Bifurcations of the Controlled Escape Equation
|
|
- Chloe Stevenson
- 5 years ago
- Views:
Transcription
1 Bifurcations of the Controlled Escape Equation Tobias Gaer Institut für Mathematik, Universität Augsburg Augsburg, German Abstract In this paper we present numerical methods for the analsis of nonlinear autonomous control sstems and two conditions, local accessibilit and an inner-pair condition, under which the can be applied. These methods can be etended to work also for sstems with time-periodic right hand side. In particular, the escape equation with sinusoidal driving term and additional control is analed. We will show that its stabilit behavior undergoes interesting bifurcations when the range of the control influence is varied. 1 Introduction In this paper we consider a famil of nonlinear control sstems ẋ(t) = f((t), u(t)) (1.1) on a connected smooth manifold M with dimension d where f : M R m TM is smooth and the controls u are taken from the set U ρ := {u L (R, R m ), u(t) ρ U for almost all t R} for a non-void, conve, and compact subset U of R m with intu and a parameter ρ [, ρ ]. Relevance of the parameter dependence will be indicated b a superfi ρ. We assume for ever initial value M and ever u U ρ that there eists a unique solution φ(t;, u), t R, with φ(;, u) =. Furthermore, b restricting ourselves to a compact invariant subset of the state space we can focus on compact M onl. Throughout we assume the sstem to be locall accessible for all ρ >, i.e. for all M and all T > one has into ρ,+ T (), and intoρ, T (), where Oρ,+ T () = { M, = φ(t;, u) with t T and u U ρ } and O ρ, T () = { M, = φ(t;, u) with t T and u U ρ } are the positive and negative reachable sets from. The limit behavior of this sstem is determined b its control sets, i.e. maimal subsets of M where complete approimate controllabilit holds, and its chain control sets (see [1]). Under an inner-pair condition the chain control sets and the control sets genericall coincide. We will see in section 2 that this inner-pair condition holds for n-th order equations with additive control and therefore under change of the parameter ρ the control sets will change continuousl for all but countabl man ρ [, ρ ]. If the right hand side of the control sstem has a t-periodic driving term as onl eplicit time dependence, the time can be interpreted as a further space dimension and control sets and chain control sets can be studied for the corresponding sstem without direct time dependence. We show that if in this case control 1
2 sets merge under variation of the parameter ρ, then the merge over the whole time-period at once. Onl in ver simple cases can control sets be found analticall. Thus numerical methods are an important tool for the analsis of control sstems. In section 3 we briefl present a method for the computation of control sets that was developed b Solnoki (see [7]) and is based on an algorithm for the computation of relative global attractors and unstable manifolds of dnamical sstems b Dellnit, Hohmann and Junge (see [3] and [4]). In section 4 finall we will clarif our findings b numerical results for the controlled escape equation. 2 Control sets and chain control sets Control sets are maimal subsets of the state space where complete approimate controllabilit holds. A control set C is called invariant if clc = cl{φ(t;, u), u U, t }. If we allow the solutions to make small jumps, we are led to the notion of chain-controllabilit: For ɛ, T >, and, M a controlled (ɛ, T )-chain from to is given b n N,, 1,..., n M, u,..., u n 1 U, t,... t n 1 T with =, n =, and d(φ(t j ; j, u j ), j+1 ) ɛ for all j =,..., n 1. A chain control set E M is a maimal subset such that (i) for all, E and all ɛ, T > there is a controlled (ɛ, T )-chain from to, and (ii) for all E there is u U such that φ(t;, u) E for all t R. Ever control set is contained in a chain control set. The following inner-pair condition guarantees much closer a relationship between control sets and chain control sets (cf. [1]): Definition 2.1. If for all ρ 1, ρ 2 [, ρ ], ρ 1 < ρ 2, and for all M, u U ρ 1, there is T > such that φ(t ;, u) into +,ρ 2 (), then the famil of control sstems (1.1) is said to fulfill the inner-pair condition. Theorem 2.1. Let E ρ 1 M be a chain control set of (1.1) ρ 1, and for each ρ [ρ 1, ρ ] let E ρ M be the unique chain control set of (1.1) ρ with E ρ 1 E ρ. Moreover let the famil (1.1) be locall accessible and fulfill the inner-pair condition. Then for each ρ there is precisel one control set D ρ with E ρ 1 intd ρ, and for all but countabl man values of ρ the following holds: (i) cld ρ = E ρ. (ii) The mapping ρ cld ρ is continuous with respect to the Hausdorff metric. At the discontinuities control sets merge and possibl change their stabilit behavior. The inner-pair condition can be etended to time dependent sstems in the obvious wa and the following holds. Proposition 2.1. Consider the famil of n-th order sstems (n) (t) + f(t,, (1),..., (n 1) ) = b(t,,..., (n 1) ) u(t) (2.2) 2
3 where f, b : R n+1 R are continuous mappings and u U ρ. If there is α > such that b(t,, 1,..., n 1 ) α holds for all (t,, 1,..., n 1 ) R n+1, then (2.2) fulfills the inner-pair condition. If the right hand side in (1.1) is generalied to be eplicitl periodicall time dependent, the control setting can still be used. Let f : R M R m TM be smooth and a sstem be given b ẋ(t) = f(t, (t), u(t)). If f is T -periodic in the first coordinate: f(t,, u) = f(t + T,, u), we can consider the associated time-independent sstem ( ẋ ż ) = ( f(,, u) T/2 ) (2.3) where we identif = and = 2. If control sets for such a sstem have contact for one R/2Z, the have contact for the whole interval. Proposition 2.2. If D 1 and D 2 are control sets for the control sstem (2.3) and if there is (, ) R/2Z M such that (, ) cld 1 cld 2, then for ever R/2Z there is M such that (, ) cld 1 cld 2. Proof. For proofs of Propositions 2.1 and 2.2 see [5]. 3 Numerical approimation of control sets Control sets can not be approimated directl but the computation of chain control sets is almost as good according to Theorem 2.1. Solnoki developed three algorithms whose combination is a powerful tool for locating chain control sets (see [7] and [8]). The have in common that a compact subset Q of the state space is chosen and successivel divided in finer and finer partitions. A selection criterion in each step defines a decreasing sequence of subsets of Q. A graph algorithm finds strongl connected components of a directed graph associated to the discretied dnamics. In principle these strongl connected components converge to the chain control sets if the partition gets finer but memor and time consumption do not allow for fine partitioning in higher dimensions. The subdivision algorithm approimates viabilit kernels V (Q) relative to the start set Q. If Q contains precisel one chain control set E, then E = V (Q). If Q possibl contains more than one chain control set, then in some cases a continuation algorithm can be used to create a covering of precisel one chain control set, which allows the subdivision algorithm to be used. A combination of these methods together with smart selection of the set Q often produces good results. But for sstems that are sensitive to small changes in their initial conditions or for high dimensions resources are limited. In the special case of time-periodic sstems (2.3) one can spare one dimension b investigating Poincaré-cuts and using Proposition 2.2 to draw conclusions for the full dimensional sstem. 3
4 4 The controlled escape equation As an eample we consider the controlled escape equation ẍ(t) + γẋ(t) + (t) (t) 2 = F sin(ωt) + u(t) γ, ω, F >, t R. This equation has been intensivel studied for instance b Soliman and Thompson (see [6]). Interpreting the controls u as bounded noise terms it models the rocking movement of a ship on sea under the influence of periodic waves and some additional disturbance. According to Proposition 2.1 the inner-pair condition is satisfied. Transferring it into the form of sstem (1.1) leads to ẋ(t) ẏ(t) ż(t) = (t) γ (t) (t) (t) 2 ω + F sin((t)) + u(t) where M = R R R/2Z, u(t) U ρ := [ ρ, ρ], ρ ρ. We choose the parameter values ω =.85, γ =.1, F =.6. The sstem proves too complicated for a direct three dimensional investigation. Therefore we look at four Poincaré-cuts, at =, =.25 2, =.5 2, and =.75 2 and 1.5 investigate the discrete dnamics induced b the time-2-map. For the uncontrolled case ρ = we find four 1 fied points. At = FP1 is located close to (;.25) and is stable,.5 FP2 is near (.2;.4) and unstable, stable FP3 is close to (.3;.4) and unstable FP4 finall is close to (1; ). Choosing small neighborhoods of these fied points and using.5 the continuation algorithm we also find the unstable and stable manifolds of the unstable fied points 1 (see Figure 1). Now if we go to =, ρ = Figure 1. For ρ = the picture shows two unstable (blue) and two stable (red) fied points. The unstable manifold of FP4 is depicted in green, its stable one in light brown. The unstable manifold of FP2 has the ellow color, its stable one is dark brown. 4, ρ =.5, according to Theorem 2.1 around these fied points develop invariant control sets D1 ρ and D3 ρ and variant control sets D2 ρ and D4 ρ for which we draw the domains of attractions as well. Finding a parameter value ρ for which there are three control sets is difficult. Onl for = /2 the graph algorithm is
5 able to resolve that at ρ =.85 the control sets D1 ρ and D2 ρ have merged whereas D3 ρ is still separated from them. But then Proposition 2.2 states that D3.85 must be separate for all and therefore we can cut out D1.5 and appl subdivision to the rest which gives the displaed pictures. At ρ =.1 the three control sets around the origin have merged to one invariant control set. At ρ =.13 the whole inner area has become transient since now the eit via the D4 control set is possible. Let A(D4 ρ ) denote the domain of attraction of D4 ρ and A(D2 ρ ) denote the domain of attraction of the variant control set that includes D2.5 (see Figure 2). We get the full 3D-pictures of the control sets b inserting the results at the selected four Poincaré-cuts, computing connecting orbits and appling subdivision afterwards (see Figure 3) =, ρ = =, ρ = =, ρ = =, ρ =.13 Figure 2. For ρ =.5 control sets (red = invariant, blue = variant) have developed around the fied points. At ρ =.85 invariant D1 ρ has merged with variant D2 ρ forming D12.85 and lost its invariance. For ρ =.1 the sets D12.85 and D3.85 have merged and form one invariant control set in the middle. At ρ =.13 this control set in the middle has merged with the variant control set near (, 1) and formed one variant control set. Eit is now possible from everwhere. A(D4 ρ ) is depicted dark brown, A(D2 ρ ) is ellow. 5
6 /2. / /2 1. / References Figure 3 The 3D control sets for ρ =.5,.85,.1,.13. [1] F. Colonius, W. Kliemann, The Dnamics of Control, Birkhäuser, Boston, 2. [2] F. Colonius, W. Kliemann, Invariance under Bounded Time-Varing Perturbations, in: Proc. 11th IFAC International Workshop Control Applications of Optimiation, V. Zakhavov, ed., St. Petersburg (2), pp [3] M. Dellnit, A. Hohmann, A Subdivision Algorithm for the Computation of Unstable Manifolds and Global Attractors, Numer.Math. 75 (1997), pp [4] M. Dellnit, O. Junge, On the Approimation of Complicated Dnamical Behavior, SIAM J.Numer.Anal. 36 (1999), pp [5] T. Gaer, PhD-thesis, in preparation. [6] M.S. Soliman, J.M. Thompson, Integrit Measures Quantifing the Erosion of Smooth and Fractal Basins of Attraction, J.Sound and Vibration, 135 (3) (1989), pp [7] D. Solnoki, Algorithms for Reachabilit Problems, Shaker, Aachen, 21. [8] D. Solnoki, Computation of Control Sets using Subdivision and Continuation Techniques, in: Proc. 39th IEEE Conference on Decision and Control, Sdne (2), pp
10 Back to planar nonlinear systems
10 Back to planar nonlinear sstems 10.1 Near the equilibria Recall that I started talking about the Lotka Volterra model as a motivation to stud sstems of two first order autonomous equations of the form
More informationResearch Article Chaotic Attractor Generation via a Simple Linear Time-Varying System
Discrete Dnamics in Nature and Societ Volume, Article ID 836, 8 pages doi:.//836 Research Article Chaotic Attractor Generation via a Simple Linear Time-Varing Sstem Baiu Ou and Desheng Liu Department of
More informationIntroduction to Differential Equations. National Chiao Tung University Chun-Jen Tsai 9/14/2011
Introduction to Differential Equations National Chiao Tung Universit Chun-Jen Tsai 9/14/011 Differential Equations Definition: An equation containing the derivatives of one or more dependent variables,
More informationInstitut für Mathematik
U n i v e r s i t ä t A u g s b u r g Institut für Mathematik Fritz Colonius, Christoph Kawan Invariance Entropy for Outputs Preprint Nr. 29/2009 04. November 2009 Institut für Mathematik, Universitätsstraße,
More informationLocal Phase Portrait of Nonlinear Systems Near Equilibria
Local Phase Portrait of Nonlinear Sstems Near Equilibria [1] Consider 1 = 6 1 1 3 1, = 3 1. ( ) (a) Find all equilibrium solutions of the sstem ( ). (b) For each equilibrium point, give the linear approimating
More informationConnecting orbits in perturbed systems
Noname manuscript No. (will be inserted by the editor) Connecting orbits in perturbed systems Fritz Colonius Thorsten Hüls Martin Rasmussen December 3, 8 Abstract We apply Newton s method in perturbed
More informationChini Equations and Isochronous Centers in Three-Dimensional Differential Systems
Qual. Th. Dyn. Syst. 99 (9999), 1 1 1575-546/99-, DOI 1.17/s12346-3- c 29 Birkhäuser Verlag Basel/Switzerland Qualitative Theory of Dynamical Systems Chini Equations and Isochronous Centers in Three-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 informationNew approach to study the van der Pol equation for large damping
Electronic Journal of Qualitative Theor of Differential Equations 2018, No. 8, 1 10; https://doi.org/10.1422/ejqtde.2018.1.8 www.math.u-szeged.hu/ejqtde/ New approach to stud the van der Pol equation for
More informationFUNDAMENTAL SEMIGROUPS FOR DYNAMICAL SYSTEMS. Fritz Colonius. Marco Spadini
FUNDAMENTAL SEMIGROUPS FOR DYNAMICAL SYSTEMS Fritz Colonius Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany Marco Spadini Dipartimento di Matematica Applicata, Via S Marta 3, 50139
More informationUsing MatContM in the study of a nonlinear map in economics
Journal of Phsics: Conference Series PAPER OPEN ACCESS Using MatContM in the stud of a nonlinear map in economics To cite this article: Niels Neirnck et al 016 J. Phs.: Conf. Ser. 69 0101 Related content
More informationFunctions of Several Variables
Chapter 1 Functions of Several Variables 1.1 Introduction A real valued function of n variables is a function f : R, where the domain is a subset of R n. So: for each ( 1,,..., n ) in, the value of f is
More informationUNFOLDING THE CUSP-CUSP BIFURCATION OF PLANAR ENDOMORPHISMS
UNFOLDING THE CUSP-CUSP BIFURCATION OF PLANAR ENDOMORPHISMS BERND KRAUSKOPF, HINKE M OSINGA, AND BRUCE B PECKHAM Abstract In man applications of practical interest, for eample, in control theor, economics,
More informationIdentification of Nonlinear Dynamic Systems with Multiple Inputs and Single Output using discrete-time Volterra Type Equations
Identification of Nonlinear Dnamic Sstems with Multiple Inputs and Single Output using discrete-time Volterra Tpe Equations Thomas Treichl, Stefan Hofmann, Dierk Schröder Institute for Electrical Drive
More informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 3. Fundamental properties IST-DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Example Consider the system ẋ = f
More informationFuzzy Topology On Fuzzy Sets: Regularity and Separation Axioms
wwwaasrcorg/aasrj American Academic & Scholarl Research Journal Vol 4, No 2, March 212 Fuzz Topolog n Fuzz Sets: Regularit and Separation Aioms AKandil 1, S Saleh 2 and MM Yakout 3 1 Mathematics Department,
More informationEE222: Solutions to Homework 2
Spring 7 EE: Solutions to Homework 1. Draw the phase portrait of a reaction-diffusion sstem ẋ 1 = ( 1 )+ 1 (1 1 ) ẋ = ( 1 )+ (1 ). List the equilibria and their tpes. Does the sstem have limit ccles? (Hint:
More informationPassivity-based Stabilization of Non-Compact Sets
Passivity-based Stabilization of Non-Compact Sets Mohamed I. El-Hawwary and Manfredi Maggiore Abstract We investigate the stabilization of closed sets for passive nonlinear systems which are contained
More informationNonlinear Systems Theory
Nonlinear Systems Theory Matthew M. Peet Arizona State University Lecture 2: Nonlinear Systems Theory Overview Our next goal is to extend LMI s and optimization to nonlinear systems analysis. Today we
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 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 informationOrdinary Differential Equations: Homework 2
Orinary Differential Equations: Homework 2 M. Gameiro, J.-P. Lessar, J.D. Mireles James, K. Mischaikow January 30, 2017 2 0.1 Eercises Eercise 0.1.1. Let (X, ) be a metric space. function (in the metric
More informationBlow-up collocation solutions of nonlinear homogeneous Volterra integral equations
Blow-up collocation solutions of nonlinear homogeneous Volterra integral equations R. Benítez 1,, V. J. Bolós 2, 1 Dpto. Matemáticas, Centro Universitario de Plasencia, Universidad de Extremadura. Avda.
More informationMAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function
MAT 275: Introduction to Mathematical Analsis Dr. A. Rozenblum Graphs and Simplest Equations for Basic Trigonometric Functions We consider here three basic functions: sine, cosine and tangent. For them,
More informationMULTI-SCROLL CHAOTIC AND HYPERCHAOTIC ATTRACTORS GENERATED FROM CHEN SYSTEM
International Journal of Bifurcation and Chaos, Vol. 22, No. 2 (212) 133 ( pages) c World Scientific Publishing Compan DOI: 1.1142/S21812741332 MULTI-SCROLL CHAOTIC AND HYPERCHAOTIC ATTRACTORS GENERATED
More informationAPPROXIMATE SIMULATION RELATIONS FOR HYBRID SYSTEMS 1. Antoine Girard A. Agung Julius George J. Pappas
APPROXIMATE SIMULATION RELATIONS FOR HYBRID SYSTEMS 1 Antoine Girard A. Agung Julius George J. Pappas Department of Electrical and Systems Engineering University of Pennsylvania Philadelphia, PA 1914 {agirard,agung,pappasg}@seas.upenn.edu
More informationUncertainty and Parameter Space Analysis in Visualization -
Uncertaint and Parameter Space Analsis in Visualiation - Session 4: Structural Uncertaint Analing the effect of uncertaint on the appearance of structures in scalar fields Rüdiger Westermann and Tobias
More informationPiecewise Smooth Solutions to the Burgers-Hilbert Equation
Piecewise Smooth Solutions to the Burgers-Hilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA e-mails: bressan@mathpsuedu, zhang
More informationNonlinear Dynamical Systems Eighth Class
Nonlinear Dynamical Systems Eighth Class Alexandre Nolasco de Carvalho September 19, 2017 Now we exhibit a Morse decomposition for a dynamically gradient semigroup and use it to prove that a dynamically
More informationNonlinear Systems Examples Sheet: Solutions
Nonlinear Sstems Eamples Sheet: Solutions Mark Cannon, Michaelmas Term 7 Equilibrium points. (a). Solving ẋ =sin 4 3 =for gives =as an equilibrium point. This is the onl equilibrium because there is onl
More informationControl Schemes to Reduce Risk of Extinction in the Lotka-Volterra Predator-Prey Model
Journal of Applied Mathematics and Phsics, 2014, 2, 644-652 Published Online June 2014 in SciRes. http://www.scirp.org/journal/jamp http://d.doi.org/10.4236/jamp.2014.27071 Control Schemes to Reduce Risk
More informationResearch Article A New Four-Scroll Chaotic Attractor Consisted of Two-Scroll Transient Chaotic and Two-Scroll Ultimate Chaotic
Mathematical Problems in Engineering Volume, Article ID 88, pages doi:.//88 Research Article A New Four-Scroll Chaotic Attractor Consisted of Two-Scroll Transient Chaotic and Two-Scroll Ultimate Chaotic
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 informationNONLINEAR DYNAMICS AND CHAOS. Numerical integration. Stability analysis
LECTURE 3: FLOWS NONLINEAR DYNAMICS AND CHAOS Patrick E McSharr Sstems Analsis, Modelling & Prediction Group www.eng.o.ac.uk/samp patrick@mcsharr.net Tel: +44 83 74 Numerical integration Stabilit analsis
More informationSystems of Differential Equations
WWW Problems and Solutions 5.1 Chapter 5 Sstems of Differential Equations Section 5.1 First-Order Sstems www Problem 1. (From Scalar ODEs to Sstems). Solve each linear ODE; construct and solve an equivalent
More informationConvergence Rate of Nonlinear Switched Systems
Convergence Rate of Nonlinear Switched Systems Philippe JOUAN and Saïd NACIRI arxiv:1511.01737v1 [math.oc] 5 Nov 2015 January 23, 2018 Abstract This paper is concerned with the convergence rate of the
More informationTHE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM
Bull. Aust. Math. Soc. 88 (2013), 267 279 doi:10.1017/s0004972713000348 THE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM MICHAEL F. BARNSLEY and ANDREW VINCE (Received 15 August 2012; accepted 21 February
More informationGEOMETRY MID-TERM CLAY SHONKWILER
GEOMETRY MID-TERM CLAY SHONKWILER. Polar Coordinates R 2 can be considered as a regular surface S by setting S = {(, y, z) R 3 : z = 0}. Polar coordinates on S are given as follows. Let U = {(ρ, θ) : ρ
More informationThe Cusp-cusp Bifurcation for noninvertible maps of the plane
The Cusp-cusp Bifurcation for noninvertible maps of the plane Bruce Peckham Dept of Mathematics and Statistics Universit of Minnesota, Duluth Duluth, Minnesota 55812 bpeckham@dumnedu Coauthors: Bernd Krauskopf
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 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 informationy R T However, the calculations are easier, when carried out using the polar set of co-ordinates ϕ,r. The relations between the co-ordinates are:
Curved beams. Introduction Curved beams also called arches were invented about ears ago. he purpose was to form such a structure that would transfer loads, mainl the dead weight, to the ground b the elements
More informationChapter 23. Predicting Chaos The Shift Map and Symbolic Dynamics
Chapter 23 Predicting Chaos We have discussed methods for diagnosing chaos, but what about predicting the existence of chaos in a dynamical system. This is a much harder problem, and it seems that the
More informationConverse Lyapunov theorem and Input-to-State Stability
Converse Lyapunov theorem and Input-to-State Stability April 6, 2014 1 Converse Lyapunov theorem In the previous lecture, we have discussed few examples of nonlinear control systems and stability concepts
More information4.7. Newton s Method. Procedure for Newton s Method HISTORICAL BIOGRAPHY
4. Newton s Method 99 4. Newton s Method HISTORICAL BIOGRAPHY Niels Henrik Abel (18 189) One of the basic problems of mathematics is solving equations. Using the quadratic root formula, we know how to
More informationA Note on Some Properties of Local Random Attractors
Northeast. Math. J. 24(2)(2008), 163 172 A Note on Some Properties of Local Random Attractors LIU Zhen-xin ( ) (School of Mathematics, Jilin University, Changchun, 130012) SU Meng-long ( ) (Mathematics
More information520 Chapter 9. Nonlinear Differential Equations and Stability. dt =
5 Chapter 9. Nonlinear Differential Equations and Stabilit dt L dθ. g cos θ cos α Wh was the negative square root chosen in the last equation? (b) If T is the natural period of oscillation, derive the
More informationExamples and counterexamples for Markus-Yamabe and LaSalle global asymptotic stability problems Anna Cima, Armengol Gasull and Francesc Mañosas
Proceedings of the International Workshop Future Directions in Difference Equations. June 13-17, 2011, Vigo, Spain. PAGES 89 96 Examples and counterexamples for Markus-Yamabe and LaSalle global asmptotic
More informationHyperbolic Systems of Conservation Laws. I - Basic Concepts
Hyperbolic Systems of Conservation Laws I - Basic Concepts Alberto Bressan Mathematics Department, Penn State University Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 1 / 27 The
More informationSolutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10)
Solutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10) Mason A. Porter 15/05/2010 1 Question 1 i. (6 points) Define a saddle-node bifurcation and show that the first order system dx dt = r x e x
More information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More informationFunctions with orthogonal Hessian
Functions with orthogonal Hessian B. Dacorogna P. Marcellini E. Paolini Abstract A Dirichlet problem for orthogonal Hessians in two dimensions is eplicitly solved, by characterizing all piecewise C 2 functions
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 informationTOWARDS AUTOMATED CHAOS VERIFICATION
TOWARDS AUTOMATED CHAOS VERIFICATION SARAH DAY CDSNS, Georgia Institute of Technology, Atlanta, GA 30332 OLIVER JUNGE Institute for Mathematics, University of Paderborn, 33100 Paderborn, Germany KONSTANTIN
More informationA comparison of estimation accuracy by the use of KF, EKF & UKF filters
Computational Methods and Eperimental Measurements XIII 779 A comparison of estimation accurac b the use of KF EKF & UKF filters S. Konatowski & A. T. Pieniężn Department of Electronics Militar Universit
More informationA NOTE ON THE DYNAMICS AROUND THE L 1,2 LAGRANGE POINTS OF THE EARTH MOON SYSTEM IN A COMPLETE SOLAR SYSTEM MODEL
IAA-AAS-DCoSS1-8-8 A NOTE ON THE DYNAMICS AROUND THE L 1,2 LAGRANGE POINTS OF THE EARTH MOON SYSTEM IN A COMPLETE SOLAR SYSTEM MODEL Lian Yijun, Gerard Góme, Josep J. Masdemont, Tang Guojian INTRODUCTION
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationOptimal Sojourn Time Control within an Interval 1
Optimal Sojourn Time Control within an Interval Jianghai Hu and Shankar Sastry Department of Electrical Engineering and Computer Sciences University of California at Berkeley Berkeley, CA 97-77 {jianghai,sastry}@eecs.berkeley.edu
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 informationDIFFERENTIATION. 3.1 Approximate Value and Error (page 151)
CHAPTER APPLICATIONS OF DIFFERENTIATION.1 Approimate Value and Error (page 151) f '( lim 0 f ( f ( f ( f ( f '( or f ( f ( f '( f ( f ( f '( (.) f ( f '( (.) where f ( f ( f ( Eample.1 (page 15): Find
More informationMañé s Conjecture from the control viewpoint
Mañé s Conjecture from the control viewpoint Université de Nice - Sophia Antipolis Setting Let M be a smooth compact manifold of dimension n 2 be fixed. Let H : T M R be a Hamiltonian of class C k, with
More information7.7 LOTKA-VOLTERRA M ODELS
77 LOTKA-VOLTERRA M ODELS sstems, such as the undamped pendulum, enabled us to draw the phase plane for this sstem and view it globall In that case we were able to understand the motions for all initial
More informationEntrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.
Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the
More information(x k ) sequence in F, lim x k = x x F. If F : R n R is a function, level sets and sublevel sets of F are any sets of the form (respectively);
STABILITY OF EQUILIBRIA AND LIAPUNOV FUNCTIONS. By topological properties in general we mean qualitative geometric properties (of subsets of R n or of functions in R n ), that is, those that don t depend
More informationRANGE CONTROL MPC APPROACH FOR TWO-DIMENSIONAL SYSTEM 1
RANGE CONTROL MPC APPROACH FOR TWO-DIMENSIONAL SYSTEM Jirka Roubal Vladimír Havlena Department of Control Engineering, Facult of Electrical Engineering, Czech Technical Universit in Prague Karlovo náměstí
More informationPERIODICALLY FORCED CHAOTIC SYSTEM WITH SIGNUM NONLINEARITY
International Journal of Bifurcation and Chaos, Vol., No. 5 () 499 57 c World Scientific Publishing Compan DOI:.4/S874664 PERIODICALLY FORCED CHAOTIC SYSTEM WITH SIGNUM NONLINEARITY KEHUI SUN,, and J.
More informationOn the dynamics of strongly tridiagonal competitive-cooperative system
On the dynamics of strongly tridiagonal competitive-cooperative system Chun Fang University of Helsinki International Conference on Advances on Fractals and Related Topics, Hong Kong December 14, 2012
More informationLimits 4: Continuity
Limits 4: Continuit 55 Limits 4: Continuit Model : Continuit I. II. III. IV. z V. VI. z a VII. VIII. IX. Construct Your Understanding Questions (to do in class). Which is the correct value of f (a) in
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 informationarxiv: v2 [nlin.cd] 24 Aug 2018
3D billiards: visualization of regular structures and trapping of chaotic trajectories arxiv:1805.06823v2 [nlin.cd] 24 Aug 2018 Markus Firmbach, 1, 2 Steffen Lange, 1 Roland Ketzmerick, 1, 2 and Arnd Bäcker
More informationDynamics of multiple pendula without gravity
Chaotic Modeling and Simulation (CMSIM) 1: 57 67, 014 Dnamics of multiple pendula without gravit Wojciech Szumiński Institute of Phsics, Universit of Zielona Góra, Poland (E-mail: uz88szuminski@gmail.com)
More informationThe Conley Index and Rigorous Numerics of Attracting Periodic Orbits
The Conley Index and Rigorous Numerics of Attracting Periodic Orbits Marian Mrozek Pawe l Pilarczyk Conference on Variational and Topological Methods in the Study of Nonlinear Phenomena (Pisa, 2000) 1
More informationNumerical Methods for Differential Games based on Partial Differential Equations
Numerical Methods for Differential Games based on Partial Differential Equations M. Falcone 29th April 25 Abstract In this paper we present some numerical methods for the solution of two-persons zero-sum
More informationRolle s Theorem. THEOREM 3 Rolle s Theorem. x x. then there is at least one number c in (a, b) at which ƒ scd = 0.
4.2 The Mean Value Theorem 255 4.2 The Mean Value Theorem f '(c) 0 f() We know that constant functions have zero derivatives, ut could there e a complicated function, with man terms, the derivatives of
More informationIn applications, we encounter many constrained optimization problems. Examples Basis pursuit: exact sparse recovery problem
1 Conve Analsis Main references: Vandenberghe UCLA): EECS236C - Optimiation methods for large scale sstems, http://www.seas.ucla.edu/ vandenbe/ee236c.html Parikh and Bod, Proimal algorithms, slides and
More informationA route to computational chaos revisited: noninvertibility and the breakup of an invariant circle
A route to computational chaos revisited: noninvertibilit and the breakup of an invariant circle Christos E. Frouzakis Combustion Research Laborator, Paul Scherrer Institute CH-5232, Villigen, Switzerland
More informationResearch Article Chaos and Control of Game Model Based on Heterogeneous Expectations in Electric Power Triopoly
Discrete Dnamics in Nature and Societ Volume 29, Article ID 469564, 8 pages doi:.55/29/469564 Research Article Chaos and Control of Game Model Based on Heterogeneous Epectations in Electric Power Triopol
More informationSection 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y
Paul J. Bruillard MATH 0.970 Problem Set 6 An Introduction to Abstract Mathematics R. Bond and W. Keane Section 3.1: 3b,c,e,i, 4bd, 6, 9, 15, 16, 18c,e, 19a, 0, 1b Section 3.: 1f,i, e, 6, 1e,f,h, 13e,
More informationThe Coupled Three-Body Problem and Ballistic Lunar Capture
The Coupled Three-Bod Problem and Ballistic Lunar Capture Shane Ross Martin Lo (JPL), Wang Sang Koon and Jerrold Marsden (Caltech) Control and Dnamical Sstems California Institute of Technolog Three Bod
More informationDynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP
Dnamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP E. Barrabés, J.M. Mondelo, M. Ollé December, 28 Abstract We consider the planar Restricted Three-Bod problem and the collinear
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 informationParameterized Joint Densities with Gaussian and Gaussian Mixture Marginals
Parameterized Joint Densities with Gaussian and Gaussian Miture Marginals Feli Sawo, Dietrich Brunn, and Uwe D. Hanebeck Intelligent Sensor-Actuator-Sstems Laborator Institute of Computer Science and Engineering
More information8 Autonomous first order ODE
8 Autonomous first order ODE There are different ways to approach differential equations. Prior to this lecture we mostly dealt with analytical methods, i.e., with methods that require a formula as a final
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 informationNonlinear Control Lecture # 1 Introduction. Nonlinear Control
Nonlinear Control Lecture # 1 Introduction Nonlinear State Model ẋ 1 = f 1 (t,x 1,...,x n,u 1,...,u m ) ẋ 2 = f 2 (t,x 1,...,x n,u 1,...,u m ).. ẋ n = f n (t,x 1,...,x n,u 1,...,u m ) ẋ i denotes the derivative
More informationAn introduction to Mathematical Theory of Control
An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018
More informationAnalytic Continuation of Analytic (Fractal) Functions
Analytic Continuation of Analytic (Fractal) Functions Michael F. Barnsley Andrew Vince (UFL) Australian National University 10 December 2012 Analytic continuations of fractals generalises analytic continuation
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 informationEffective dynamics of many-particle systems with dynamical constraint
Michael Herrmann Effective dnamics of man-particle sstems with dnamical constraint joint work with Barbara Niethammer and Juan J.L. Velázquez Workshop From particle sstems to differential equations WIAS
More informationThe Big Picture. Discuss Examples of unpredictability. Odds, Stanisław Lem, The New Yorker (1974) Chaos, Scientific American (1986)
The Big Picture Discuss Examples of unpredictability Odds, Stanisław Lem, The New Yorker (1974) Chaos, Scientific American (1986) Lecture 2: Natural Computation & Self-Organization, Physics 256A (Winter
More informationComputer Vision & Digital Image Processing
Computer Vision & Digital Image Processing Image Segmentation Dr. D. J. Jackson Lecture 6- Image segmentation Segmentation divides an image into its constituent parts or objects Level of subdivision depends
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 informationNOTES AND ERRATA FOR RECURRENCE AND TOPOLOGY. (τ + arg z) + 1
NOTES AND ERRATA FOR RECURRENCE AND TOPOLOGY JOHN M. ALONGI p.11 Replace ρ(t, z) = ρ(t, z) = p.11 Replace sin 2 z e τ ] 2 (τ + arg z) + 1 1 + z (e τ dτ. sin 2 (τ + arg z) + 1 z 2 z e τ ] 2 1 + z (e τ dτ.
More information(4p) See Theorem on pages in the course book. 3. Consider the following system of ODE: z 2. n j 1
MATEMATIK Datum -8- Tid eftermiddag GU, Chalmers Hjälpmedel inga A.Heintz Telefonvakt Aleei Heintz Tel. 76-786. Tenta i ODE och matematisk modellering, MMG, MVE6 Answer rst those questions that look simpler,
More informationPERIODIC SOLUTIONS WITH NONCONSTANT SIGN IN ABEL EQUATIONS OF THE SECOND KIND
PERIODIC SOLUTIONS WITH NONCONSTANT SIGN IN ABEL EQUATIONS OF THE SECOND KIND JOSEP M. OLM, XAVIER ROS-OTON, AND TERE M. SEARA Abstract. The study of periodic solutions with constant sign in the Abel equation
More informationChapter 8 Equilibria in Nonlinear Systems
Chapter 8 Equilibria in Nonlinear Sstems Recall linearization for Nonlinear dnamical sstems in R n : X 0 = F (X) : if X 0 is an equilibrium, i.e., F (X 0 ) = 0; then its linearization is U 0 = AU; A =
More informationKrylov Integration Factor Method on Sparse Grids for High Spatial Dimension Convection Diffusion Equations
DOI.7/s95-6-6-7 Krlov Integration Factor Method on Sparse Grids for High Spatial Dimension Convection Diffusion Equations Dong Lu Yong-Tao Zhang Received: September 5 / Revised: 9 March 6 / Accepted: 8
More informationSEPARABLE EQUATIONS 2.2
46 CHAPTER FIRST-ORDER DIFFERENTIAL EQUATIONS 4. Chemical Reactions When certain kinds of chemicals are combined, the rate at which the new compound is formed is modeled b the autonomous differential equation
More informationState Estimation of Continuous-Time Systems with Implicit Outputs from Discrete Noisy Time-Delayed Measurements
State Estimation of Continuous-Time Sstems with Implicit Outputs from Discrete Nois Time-Delaed Measurements A. Pedro Aguiar and João P. Hespanha Abstract We address the state estimation of a class of
More information2.2 SEPARABLE VARIABLES
44 CHAPTER FIRST-ORDER DIFFERENTIAL EQUATIONS 6 Consider the autonomous DE 6 Use our ideas from Problem 5 to find intervals on the -ais for which solution curves are concave up and intervals for which
More information