Time-Series Based Prediction of Dynamical Systems and Complex. Networks
|
|
- Oliver Briggs
- 5 years ago
- Views:
Transcription
1 Time-Series Based Prediction of Dynamical Systems and Complex Collaborators Networks Ying-Cheng Lai ECEE Arizona State University Dr. Wenxu Wang, ECEE, ASU Mr. Ryan Yang, ECEE, ASU Mr. Riqi Su, ECEE, ASU Prof. Celso Grebogi, Univ. of Aberdeen Dr. Vassilios Kovanis, AFRL, Wright-Patterson AFB
2 Problem 1: Can catastrophic events in dynamical systems be predicted in advance? Early Warning? A related problem: Can future behaviors of time-varying dynamical systems be forecasted?
3 Problem 2: Reverse-engineering of complex networks network Measured Time Series Assumption: all nodes are externally accessible Full network topology?
4 Problem 3: Detecting hidden nodes No information is available from the black node. How can we ascertain its existence and its location in the network?
5 Basic idea (1 Dynamical system: dx/dt = F(x, x! R m Goal: to determine F(x from measured time series x(t! Power-series expansion of jth component of vector field F(x n n " l 1 =0 l 2 =0 n " l m =0 [F(x] j = " l... (a j l1 l 2...l m x 1 l 1 x 2 l 2...x m m x k kth component of x; Highest-order power: n (a j l1 l 2...l m - coefficients to be estimated from time series - (1+n m coefficients altogether If F(x contains only a few power-series terms, most of the coefficients will be zero.
6 Basic idea (2 Concrete example: m = 3 (phase-space dimension: (x,y,z n = 3 (highest order in power-series expansion total (1 + n m = ( = 64 unknown coefficients [F(x] 1 = (a 1 0,0,0 x 0 y 0 z 0 + (a 1 1,0,0 x 1 y 0 z (a 1 3,3,3 x 3 y 3 z 3! Coefficient vector a 1 = " (a 1 0,0,0 (a 1 1,0,0... (a 1 3,3,3-64 '1 Measurement vector g(t = [x(t 0 y(t 0 z(t 0, x(t 1 y(t 0 z(t 0,..., x(t 3 y(t 3 z(t 3 ] So [F(x(t] 1 = g(t a 1 1 ' 64
7 Basic idea (3 Suppose x(t is available at times t 0, t 1,t 2,...,t 10 (11 vector data points dx dt (t 1 = [F(x(t 1 ] 1 = g(t 1 a 1 dx dt (t 2 = [F(x(t 2 ] 1 = g(t 2 a 1... dx dt (t 10 = [F(x(t 10 ] 1 = g(t 10 a 1 Derivative vector dx =! " (dx/dt(t 1 (dx/dt(t 2... (dx/dt(t 10 10'1 ; Measurement matrix G =! " g(t 1 g(t 2! g(t 10 10'64 We finally have dx = G a 1 or dx 10'1 = G 10'64 (a 1 64'1
8 Basic idea (4 dx = G a 1 or dx 10!1 = G 10!64 (a 1 64!1 Reminder: a 1 is the coefficient vector for the first dynamical variable x. To obtain [F(x] 2, we expand [F(x] 2 = (a 2 0,0,0 x 0 y 0 z 0 + (a 2 1,0,0 x 1 y 0 z (a 2 3,3,3 x 3 y 3 z 3 with a 2, the coefficient vector for the second dynamical variable y. We have dy = G a 2 or dy 10!1 = G 10!64 (a 2 64!1 where dy = " (dy/dt(t 1 (dy/dt(t 2... (dy/dt(t 10 ' ' ' ' 10!1 Note: the measurement matrix G is the same.. Similar expressions can be obtained for all components of the velocity field.
9 Compressive sensing (1 Look at dx = G a 1 or dx 10!1 = G 10!64 (a 1 64!1 Note that a 1 is sparse - Compressive sensing! Data/Image compression:! : Random projection (not full rank x - sparse vector to be recovered Goal of compressive sensing: Find a vector x with minimum number of entries subject to the constraint y =! x
10 Compressive Sensing (2 Find a vector x with minimum number of entries subject to the constraint y =! x: l 1 " norm Why l 1! norm? - Simple example in three dimensions E. Candes, J. Romberg, and T. Tao, IEEE Trans. Information Theory 52, 489 (2006, Comm. Pure. Appl. Math. 59, 1207 (2006; D. Donoho, IEEE Trans. Information Theory 52, 1289 (2006; Special review: IEEE Signal Process. Mag. 24, 2008
11 Predicting catastrophe (1 Henon map: (x n+1, y n+1 = (1- ax 2 n + y n, bx n Say the system operates at parameter values: a = 1.2 and b = 0.3. There is a chaotic attractor. Can we assess if a catastrophic bifurcation (e.g., crisis is imminent based on a limited set of measurements? Step 1: Predicting system equations Distribution of predicted values of ten power-series coefficients: constant, y, y 2, y 3, x, xy, xy 2, x 2, x 2 y, x 3 of data points used: 8
12 Predicting catastrophe (2 Step 2: Performing numerical bifurcation analysis Current operation point Boundary Crisis
13 Predicting catastrophe (3 Examples of predicting continuous-time dynamical systems Classical Lorenz system dx/dt = 10y - 10x dy/dt = x(a - z - y dz/dt = xy - (2/3z Classical Rossler system dx/dt = -y - z dy/dt = x + 0.2y dz/dt = z(x - a of data points used: 18
14 Performance analysis Standard map Lorenz system n m! of measurements n nz! of non-zero coefficients; n z! of zero coefficients (n nz + n z! total of coefficients to be determined n t! minimum of measurements required for accurate prediction
15 Effect of noise Henon map Standard map n m = 8 (n nz + n z =16 n m =10 (n nz + n z = 20 W.-X. Wang, R. Yang, Y.-C. Lai, V. Kovanis, and C. Grebogi, Physical Review Letters 106, (2011.
16 Predicting future attractors of time-varying dynamical systems (1 Dynamical system: dx/dt = F[x,p(t], p(t - parameters varying slowly with time T M " measurement time period; x! R m x(t - available in time interval: t M -T M! t! t M Goal: to determine both F[x,p(t] and p(t from available time series x(t so that the nature of the attractor for t > t M can be assessed. Power-series expansion n n n l [F(x] j =...(! j l1 l 2...l m x 1 l 1 x 2 l 2...x m m (! j w t w n v l 1 =0 l 2 =0 l m =0 = (c j x l 1 l l1,...,l m ;w 1 x 2 l 2...x m m t w CS framework l 1,...,l m =0 w=0 v w=0
17 Predicting future attractors of time-varying dynamical systems (2 Formulated as a CS problem:
18 Predicting future attractors of time-varying dynamical systems (3 z z (b time (a time Measurements Actual time series Predicted time series k 1 (t = 0 k 2 (t = 0 k 3 (t = 0 k 4 (t = 0 k 1 (t = - t 2 k 2 (t = 0.5t k 3 (t = t k 4 (t = - 0.5t 2 Time-varying Lorenz system dx/dt = -10(x - y + k 1 (t! y dy/dt = 28x - y - xz + k 2 (t! z dz/dt = xy - (8/3z + [k 3 (t + k 4 (t]! y
19 Uncovering full topology of oscillator networks (1 A class of commonly studied oscillator -network models: dx N i = F i (x i +! C ij (x j - x i (i = 1,..., N dt j=1,j!i - dynamical equation of node i N - size of network, x i " R m, C ij is the local coupling matrix C ij = C ij 1,1 C ij 2,1 C ij 1,2 C ij 2,2! C ij 1,m! C ij 2,m " " " " C ij m,1 C ij m,2! C ij m,m ( ( ( ( ( ' - determines full topology If there is at least one nonzero element in C ij, nodes i and j are coupled. Goal: to determine all F i (x i and C ij from time series.
20 X =! " x 1 x 2... x N Uncovering full topology of oscillator networks (2 Nm'1 ( Network equation is dx dt [G(X] i = F i (x i + = G(X, where (x j - x i N C ij j=1,j!i A very high-dimensional (Nm-dimensional dynamical system; For complex networks (e.g, random, small-world, scale-free, node-to-node connections are typically sparse; In power-series expansion of [G(X] i, most coefficients will be zero - guaranteeing sparsity condition for compressive sensing. W.-X. Wang, R. Yang, Y.-C. Lai, V. Kovanis, M. A. F. Harrison, Time-series based prediction of complex oscillator networks via compressive sensing, Europhysics Letters 94, (2011.
21 Evolutionary-game dynamics Example: Prisoner s dilemma game! Strategies: cooperation S(C = "! Payoff matrix: P(PD = " 1 0 b 0 1 0! ; defection S(D = " b - parameter Payoff of agent x from playing PDG with agent y: For example, M C'C = 1 M D'D = 0 M C'D = 0 M D'C = b 0 1 M x'y = S x T PS y
22 Evolutionary game on network (social and economical networks A network of agents playing games with one another:! Adjacency matrix = " a xy : Payoff of agent x from agent y: M x*y = a xy S x T PS y ' a xy =1 if x connects with y ( a xy = 0 if no connection Time series of agents (Detectable (1 payoffs (2 strategies Compressive sensing Full social network structure
23 Prediction as a CS Problem Payoff of x at time t: M x (t = a x1 S x T (tps 1 (t + a x2 S x T (tps 2 (t +!+ a xn S x T (tps N (t! Y = "! ' = " M x (t 1 M x (t 2! M x (t m! X = " a x1 a x2! a xn S x T (t 1 PS 1 (t 1 S x T (t 1 PS 2 (t 1! S x T (t 1 PS N (t 1 S x T (t 2 PS 1 (t 2 S x T (t 2 PS 2 (t 2! S x T (t 2 PS N (t 2!!!! S x T (t m PS 1 (t m S x T (t m PS 2 (t m! S x T (t m PS N (t m X : connection vector of agent x (to be predicted W.-X. Wang, Y.-C. Lai, C. Grebogi, and J.-P. Ye, Network reconstruction based on evolutionary-game data, submitted Y = ' X Y,': obtainable from time series Compressive sensing a a X = a x1 x2 xn x y N matching Full network structure Neighbors of x
24 Success rate for model networks
25 22 students play PDG together and write down their payoffs and strategies Friendship network Reverse engineering of a real social network Experimental record of two players Observation: Large-degree nodes are not necessarily winners
26 20 Detecting hidden nodes (a Idea Two green nodes: immediate neighbors of hidden node Information from green nodes is not complete Anomalies in the prediction of connections of green nodes Node Node (b Node Node Directed/weighted network: adjacency matrix not symmetric Node (c 2 Variance of predicted coefficients
27 Distinguishing between effects of hidden node and noise ( Say, h - hidden node; i, j - two neighboring nodes of h Variance of predicted coefficients Hidden node: as N m is increased through a threshold value, variance of predicted coefficients of i and j will decrease 16 7 Noise - variance will not change 2. Cancellation factor w ih! weight of link between i and h; similarly for w jh Cancellation factor:! ij = w ih / w jh Hidden node:! ij will change and then plateaued Noise:! ij will remain approximately constant
28 Distinguishing between effects of hidden node and noise (2 1 (a Hidden 10 5 (b 0.5 Noise Hidden ij Noise (c R m (d R m R.-Q. Su, W.-X. Wang, Y.-C. Lai, and C. Grebogi, Reverse engineering of complex networks: detecting hidden nodes, submitted
29 Limitations (1 1. Key requirement of compressive sensing the vector to be determined must be sparse. Dynamical systems - three cases: Vector field/map contains a few Fourier-series terms - Yes Vector field/map contains a few power-series terms - Yes Vector field /map contains many terms not known Ikeda Map: F(x, y= [A + B(x cos!! ysin!, B(xsin! + ycos!] where! " p! k - describes dynamics in an optical cavity 1+ x y Mathematical question: given an arbitrary function, can one find a suitable base of expansion so that the function can be represented by a limited number of terms?
30 Limitations (2 2. Networked systems described by evolutionary games Yes 3. Measurements of ALL dynamical variables are needed. Outstanding issue If this is not the case, say, if only one dynamical variable can be measured, the CS-based method would not work. Delay-coordinate embedding method? - gives only a topological equivalent of the underlying dynamical system (e.g., Takens embedding theorem guarantees only a one-to-one correspondence between the true system and the reconstructed system. 4. In Conclusion, Much work is needed!
Chain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics
3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)
More informationComplex system approach to geospace and climate studies. Tatjana Živković
Complex system approach to geospace and climate studies Tatjana Živković 30.11.2011 Outline of a talk Importance of complex system approach Phase space reconstruction Recurrence plot analysis Test for
More informationSignal Recovery from Permuted Observations
EE381V Course Project Signal Recovery from Permuted Observations 1 Problem Shanshan Wu (sw33323) May 8th, 2015 We start with the following problem: let s R n be an unknown n-dimensional real-valued signal,
More informationAN INTRODUCTION TO COMPRESSIVE SENSING
AN INTRODUCTION TO COMPRESSIVE SENSING Rodrigo B. Platte School of Mathematical and Statistical Sciences APM/EEE598 Reverse Engineering of Complex Dynamical Networks OUTLINE 1 INTRODUCTION 2 INCOHERENCE
More informationReconstructing and Controlling Nonlinear Complex Systems. Ri-Qi Su
Reconstructing and Controlling Nonlinear Complex Systems by Ri-Qi Su A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Approved October 2015 by the
More informationNew Coherence and RIP Analysis for Weak. Orthogonal Matching Pursuit
New Coherence and RIP Analysis for Wea 1 Orthogonal Matching Pursuit Mingrui Yang, Member, IEEE, and Fran de Hoog arxiv:1405.3354v1 [cs.it] 14 May 2014 Abstract In this paper we define a new coherence
More informationLarge-Scale L1-Related Minimization in Compressive Sensing and Beyond
Large-Scale L1-Related Minimization in Compressive Sensing and Beyond Yin Zhang Department of Computational and Applied Mathematics Rice University, Houston, Texas, U.S.A. Arizona State University March
More informationarxiv: v2 [q-bio.pe] 18 Dec 2007
The Effect of a Random Drift on Mixed and Pure Strategies in the Snowdrift Game arxiv:0711.3249v2 [q-bio.pe] 18 Dec 2007 André C. R. Martins and Renato Vicente GRIFE, Escola de Artes, Ciências e Humanidades,
More informationSystem Identification of an Inverted Pendulum on a Cart using Compressed Sensing
System Identification of an Inverted Pendulum on a Cart using Compressed Sensing Manjish Naik, Douglas Cochran, mnaik@asu.edu, cochran@asu.edu School of Electrical and Computer Engineering Arizona State
More informationSparse Solutions of an Undetermined Linear System
1 Sparse Solutions of an Undetermined Linear System Maddullah Almerdasy New York University Tandon School of Engineering arxiv:1702.07096v1 [math.oc] 23 Feb 2017 Abstract This work proposes a research
More informationNo. 6 Determining the input dimension of a To model a nonlinear time series with the widely used feed-forward neural network means to fit the a
Vol 12 No 6, June 2003 cfl 2003 Chin. Phys. Soc. 1009-1963/2003/12(06)/0594-05 Chinese Physics and IOP Publishing Ltd Determining the input dimension of a neural network for nonlinear time series prediction
More informationMath 212-Lecture 8. The chain rule with one independent variable
Math 212-Lecture 8 137: The multivariable chain rule The chain rule with one independent variable w = f(x, y) If the particle is moving along a curve x = x(t), y = y(t), then the values that the particle
More informationRobust Principal Component Analysis
ELE 538B: Mathematics of High-Dimensional Data Robust Principal Component Analysis Yuxin Chen Princeton University, Fall 2018 Disentangling sparse and low-rank matrices Suppose we are given a matrix M
More informationExponential decay of reconstruction error from binary measurements of sparse signals
Exponential decay of reconstruction error from binary measurements of sparse signals Deanna Needell Joint work with R. Baraniuk, S. Foucart, Y. Plan, and M. Wootters Outline Introduction Mathematical Formulation
More informationISSN Article. Identifying Chaotic FitzHugh Nagumo Neurons Using Compressive Sensing
Entropy 214, 16, 3889-392; doi:1.339/e1673889 OPEN ACCESS entropy ISSN 199-43 www.mdpi.com/journal/entropy Article Identifying Chaotic FitzHugh Nagumo Neurons Using Compressive Sensing Ri-Qi Su 1, Ying-Cheng
More informationCharacterizing chaotic time series
Characterizing chaotic time series Jianbo Gao PMB InTelliGence, LLC, West Lafayette, IN 47906 Mechanical and Materials Engineering, Wright State University jbgao.pmb@gmail.com http://www.gao.ece.ufl.edu/
More informationEvolutionary Games on Networks. Wen-Xu Wang and G Ron Chen Center for Chaos and Complex Networks
Evolutionary Games on Networks Wen-Xu Wang and G Ron Chen Center for Chaos and Complex Networks Email: wenxuw@gmail.com; wxwang@cityu.edu.hk Cooperative behavior among selfish individuals Evolutionary
More informationMultiplicative and Additive Perturbation Effects on the Recovery of Sparse Signals on the Sphere using Compressed Sensing
Multiplicative and Additive Perturbation Effects on the Recovery of Sparse Signals on the Sphere using Compressed Sensing ibeltal F. Alem, Daniel H. Chae, and Rodney A. Kennedy The Australian National
More informationLecture 31. Basic Theory of First Order Linear Systems
Math 245 - Mathematics of Physics and Engineering I Lecture 31. Basic Theory of First Order Linear Systems April 4, 2012 Konstantin Zuev (USC) Math 245, Lecture 31 April 4, 2012 1 / 10 Agenda Existence
More information25. Chain Rule. Now, f is a function of t only. Expand by multiplication:
25. Chain Rule The Chain Rule is present in all differentiation. If z = f(x, y) represents a two-variable function, then it is plausible to consider the cases when x and y may be functions of other variable(s).
More informationCompressibility of Infinite Sequences and its Interplay with Compressed Sensing Recovery
Compressibility of Infinite Sequences and its Interplay with Compressed Sensing Recovery Jorge F. Silva and Eduardo Pavez Department of Electrical Engineering Information and Decision Systems Group Universidad
More informationHW2 Solutions. MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22]
HW2 Solutions MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, 2013 Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22] Section 3.1: 1, 2, 3, 9, 16, 18, 20, 23 Section 3.2: 1, 2,
More informationEvolution & Learning in Games
1 / 27 Evolution & Learning in Games Econ 243B Jean-Paul Carvalho Lecture 2. Foundations of Evolution & Learning in Games II 2 / 27 Outline In this lecture, we shall: Take a first look at local stability.
More informationWhy are Discrete Maps Sufficient?
Why are Discrete Maps Sufficient? Why do dynamical systems specialists study maps of the form x n+ 1 = f ( xn), (time is discrete) when much of the world around us evolves continuously, and is thus well
More informationCS168: The Modern Algorithmic Toolbox Lecture #6: Regularization
CS168: The Modern Algorithmic Toolbox Lecture #6: Regularization Tim Roughgarden & Gregory Valiant April 18, 2018 1 The Context and Intuition behind Regularization Given a dataset, and some class of models
More informationBayesian Methods for Sparse Signal Recovery
Bayesian Methods for Sparse Signal Recovery Bhaskar D Rao 1 University of California, San Diego 1 Thanks to David Wipf, Jason Palmer, Zhilin Zhang and Ritwik Giri Motivation Motivation Sparse Signal Recovery
More informationEen vlinder in de wiskunde: over chaos en structuur
Een vlinder in de wiskunde: over chaos en structuur Bernard J. Geurts Enschede, November 10, 2016 Tuin der Lusten (Garden of Earthly Delights) In all chaos there is a cosmos, in all disorder a secret
More informationBayesian Inference and the Symbolic Dynamics of Deterministic Chaos. Christopher C. Strelioff 1,2 Dr. James P. Crutchfield 2
How Random Bayesian Inference and the Symbolic Dynamics of Deterministic Chaos Christopher C. Strelioff 1,2 Dr. James P. Crutchfield 2 1 Center for Complex Systems Research and Department of Physics University
More informationTime-delay feedback control in a delayed dynamical chaos system and its applications
Time-delay feedback control in a delayed dynamical chaos system and its applications Ye Zhi-Yong( ), Yang Guang( ), and Deng Cun-Bing( ) School of Mathematics and Physics, Chongqing University of Technology,
More informationDr. Allen Back. Oct. 6, 2014
Dr. Allen Back Oct. 6, 2014 Distribution Min=48 Max=100 Q1=74 Median=83 Q3=91.5 mean = 81.48 std. dev. = 12.5 Distribution Distribution: (The letters will not be directly used.) Scores Freq. 100 2 98-99
More informationA new method on deterministic construction of the measurement matrix in compressed sensing
A new method on deterministic construction of the measurement matrix in compressed sensing Qun Mo 1 arxiv:1503.01250v1 [cs.it] 4 Mar 2015 Abstract Construction on the measurement matrix A is a central
More informationLecture 22: More On Compressed Sensing
Lecture 22: More On Compressed Sensing Scribed by Eric Lee, Chengrun Yang, and Sebastian Ament Nov. 2, 207 Recap and Introduction Basis pursuit was the method of recovering the sparsest solution to an
More informationPrisoner s Dilemma. Veronica Ciocanel. February 25, 2013
n-person February 25, 2013 n-person Table of contents 1 Equations 5.4, 5.6 2 3 Types of dilemmas 4 n-person n-person GRIM, GRIM, ALLD Useful to think of equations 5.4 and 5.6 in terms of cooperation and
More informationCombining geometry and combinatorics
Combining geometry and combinatorics A unified approach to sparse signal recovery Anna C. Gilbert University of Michigan joint work with R. Berinde (MIT), P. Indyk (MIT), H. Karloff (AT&T), M. Strauss
More informationGREEDY SIGNAL RECOVERY REVIEW
GREEDY SIGNAL RECOVERY REVIEW DEANNA NEEDELL, JOEL A. TROPP, ROMAN VERSHYNIN Abstract. The two major approaches to sparse recovery are L 1-minimization and greedy methods. Recently, Needell and Vershynin
More informationCS 229r: Algorithms for Big Data Fall Lecture 19 Nov 5
CS 229r: Algorithms for Big Data Fall 215 Prof. Jelani Nelson Lecture 19 Nov 5 Scribe: Abdul Wasay 1 Overview In the last lecture, we started discussing the problem of compressed sensing where we are given
More information8-1: Backpropagation Prof. J.C. Kao, UCLA. Backpropagation. Chain rule for the derivatives Backpropagation graphs Examples
8-1: Backpropagation Prof. J.C. Kao, UCLA Backpropagation Chain rule for the derivatives Backpropagation graphs Examples 8-2: Backpropagation Prof. J.C. Kao, UCLA Motivation for backpropagation To do gradient
More informationRecovery of Low-Rank Plus Compressed Sparse Matrices with Application to Unveiling Traffic Anomalies
July 12, 212 Recovery of Low-Rank Plus Compressed Sparse Matrices with Application to Unveiling Traffic Anomalies Morteza Mardani Dept. of ECE, University of Minnesota, Minneapolis, MN 55455 Acknowledgments:
More informationLecture: Introduction to Compressed Sensing Sparse Recovery Guarantees
Lecture: Introduction to Compressed Sensing Sparse Recovery Guarantees http://bicmr.pku.edu.cn/~wenzw/bigdata2018.html Acknowledgement: this slides is based on Prof. Emmanuel Candes and Prof. Wotao Yin
More informationLB 220 Homework 4 Solutions
LB 220 Homework 4 Solutions Section 11.4, # 40: This problem was solved in class on Feb. 03. Section 11.4, # 42: This problem was also solved in class on Feb. 03. Section 11.4, # 43: Also solved in class
More informationNear Ideal Behavior of a Modified Elastic Net Algorithm in Compressed Sensing
Near Ideal Behavior of a Modified Elastic Net Algorithm in Compressed Sensing M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas M.Vidyasagar@utdallas.edu www.utdallas.edu/ m.vidyasagar
More informationControlling a Novel Chaotic Attractor using Linear Feedback
ISSN 746-7659, England, UK Journal of Information and Computing Science Vol 5, No,, pp 7-4 Controlling a Novel Chaotic Attractor using Linear Feedback Lin Pan,, Daoyun Xu 3, and Wuneng Zhou College of
More informationAn Introduction to Evolutionary Game Theory: Lecture 2
An Introduction to Evolutionary Game Theory: Lecture 2 Mauro Mobilia Lectures delivered at the Graduate School on Nonlinear and Stochastic Systems in Biology held in the Department of Applied Mathematics,
More informationProblems on Evolutionary dynamics
Problems on Evolutionary dynamics Doctoral Programme in Physics José A. Cuesta Lausanne, June 10 13, 2014 Replication 1. Consider the Galton-Watson process defined by the offspring distribution p 0 =
More informationLIMITATION OF LEARNING RANKINGS FROM PARTIAL INFORMATION. By Srikanth Jagabathula Devavrat Shah
00 AIM Workshop on Ranking LIMITATION OF LEARNING RANKINGS FROM PARTIAL INFORMATION By Srikanth Jagabathula Devavrat Shah Interest is in recovering distribution over the space of permutations over n elements
More informationSPARSE SIGNAL RESTORATION. 1. Introduction
SPARSE SIGNAL RESTORATION IVAN W. SELESNICK 1. Introduction These notes describe an approach for the restoration of degraded signals using sparsity. This approach, which has become quite popular, is useful
More informationEnhanced Compressive Sensing and More
Enhanced Compressive Sensing and More Yin Zhang Department of Computational and Applied Mathematics Rice University, Houston, Texas, U.S.A. Nonlinear Approximation Techniques Using L1 Texas A & M University
More informationMultipath Matching Pursuit
Multipath Matching Pursuit Submitted to IEEE trans. on Information theory Authors: S. Kwon, J. Wang, and B. Shim Presenter: Hwanchol Jang Multipath is investigated rather than a single path for a greedy
More informationINDUSTRIAL MATHEMATICS INSTITUTE. B.S. Kashin and V.N. Temlyakov. IMI Preprint Series. Department of Mathematics University of South Carolina
INDUSTRIAL MATHEMATICS INSTITUTE 2007:08 A remark on compressed sensing B.S. Kashin and V.N. Temlyakov IMI Preprint Series Department of Mathematics University of South Carolina A remark on compressed
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 15: Nonlinear Systems and Lyapunov Functions Overview Our next goal is to extend LMI s and optimization to nonlinear
More informationVII. Cooperation & Competition
VII. Cooperation & Competition A. The Iterated Prisoner s Dilemma Read Flake, ch. 17 4/23/18 1 The Prisoners Dilemma Devised by Melvin Dresher & Merrill Flood in 1950 at RAND Corporation Further developed
More informationMath 1060 Linear Algebra Homework Exercises 1 1. Find the complete solutions (if any!) to each of the following systems of simultaneous equations:
Homework Exercises 1 1 Find the complete solutions (if any!) to each of the following systems of simultaneous equations: (i) x 4y + 3z = 2 3x 11y + 13z = 3 2x 9y + 2z = 7 x 2y + 6z = 2 (ii) x 4y + 3z =
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 informationSpatial three-player prisoners dilemma
Spatial three-player prisoners dilemma Rui Jiang, 1 Hui Deng, 1 Mao-Bin Hu, 1,2 Yong-Hong Wu, 2 and Qing-Song Wu 1 1 School of Engineering Science, University of Science and Technology of China, Hefei
More informationEvolution of cooperation in finite populations. Sabin Lessard Université de Montréal
Evolution of cooperation in finite populations Sabin Lessard Université de Montréal Title, contents and acknowledgements 2011 AMS Short Course on Evolutionary Game Dynamics Contents 1. Examples of cooperation
More informationBabu Banarasi Das Northern India Institute of Technology, Lucknow
Babu Banarasi Das Northern India Institute of Technology, Lucknow B.Tech Second Semester Academic Session: 2012-13 Question Bank Maths II (EAS -203) Unit I: Ordinary Differential Equation 1-If dy/dx +
More informationModel Building: Selected Case Studies
Chapter 2 Model Building: Selected Case Studies The goal of Chapter 2 is to illustrate the basic process in a variety of selfcontained situations where the process of model building can be well illustrated
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 informationProject 1 Modeling of Epidemics
532 Chapter 7 Nonlinear Differential Equations and tability ection 7.5 Nonlinear systems, unlike linear systems, sometimes have periodic solutions, or limit cycles, that attract other nearby solutions.
More information3.2 Systems of Two First Order Linear DE s
Agenda Section 3.2 Reminders Lab 1 write-up due 9/26 or 9/28 Lab 2 prelab due 9/26 or 9/28 WebHW due 9/29 Office hours Tues, Thurs 1-2 pm (5852 East Hall) MathLab office hour Sun 7-8 pm (MathLab) 3.2 Systems
More informationCompressed Sensing and Neural Networks
and Jan Vybíral (Charles University & Czech Technical University Prague, Czech Republic) NOMAD Summer Berlin, September 25-29, 2017 1 / 31 Outline Lasso & Introduction Notation Training the network Applications
More informationAn Application of Perturbation Methods in Evolutionary Ecology
Dynamics at the Horsetooth Volume 2A, 2010. Focused Issue: Asymptotics and Perturbations An Application of Perturbation Methods in Evolutionary Ecology Department of Mathematics Colorado State University
More informationConditions for a Unique Non-negative Solution to an Underdetermined System
Conditions for a Unique Non-negative Solution to an Underdetermined System Meng Wang and Ao Tang School of Electrical and Computer Engineering Cornell University Ithaca, NY 14853 Abstract This paper investigates
More informationNumerical Analysis: Interpolation Part 1
Numerical Analysis: Interpolation Part 1 Computer Science, Ben-Gurion University (slides based mostly on Prof. Ben-Shahar s notes) 2018/2019, Fall Semester BGU CS Interpolation (ver. 1.00) AY 2018/2019,
More informationConstructing Explicit RIP Matrices and the Square-Root Bottleneck
Constructing Explicit RIP Matrices and the Square-Root Bottleneck Ryan Cinoman July 18, 2018 Ryan Cinoman Constructing Explicit RIP Matrices July 18, 2018 1 / 36 Outline 1 Introduction 2 Restricted Isometry
More informationPackage R1magic. April 9, 2013
Package R1magic April 9, 2013 Type Package Title Compressive Sampling: Sparse signal recovery utilities Version 0.2 Date 2013-04-09 Author Maintainer Depends stats, utils Provides
More informationa k 0, then k + 1 = 2 lim 1 + 1
Math 7 - Midterm - Form A - Page From the desk of C. Davis Buenger. https://people.math.osu.edu/buenger.8/ Problem a) [3 pts] If lim a k = then a k converges. False: The divergence test states that if
More informationLecture 13 - Wednesday April 29th
Lecture 13 - Wednesday April 29th jacques@ucsdedu Key words: Systems of equations, Implicit differentiation Know how to do implicit differentiation, how to use implicit and inverse function theorems 131
More informationMATH 251 MATH 251: Multivariate Calculus MATH 251 FALL 2005 FINAL EXAM FALL 2005 FINAL EXAM EXAMINATION COVER PAGE Professor Moseley
MATH 5 MATH 5: Multivariate Calculus MATH 5 FALL 5 FINAL EXAM FALL 5 FINAL EXAM EXAMINATION COVER PAGE Professor Moseley PRINT NAME ( ) Last Name, First Name MI (What you wish to be called) ID # EXAM DATE
More informationISE I Brief Lecture Notes
ISE I Brief Lecture Notes 1 Partial Differentiation 1.1 Definitions Let f(x, y) be a function of two variables. The partial derivative f/ x is the function obtained by differentiating f with respect to
More informationData assimilation with and without a model
Data assimilation with and without a model Tim Sauer George Mason University Parameter estimation and UQ U. Pittsburgh Mar. 5, 2017 Partially supported by NSF Most of this work is due to: Tyrus Berry,
More informationRui ZHANG Song LI. Department of Mathematics, Zhejiang University, Hangzhou , P. R. China
Acta Mathematica Sinica, English Series May, 015, Vol. 31, No. 5, pp. 755 766 Published online: April 15, 015 DOI: 10.1007/s10114-015-434-4 Http://www.ActaMath.com Acta Mathematica Sinica, English Series
More informationEngg. Math. I. Unit-I. Differential Calculus
Dr. Satish Shukla 1 of 50 Engg. Math. I Unit-I Differential Calculus Syllabus: Limits of functions, continuous functions, uniform continuity, monotone and inverse functions. Differentiable functions, Rolle
More informationSensing systems limited by constraints: physical size, time, cost, energy
Rebecca Willett Sensing systems limited by constraints: physical size, time, cost, energy Reduce the number of measurements needed for reconstruction Higher accuracy data subject to constraints Original
More informationChapter 6 Nonlinear Systems and Phenomena. Friday, November 2, 12
Chapter 6 Nonlinear Systems and Phenomena 6.1 Stability and the Phase Plane We now move to nonlinear systems Begin with the first-order system for x(t) d dt x = f(x,t), x(0) = x 0 In particular, consider
More informationAn Introduction to Sparse Approximation
An Introduction to Sparse Approximation Anna C. Gilbert Department of Mathematics University of Michigan Basic image/signal/data compression: transform coding Approximate signals sparsely Compress images,
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation Instructor: Moritz Hardt Email: hardt+ee227c@berkeley.edu Graduate Instructor: Max Simchowitz Email: msimchow+ee227c@berkeley.edu
More informationMATH 317 Fall 2016 Assignment 5
MATH 37 Fall 26 Assignment 5 6.3, 6.4. ( 6.3) etermine whether F(x, y) e x sin y îı + e x cos y ĵj is a conservative vector field. If it is, find a function f such that F f. enote F (P, Q). We have Q x
More informationTHEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.)
4 Vector fields Last updated: November 26, 2009. (Under construction.) 4.1 Tangent vectors as derivations After we have introduced topological notions, we can come back to analysis on manifolds. Let M
More informationSome explicit formulas of Lyapunov exponents for 3D quadratic mappings
Some explicit formulas of Lyapunov exponents for 3D quadratic mappings Zeraoulia Elhadj 1,J.C.Sprott 2 1 Department of Mathematics, University of Tébessa, (12002), Algeria. E-mail: zeraoulia@mail.univ-tebessa.dz
More informationLecture two. January 17, 2019
Lecture two January 17, 2019 We will learn how to solve rst-order linear equations in this lecture. Example 1. 1) Find all solutions satisfy the equation u x (x, y) = 0. 2) Find the solution if we know
More informationSynchronization between different motifs
Synchronization between different motifs Li Ying( ) a) and Liu Zeng-Rong( ) b) a) College of Information Technology, Shanghai Ocean University, Shanghai 201306, China b) Institute of Systems Biology, Shanghai
More informationCalculus for the Life Sciences II Assignment 6 solutions. f(x, y) = 3π 3 cos 2x + 2 sin 3y
Calculus for the Life Sciences II Assignment 6 solutions Find the tangent plane to the graph of the function at the point (0, π f(x, y = 3π 3 cos 2x + 2 sin 3y Solution: The tangent plane of f at a point
More informationHOMEWORK 7 SOLUTIONS
HOMEWORK 7 SOLUTIONS MA11: ADVANCED CALCULUS, HILARY 17 (1) Using the method of Lagrange multipliers, find the largest and smallest values of the function f(x, y) xy on the ellipse x + y 1. Solution: The
More informationSpace curves, vector fields and strange surfaces. Simon Salamon
Space curves, vector fields and strange surfaces Simon Salamon Lezione Lagrangiana, 26 May 2016 Curves in space 1 A curve is the path of a particle moving continuously in space. Its position at time t
More informationMethod of Successive Approximations for Solving the Multi-Pantograph Delay Equations
Gen. Math. Notes, Vol. 8, No., January, pp.3-8 ISSN 9-784; Copyright c ICSRS Publication, www.i-csrs.org Available free online at http://www.geman.in Method of Successive Approximations for Solving the
More informationMultiscale Entropy Analysis: A New Method to Detect Determinism in a Time. Series. A. Sarkar and P. Barat. Variable Energy Cyclotron Centre
Multiscale Entropy Analysis: A New Method to Detect Deterinis in a Tie Series A. Sarkar and P. Barat Variable Energy Cyclotron Centre /AF Bidhan Nagar, Kolkata 700064, India PACS nubers: 05.45.Tp, 89.75.-k,
More informationIntroduction How it works Theory behind Compressed Sensing. Compressed Sensing. Huichao Xue. CS3750 Fall 2011
Compressed Sensing Huichao Xue CS3750 Fall 2011 Table of Contents Introduction From News Reports Abstract Definition How it works A review of L 1 norm The Algorithm Backgrounds for underdetermined linear
More informationSparse analysis Lecture VII: Combining geometry and combinatorics, sparse matrices for sparse signal recovery
Sparse analysis Lecture VII: Combining geometry and combinatorics, sparse matrices for sparse signal recovery Anna C. Gilbert Department of Mathematics University of Michigan Sparse signal recovery measurements:
More informationg(t) = f(x 1 (t),..., x n (t)).
Reading: [Simon] p. 313-333, 833-836. 0.1 The Chain Rule Partial derivatives describe how a function changes in directions parallel to the coordinate axes. Now we shall demonstrate how the partial derivatives
More informationNoise-induced unstable dimension variability and transition to chaos in random dynamical systems
Noise-induced unstable dimension variability and transition to chaos in random dynamical systems Ying-Cheng Lai, 1,2 Zonghua Liu, 1 Lora Billings, 3 and Ira B. Schwartz 4 1 Department of Mathematics, Center
More informationLecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI
Lecture 8 Analyzing the diffusion weighted signal Room CSB 272 this week! Please install AFNI http://afni.nimh.nih.gov/afni/ Next lecture, DTI For this lecture, think in terms of a single voxel We re still
More informationGeneralized Function Projective Lag Synchronization in Fractional-Order Chaotic Systems
Generalized Function Projective Lag Synchronization in Fractional-Order Chaotic Systems Yancheng Ma Guoan Wu and Lan Jiang denotes fractional order of drive system Abstract In this paper a new synchronization
More informationAverage-Consensus of Multi-Agent Systems with Direct Topology Based on Event-Triggered Control
Outline Background Preliminaries Consensus Numerical simulations Conclusions Average-Consensus of Multi-Agent Systems with Direct Topology Based on Event-Triggered Control Email: lzhx@nankai.edu.cn, chenzq@nankai.edu.cn
More informationSynchronization and Bifurcation Analysis in Coupled Networks of Discrete-Time Systems
Commun. Theor. Phys. (Beijing, China) 48 (2007) pp. 871 876 c International Academic Publishers Vol. 48, No. 5, November 15, 2007 Synchronization and Bifurcation Analysis in Coupled Networks of Discrete-Time
More informationPackage R1magic. August 29, 2016
Type Package Package R1magic August 29, 2016 Title Compressive Sampling: Sparse Signal Recovery Utilities Version 0.3.2 Date 2015-04-19 Maintainer Depends stats, utils Utilities
More informationCompressed Sensing: Lecture I. Ronald DeVore
Compressed Sensing: Lecture I Ronald DeVore Motivation Compressed Sensing is a new paradigm for signal/image/function acquisition Motivation Compressed Sensing is a new paradigm for signal/image/function
More informationCONTROLLING IN BETWEEN THE LORENZ AND THE CHEN SYSTEMS
International Journal of Bifurcation and Chaos, Vol. 12, No. 6 (22) 1417 1422 c World Scientific Publishing Company CONTROLLING IN BETWEEN THE LORENZ AND THE CHEN SYSTEMS JINHU LÜ Institute of Systems
More informationExact Low-rank Matrix Recovery via Nonconvex M p -Minimization
Exact Low-rank Matrix Recovery via Nonconvex M p -Minimization Lingchen Kong and Naihua Xiu Department of Applied Mathematics, Beijing Jiaotong University, Beijing, 100044, People s Republic of China E-mail:
More informationCOMPRESSED SENSING IN PYTHON
COMPRESSED SENSING IN PYTHON Sercan Yıldız syildiz@samsi.info February 27, 2017 OUTLINE A BRIEF INTRODUCTION TO COMPRESSED SENSING A BRIEF INTRODUCTION TO CVXOPT EXAMPLES A Brief Introduction to Compressed
More information