ml doi: 10.11648/j.ml.20160205.12 Complete Synchronization, Anti-synchronization and Hybrid Synchronization Between Two Different
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 informationBackstepping synchronization of uncertain chaotic systems by a single driving variable
Vol 17 No 2, February 2008 c 2008 Chin. Phys. Soc. 1674-1056/2008/17(02)/0498-05 Chinese Physics B and IOP Publishing Ltd Backstepping synchronization of uncertain chaotic systems by a single driving variable
More informationDynamical Systems and Chaos Part I: Theoretical Techniques. Lecture 4: Discrete systems + Chaos. Ilya Potapov Mathematics Department, TUT Room TD325
Dynamical Systems and Chaos Part I: Theoretical Techniques Lecture 4: Discrete systems + Chaos Ilya Potapov Mathematics Department, TUT Room TD325 Discrete maps x n+1 = f(x n ) Discrete time steps. x 0
More informationI NONLINEAR EWORKBOOK
I NONLINEAR EWORKBOOK Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Wavelets, Fuzzy Logic with C++, Java and SymbolicC++ Programs Willi-Hans Steeb
More informationOn the dynamics of a vertically driven damped planar pendulum
.98/rspa..8 On the dynamics of a vertically driven damped planar pendulum By M. V. Bartuccelli, G. Gentile and K. V. Georgiou Department of Mathematics and Statistics, University of Surrey, Guildford GU
More informationStabilizing and Destabilizing Control for a Piecewise-Linear Circuit
172 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 2, FEBRUARY 1998 Stabilizing and Destabilizing Control for a Piecewise-Linear Circuit Tadashi Tsubone
More informationAdaptive feedback synchronization of a unified chaotic system
Physics Letters A 39 (4) 37 333 www.elsevier.com/locate/pla Adaptive feedback synchronization of a unified chaotic system Junan Lu a, Xiaoqun Wu a, Xiuping Han a, Jinhu Lü b, a School of Mathematics and
More informationHigh-Dimensional Dynamics in the Delayed Hénon Map
EJTP 3, No. 12 (2006) 19 35 Electronic Journal of Theoretical Physics High-Dimensional Dynamics in the Delayed Hénon Map J. C. Sprott Department of Physics, University of Wisconsin, Madison, WI 53706,
More informationDouble Transient Chaotic Behaviour of a Rolling Ball
Open Access Journal of Physics Volume 2, Issue 2, 2018, PP 11-16 Double Transient Chaotic Behaviour of a Rolling Ball Péter Nagy 1 and Péter Tasnádi 2 1 GAMF Faculty of Engineering and Computer Science,
More informationOrdinal Analysis of Time Series
Ordinal Analysis of Time Series K. Keller, M. Sinn Mathematical Institute, Wallstraße 0, 2552 Lübeck Abstract In order to develop fast and robust methods for extracting qualitative information from non-linear
More informationLECTURE 8: DYNAMICAL SYSTEMS 7
15-382 COLLECTIVE INTELLIGENCE S18 LECTURE 8: DYNAMICAL SYSTEMS 7 INSTRUCTOR: GIANNI A. DI CARO GEOMETRIES IN THE PHASE SPACE Damped pendulum One cp in the region between two separatrix Separatrix Basin
More informationHow does a diffusion coefficient depend on size and position of a hole?
How does a diffusion coefficient depend on size and position of a hole? G. Knight O. Georgiou 2 C.P. Dettmann 3 R. Klages Queen Mary University of London, School of Mathematical Sciences 2 Max-Planck-Institut
More informationGLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN SPROTT J AND K SYSTEMS BY ADAPTIVE CONTROL
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN SPROTT J AND K SYSTEMS BY ADAPTIVE CONTROL Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University Avadi,
More information698 Zou Yan-Li et al Vol. 14 and L 2, respectively, V 0 is the forward voltage drop across the diode, and H(u) is the Heaviside function 8 < 0 u < 0;
Vol 14 No 4, April 2005 cfl 2005 Chin. Phys. Soc. 1009-1963/2005/14(04)/0697-06 Chinese Physics and IOP Publishing Ltd Chaotic coupling synchronization of hyperchaotic oscillators * Zou Yan-Li( ΠΛ) a)y,
More informationNonsmooth systems: synchronization, sliding and other open problems
John Hogan Bristol Centre for Applied Nonlinear Mathematics, University of Bristol, England Nonsmooth systems: synchronization, sliding and other open problems 2 Nonsmooth Systems 3 What is a nonsmooth
More informationHYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEMS BY ACTIVE NONLINEAR CONTROL
HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEMS BY ACTIVE NONLINEAR CONTROL Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical
More informationGenerating a Complex Form of Chaotic Pan System and its Behavior
Appl. Math. Inf. Sci. 9, No. 5, 2553-2557 (2015) 2553 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/090540 Generating a Complex Form of Chaotic Pan
More informationDouble crises in fuzzy chaotic systems
Int. J. Dynam. Control (2013) 1:32 40 DOI 10.1007/s40435-013-0004-2 Double crises in fuzzy chaotic systems Ling Hong Jian-Qiao Sun Received: 19 November 2012 / Revised: 19 February 2013 / Accepted: 19
More informationME 680- Spring Representation and Stability Concepts
ME 680- Spring 014 Representation and Stability Concepts 1 3. Representation and stability concepts 3.1 Continuous time systems: Consider systems of the form x F(x), x n (1) where F : U Vis a mapping U,V
More informationPhase Desynchronization as a Mechanism for Transitions to High-Dimensional Chaos
Commun. Theor. Phys. (Beijing, China) 35 (2001) pp. 682 688 c International Academic Publishers Vol. 35, No. 6, June 15, 2001 Phase Desynchronization as a Mechanism for Transitions to High-Dimensional
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 informationADAPTIVE DESIGN OF CONTROLLER AND SYNCHRONIZER FOR LU-XIAO CHAOTIC SYSTEM
ADAPTIVE DESIGN OF CONTROLLER AND SYNCHRONIZER FOR LU-XIAO CHAOTIC SYSTEM WITH UNKNOWN PARAMETERS Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
More informationTHE theory of dynamical systems is an extremely
Attractors: Nonstrange to Chaotic Robert L. V. Taylor The College of Wooster rtaylor@wooster.edu Advised by Dr. John David The College of Wooster jdavid@wooster.edu Abstract The theory of chaotic dynamical
More informationCHARACTERIZATION OF NON-IDEAL OSCILLATORS IN PARAMETER SPACE
CHARACTERIZATION OF NON-IDEAL OSCILLATORS IN PARAMETER SPACE Silvio L. T. de Souza 1, Iberê L. Caldas 2, José M. Balthazar 3, Reyolando M. L. R. F. Brasil 4 1 Universidade Federal São João del-rei, Campus
More informationChaotic Vibrations. An Introduction for Applied Scientists and Engineers
Chaotic Vibrations An Introduction for Applied Scientists and Engineers FRANCIS C. MOON Theoretical and Applied Mechanics Cornell University Ithaca, New York A WILEY-INTERSCIENCE PUBLICATION JOHN WILEY
More informationMathematical Model of Forced Van Der Pol s Equation
Mathematical Model of Forced Van Der Pol s Equation TO Tsz Lok Wallace LEE Tsz Hong Homer December 9, Abstract This work is going to analyze the Forced Van Der Pol s Equation which is used to analyze the
More informationMechanism for boundary crises in quasiperiodically forced period-doubling systems
Physics Letters A 334 (2005) 160 168 www.elsevier.com/locate/pla Mechanism for boundary crises in quasiperiodically forced period-doubling systems Sang-Yoon Kim, Woochang Lim Department of Physics, Kangwon
More informationA Search for the Simplest Chaotic Partial Differential Equation
A Search for the Simplest Chaotic Partial Differential Equation C. Brummitt University of Wisconsin-Madison, Department of Physics cbrummitt@wisc.edu J. C. Sprott University of Wisconsin-Madison, Department
More informationDefining Chaos. Brittany Rock
Defining Chaos by Brittany Rock A thesis submitted to the Graduate Faculty of Auburn University in partial fulfillment of the requirements for the Degree of Master of Science Auburn, Alabama August, 207
More informationChaos in open Hamiltonian systems
Chaos in open Hamiltonian systems Tamás Kovács 5th Austrian Hungarian Workshop in Vienna April 9. 2010 Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 1 / 13 What s going
More informationADAPTIVE CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC NEWTON-LEIPNIK SYSTEM
ADAPTIVE CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC NEWTON-LEIPNIK SYSTEM Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University Avadi, Chennai-600
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 informationA MINIMAL 2-D QUADRATIC MAP WITH QUASI-PERIODIC ROUTE TO CHAOS
International Journal of Bifurcation and Chaos, Vol. 18, No. 5 (2008) 1567 1577 c World Scientific Publishing Company A MINIMAL 2-D QUADRATIC MAP WITH QUASI-PERIODIC ROUTE TO CHAOS ZERAOULIA ELHADJ Department
More informationxt+1 = 1 ax 2 t + y t y t+1 = bx t (1)
Exercise 2.2: Hénon map In Numerical study of quadratic area-preserving mappings (Commun. Math. Phys. 50, 69-77, 1976), the French astronomer Michel Hénon proposed the following map as a model of the Poincaré
More informationFinite-time hybrid synchronization of time-delay hyperchaotic Lorenz system
ISSN 1746-7659 England UK Journal of Information and Computing Science Vol. 10 No. 4 2015 pp. 265-270 Finite-time hybrid synchronization of time-delay hyperchaotic Lorenz system Haijuan Chen 1 * Rui Chen
More informationDoes the transition of the interval in perceptional alternation have a chaotic rhythm?
Does the transition of the interval in perceptional alternation have a chaotic rhythm? Yasuo Itoh Mayumi Oyama - Higa, Member, IEEE Abstract This study verified that the transition of the interval in perceptional
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 informationTHE ACTIVE CONTROLLER DESIGN FOR ACHIEVING GENERALIZED PROJECTIVE SYNCHRONIZATION OF HYPERCHAOTIC LÜ AND HYPERCHAOTIC CAI SYSTEMS
THE ACTIVE CONTROLLER DESIGN FOR ACHIEVING GENERALIZED PROJECTIVE SYNCHRONIZATION OF HYPERCHAOTIC LÜ AND HYPERCHAOTIC CAI SYSTEMS Sarasu Pakiriswamy 1 and Sundarapandian Vaidyanathan 1 1 Department of
More informationGIANT SUPPRESSION OF THE ACTIVATION RATE IN DYNAMICAL SYSTEMS EXHIBITING CHAOTIC TRANSITIONS
Vol. 39 (2008) ACTA PHYSICA POLONICA B No 5 GIANT SUPPRESSION OF THE ACTIVATION RATE IN DYNAMICAL SYSTEMS EXHIBITING CHAOTIC TRANSITIONS Jakub M. Gac, Jan J. Żebrowski Faculty of Physics, Warsaw University
More informationDYNAMICS AND BEHAVIOR OF HIGHER ORDER AND NONLINEAR RATIONAL DIFFERENCE EQUATION
DYNAMICS AND BEHAVIOR OF HIGHER ORDER AND NONLINEAR RATIONAL DIFFERENCE EQUATION Nirmaladevi.S 1, Karthikeyan.N 2 1 Research scholar in mathematics Vivekanha college of Arts Science for Women (Autonomous)
More informationLECTURE 10: REVIEW OF POWER SERIES. 1. Motivation
LECTURE 10: REVIEW OF POWER SERIES By definition, a power series centered at x 0 is a series of the form where a 0, a 1,... and x 0 are constants. For convenience, we shall mostly be concerned with the
More informationA Trivial Dynamics in 2-D Square Root Discrete Mapping
Applied Mathematical Sciences, Vol. 12, 2018, no. 8, 363-368 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.8121 A Trivial Dynamics in 2-D Square Root Discrete Mapping M. Mammeri Department
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 informationarxiv: v2 [nlin.cd] 8 Sep 2012
An analytical limitation for time-delayed feedback control in autonomous systems Edward W. Hooton 1 1, 2, and Andreas Amann 1 School of Mathematical Sciences, University College Cork, Ireland 2 Tyndall
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 informationComment on \Bouncing ball with nite restitution: Chattering, locking, and chaos" Nicholas B. Tullaro
Comment on \Bouncing ball with nite restitution: Chattering, locking, and chaos" Nicholas B. Tullaro Center for Nonlinear Studies and Theoretical Division, MS-B258, Los lamos National Laboratories Los
More information2 Problem Set 2 Graphical Analysis
2 PROBLEM SET 2 GRAPHICAL ANALYSIS 2 Problem Set 2 Graphical Analysis 1. Use graphical analysis to describe all orbits of the functions below. Also draw their phase portraits. (a) F(x) = 2x There is only
More informationChaos Control for the Lorenz System
Advanced Studies in Theoretical Physics Vol. 12, 2018, no. 4, 181-188 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2018.8413 Chaos Control for the Lorenz System Pedro Pablo Cárdenas Alzate
More informationApril 13, We now extend the structure of the horseshoe to more general kinds of invariant. x (v) λ n v.
April 3, 005 - Hyperbolic Sets We now extend the structure of the horseshoe to more general kinds of invariant sets. Let r, and let f D r (M) where M is a Riemannian manifold. A compact f invariant set
More informationADAPTIVE STABILIZATION AND SYNCHRONIZATION OF HYPERCHAOTIC QI SYSTEM
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF HYPERCHAOTIC QI SYSTEM Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR Dr. SR Technical University Avadi, Chennai-600 062,
More informationSynchronization of Chaotic Fractional-Order LU-LU System with Different Orders via Active Sliding Mode Control
Synchronization of Chaotic Fractional-Order LU-LU System with Different Orders via Active Sliding Mode Control Samaneh Jalalian MEM student university of Wollongong in Dubai samaneh_jalalian@yahoo.com
More informationFRACTAL GEOMETRY, DYNAMICAL SYSTEMS AND CHAOS
FRACTAL GEOMETRY, DYNAMICAL SYSTEMS AND CHAOS MIDTERM SOLUTIONS. Let f : R R be the map on the line generated by the function f(x) = x 3. Find all the fixed points of f and determine the type of their
More information