On the asymptotic behavior for a damped oscillator under a sublinear friction. J. I. Díaz and A. Liñán


 Bertha Carpenter
 1 years ago
 Views:
Transcription
1 h corresponds RAC Rev. R. Acad. Cien. Serie A. Mat. VOL. 95 (1), 2001, pp Matemática Aplicada / Applied Mathematics Comunicación Preliminar / Preliminary Communication On the asymptotic behavior for a damped oscillator under a sublinear friction. J. I. Díaz and A. Liñán equation! " where#%$ &" (')* Abstract. We show that there are two curves of initial data for which the solutions of the corresponding Cauchy problem associated to the as+",(."/10 vanish after a finite time. By using asymptotic and methods and comparison arguments we show that for many other initial data the solutions decay to, in an infinite time, Sobre el comportamiento asintótico de un oscilador amortiguado con un término de fricción sublineal iniciales ecuación 2 3 % 45 " Resumen. Mostramos la existencia de dos curvas de datos para las cuales las soluciones correspondientes del problema de Cauchy asociado a la supuesto#$6 &7 (', se anulan como+",(+"/0 identicamente despues de un tiempo finito. Mediante métodos asintóticos y argumentos de comparación mostramos que para muchos otros datos iniciales las soluciones decaen a " en un tiempo infinito, 1. Introduction We study the asymptotic behavior of solutions of the equation 8:93; (1) wherei!jlkg2mnpo and>qmre:stg UWe shall 92; ;D<V?93;)?A2B=C93;W<59:FHG work with the formulation 9 KYOZF\[ C*]^_A2B=C*`9 KbaYO (2) and[gf wherea Fdcce f ikj*lm n.opje mqopj h f UNotice c fe remaining 9 ; ; <ty zb{+ K9 ;OD<!9}~G (3) which is attaint by dividing bye and by introducing the rescaling that the9 rescaling fails fori FrN. In that case there is no well defined scale for9 to9; ;s<t[d92;s<9ufvg and the equation is reduced with[f as a parameter to characterize the dynamics. The limit caseixw G to the Coulomb friction equation Presentado por Gregorio Millán Recibido: 16 de Abril Aceptado: 10 de Octubre Palabras clave / Keywords: sublinear damped oscillator, Coulomb friction, extinction in a finite time. Mathematics Subject Classifications: 34D05, 70K05,70K99 c 2001 Real Academia de Ciencias, España. 155
2 is the andœ corresponding corresponds it andœ G). is, J. I. Díaz and A. Liñán wherey z { of ƒ byy z { is the maximal monotone graph given if if ˆ G2M Š_ N"M N) if FHG MandN K& OQF N, StG UThe limit equation wheniuw N with the linear damping equation We recall that, even if the nonlinear term?9 ;?A BDC9 ; is not a Lipschitz continuous function of9 ; the existence and uniqueness of solutions Œ of the associate Cauchy problem AŽ 99; ; KG+OQF 92 +MW9 <V?9 ;?A2B=C 9 ;KGOQF P ; <!9:FHG Y S G M (and of the limit problemsœ C to the equations (3) and (4) respectively) is well known in the literature: see, e.g. Brezis [1]. An easy application of the results of the above reference yields to a rigorous proof of the convergence of solutions whenixw G andixw N. The asymptotic behavior, fory w Mof solutions of the limit problemsœ C well known (see, 9 š for instance, Jordan Smith [5]). In the first case the decay is exponential. In the second one it is easy to see that given93 anyyz and P there exist a finite time F 4K9 "M* P PO and a number LJ Š NMN) such that for 4K93 +M* P PO. For problemœ (ŒA is wellknown thatk9 (M*9 ;*O wœkg MRG+O asy w (see, e.g. Haraux [4]). For a numerical study of A) see [6]. The main result of this paper is to show that the generic asymptotic behavior above described for the limit caseœ tokg MRG+O caseižjžkg M NqO tokg MGO only exceptional for the sublinear since the generic orbitsk9 ŸM*9 ;O decay in a infinite time and only two uniparametric families of them decay in a finite time: in other words, whenixw G exceptional behavior becomes generic. 9 ; ; <59 ; <!9:FHG U (4) 2. Formal results via asymptotic arguments We can rewrite the equation (2) in as the planar system 9 ; F which, by eliminating the time variable, forg F G, leads to the differential equation of the orbits in the phase plane 4F BDC (6) F š K 9O ifžf K9=O and that allows us to carry out a phase plane description of the dynamics. We remark that the plane phase is antisymmetric since is a solution of (6) then the function is also solution. So, it is enough to describe a semiplane (for instance9 By multiplying by9 andm respectively, we get thatk9 <V O; 9 KNP q9wm Nq P O pointkg2mgo <! S SHN F š?d?a+ sc. On the other hand, it is easy to see that satisfy a system which has the as a spiral unstable critical point. For values of the orbits of the system are given, in first approximation, by9 < FH because? =?A2B=C is small compared with9. The effect of this term is to decrease slowly For[5S[ªš«FV with time giving the trajectory a spiral character. ForI FvN the character of the trayectories close to the origin depends on the parameter[ƒu the origin is a stable mode and for[ ˆ [ª Iuw < is a stable spiral corresponding to underdamped oscillations. It should be noticed that fori S N originkg MGO the origin becomes a stable spiral point. The limit case can be described analytically with twotime scale methods (see [3]). We shall prove that there are two modes of approach to the origin and so that the is a node for the system (5). The lines of zero slope are given by ; F BDC (5) 9~F? =?A2B=CU (7) 156
3 G. On the asymptotic behavior for a damped oscillator 9 ˆ G MTS tokg2mg+o regions +K9WM Oš«9LS G MP ˆ 9 C*]RA= +K9WM*2Oš«So the convergence is only possible through the of9us G K 9=OC]AW. Let us see that the ordinary mode corresponds ˆ Let ŽF to orbits that are very close to the ones corresponding to small effects of the inertia. Due to the symmetry it is enough to describe this behavior for the orbits approaching the origin with values and 6S G UEquation (6) takes the form F²9 C]A < ³ K9=O isuf±9 C]RA The line of zero slope and we search for orbits obeying, forg ˆ 9 ˆ ˆ N, to the expression for some function³ K9O. If we anticipate the conditiong ˆ ³ K9O ˆ ˆ 9 C]RA, equation (6) takes the linearized form C A 9 ^q m B=C*`³µ<9 m ³ I 9 ^ CŸB m `³ŽFgG UThus the first term can be neglected, compared with the last one, and then the solution can be written with as³ K9OQ V u¹)º» ŠIƒ " KN+ I Ob 9 B ji lm n m thatž 9 an arbitrary C]A constant ¼; (which explain the name of ordinary orbits). This type of orbits are given, close to the origin, by the approximate equation ½9 (7), which for the orbits that reach the origin from below implies and so, integrating the simplified equation F 9 < A U (8) ½Y F 9 C]A (9) we get that 9 ¾ ŠKN I OKY I < YCO A]^CŸBA` (10) and so that it takes an infinite time to reach the origin. for#:5 0 Figure 1. The two exceptional orbits and, in the small figure, several ordinary orbits entering to the origin tangentially to the zero slop curve Some different orbits approaching the origin can be found by searching among solutions with large 157
4 G) and we, andî andî of can grows J. I. Díaz and A. Liñán with?9à?c*]ra values of?d?compared. Thus, close to the origin, the orbits with negative are very close to the solutions of the equation found by replacing (8) by the simplified the equation F A (11) ( corresponding to a balance of inertia and damping. The solution ending at the origin KGOšF\G) is given by for5s G Notice that it involves no arbitrary constant. So this curve is unique (a symmetric curve arises and9 9 ˆ ˆ ˆ which justifies the term of extraordinary orbit. The time evolution of this orbit is given, for N, by integrating the equation ½9 and so9 ŽF ^ BA+`Š^ BA`p^ CŸBA` C A KY YO ^ BA+Ầ]^C(BA+`Mwhere in generalãs FÅÄŽÆ7º that " ±Ç ŠK I OD?93? C*]^ BA+` G MÃs. This indicate that the motion (of this approximated solution) ends at a finite time,y, determined by the initial conditions which, by (13) must satisfy We point out that the two exceptional orbits emanating from the origin spiral around the origin when9 < toward infinity and È so G each of them is a separatrix curve in the phase plane. Notice that due to the autonomus nature of the equation, if9 (Œ datak93 +M* P 7O is the solution of the Cauchy problem A) of initial then for any parameter the function 9 «kfž9 KY < È O (Œ coincides with the solution of A) of initial datak9 KÈ O(M*93;KÈ (Œ O*OŸUIn (Œ this way, the above extraordinary orbits give rise to two curves of initial data which the corresponding solutions of A) vanish after a finite time. We end this section by pointing out that the solution of problem A) forg ˆÉIÊˆ ˆ N andi w G % F G takes an asymptotic form which can be easily described. The differential equations of the orbits simplify if is finite to " 6F² 96 HN forls G and 6F² 9Ë<gN at9ufž N at9f\ N or9ufín if6s G forìf² xs G2UThe solutions are circles with center and center9f±n intervalkg MNPO ˆ G if joined. An orbit formed with half circles with centers when it hits the from below it is transformed into an orbit that reaches the origin following very closely that segment, governed by the equation (9) of solution (10). In the limitilw G found that any point J (Œ Š_ N"M N) is an asymptotically stable stationary state of ). 3. Estimates on the decay K9=O¾Fv +K µ I Op9 C*]^ BA+`U (12) ½Y FvŠK Á I O9 C*]^ BA+` (13) In a previous work, by using a fixed point argument, we show Theorem 1 ([2]) There exists two curvesî B datak93 +M* P qo initial for which the solutions9 of the corresponding Cauchy problemkœ A O vanish after a finite time. andžfv It is possible to give some additional results on these two curves: Theorem 2 (i) Near the origin the curvesî B be represented by two functions,ufï K9=O someð4s G U B K9O(Msolutions of the equation (6), where «ŠG2M1Ðq w K MG7 and B «Š_ ÑÐMRGP w ŠG MŸ< O, for (ii) Functions B satisfy that ƒò KGOQF G Óˆ Ô!Õ ½y KyPO andô ½y B Õ In particular, ÀÖ K9OØ Ž when9ž G and QÖB K9=OØÙ < when9ëùág2u B KyPO ˆ < U (14) 158
5 implies andî by On the asymptotic behavior for a damped oscillator Ú µ9 K 9O j*lm Û K9O Û 9 m for9 m Û J B K9O Û K 9O j*lm for9 ŠG M1Ðq and J Š ÑÐMRGP pm JÝKG M*Ðq K9Ü OÌFÞ K9ÜO m andk 9ÜO m F B for some \STG. K 9=O «F 9 J ŠG M*9 Ü such K9Ü O. Moreover and K9O Û Û 9 m,ß B «F 9 J Š 9 ÜMG7 m Û Û B K9O +K9sM*2O: K9sM*2O: and are (iii) We have a9ü (iv) There exists regionsß that time invariants for equation (6). the In order to prove that the decay to zero in an infinite time is more generic than the decay to zero in a finite time we need to obtain sharper invariants regions 9 aàtj±kg M9ÜO Theorem 3 (i) There exists small enough such that the regionsß:á «F +K9WM*2O Jâß «J ŠG2M*9 Ü à and 9 m à ¹)º» ŠIØ 7 KN I Op.9 B ji lm n m ˆ Û 9 m «F K9WM2O Jgß B «9 J Š_ 9 Ü < àmrgp andk 9=O m Û Û K 9=O m < à ¹)º» ŠI " KNZ I Ob 9 B jpi lm n m O IfK93 M P qo J are (ii) (respectivelyßëáb (Œ Û Ú some \S G YBA])^ CŸBA` YBA])^ CŸBA`) for B :Fg K9O :Fg B and K9=OŸMis clear and92; Fž K9 O K9 *O F š respectively. A FÅ 9 m and µ9 K9OãS G, with is J KG M1ÐOj*lm M. respectively, ß assured andß B å LetK93 +M* P 7O å å A Û 9 onkg M*93 7O PROOF OF THEOREM 3.. and å K93 PO å K9=OŽFçæ+9 K9O Û å å C]RA <V³ K9O K9O P. Then, onkg2m*93 qo. Such can with³ K9=O FÊ u¹)º» ŠIØ 7 KN I Ob 9 B jpi lm n m with ès²g ³ K9O A 9 j*lm m < 9 æa9 C ˆ G arbitrary for9 JâKG2M*9 ÜO ifæ!sïg ˆ æ+9 is C]A). Finally, for initial data inß á Ú P9 m fory: G m Uä we and Acknowledgement. The research of JID was partially supported by the DGICYT, project REN Both authors thank to Miguel Hermanns for the elaboration of the Figure. time invariants for equation (6). ) then the solution9 of A) satisfies (respectively9 that9 and anyy STG. PROOF OF THEOREM 2. Since the solutions of equation (6) converge to the origin with zero orç (Œ slope the representation of curvesî once we assume?9ø?small enough. Then withy fory ÍY, large enough the solutions9 equation9=;4f² of the corresponding Cauchy problem A) vanish after a finite time.and satisfy the first order ordinary differential Therefore the necessary conditions (14) holds by (easy) well known results. To prove (iii) we call (as in the previous So, for instance,žf a solution of the new orbits equation It is easy to see that this equation admits some explicit sub and supersolutions of the form9 and so the result holds by a comparison argument. Finally, the existence of9 Ü in (iv) can be obtained, for instance, by using the inequalities of (iii) and equation (6). The direction of the flow at the boundaries of the time invariance of these regions.ä let us construct an auxiliary function satisfying that by comparison arguments (forward in the9 direction) we get that necesarily a function be constructed in the form but small enough (use the fact thatéj suitably chosen and that get that9 ; the decay inequality follows from the comparison with the exact solution of9 ;QFv Ú 79 References [1] Brezis, H. (1972). Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, NorthHolland, Amsterdam. [2] Díaz, J. I. and Liñán, A. (2002). On the asymptotic behavior of solutions of a damped oscillator under a sublinear friction term: from the exceptional to the generic behaviors. To appear in Advences in PDE, Lecture Notes in Pure and Applied Mathematics (A. Benkirane and A. Touzani. eds.), Marcel Dekker. 159
6 J. I. Díaz and A. Liñán [3] Díaz, J. I. and Liñán, A., On the dynamics of a constrained oscillator as limit of oscillators under an increasing superlinear friction. To appear. [4] Haraux, A. (1979). Comportement à l infini pour certains systèmes dissipatifs non linéaires, Proc. Roy. Soc. Edinburgh, 84A, [5] Jordan, D. W. and Smith, P. (1979). Nonlinear Ordinary Differential Equations, (Second Edition), Clarendon Press, Oxford. [6] Kuo PenYu and Vazquez, L. (1982). Numerical solution of an ordinary differential equation, Anales de Física, Serie B, 78, J. I. Díaz A. Liñán Facultad de Matemáticas E. T. S. I. Aeronaúticos Universidad Complutense de Madrid Universidad Politécnica de Madrid Madrid, Spain Madrid, Spain 160
New Results on the Burgers and the Linear Heat Equations in Unbounded Domains. J. I. Díaz and S. González
RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 99 (2), 25, pp. 219 225 Matemática Aplicada / Applied Mathematics Comunicación Preliminar / Preliminary Communication New Results on the Burgers and the Linear
More informationAn Introduction to the NavierStokes InitialBoundary Value Problem
An Introduction to the NavierStokes InitialBoundary Value Problem Giovanni P. Galdi Department of Mechanical Engineering University of Pittsburgh, USA Rechts auf zwei hohen Felsen befinden sich Schlösser,
More informationControllability and Observability of Partial Differential Equations: Some results and open problems
Controllability and Observability of Partial Differential Equations: Some results and open problems Enrique ZUAZUA Departamento de Matemáticas Universidad Autónoma 2849 Madrid. Spain. enrique.zuazua@uam.es
More informationA UNIQUENESS RESULT FOR THE CONTINUITY EQUATION IN TWO DIMENSIONS. Dedicated to Constantine Dafermos on the occasion of his 70 th birthday
A UNIQUENESS RESULT FOR THE CONTINUITY EQUATION IN TWO DIMENSIONS GIOVANNI ALBERTI, STEFANO BIANCHINI, AND GIANLUCA CRIPPA Dedicated to Constantine Dafermos on the occasion of his 7 th birthday Abstract.
More informationRegularity of the solutions for a Robin problem and some applications
Revista Colombiana de Matemáticas Volumen 42(2008)2, páginas 127144 Regularity of the solutions for a Robin problem and some applications Regularidad de las soluciones para un problema de Robin y algunas
More informationTHE PROBLEM OF finding localized energy solutions
600 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 3, MARCH 1997 Sparse Signal Reconstruction from Limited Data Using FOCUSS: A Reweighted Minimum Norm Algorithm Irina F. Gorodnitsky, Member, IEEE,
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More informationLecture notes on the Stefan problem
Lecture notes on the Stefan problem Daniele Andreucci Dipartimento di Metodi e Modelli Matematici Università di Roma La Sapienza via Antonio Scarpa 16 161 Roma, Italy andreucci@dmmm.uniroma1.it Introduction
More informationConservative nonlinear oscillators in Abel s mechanical problem
ENSEÑANZA REVISTA MEXICANA DE FÍSICA E 57 (1) 67 72 JUNIO 211 Conservative nonlinear oscillators in Abel s mechanical problem R. Muñoz Universidad Autónoma de la Ciudad de México, Centro istórico, Fray
More informationFamilies of symmetric periodic orbits in the three body problem and the figure eight
Monografías de la Real Academia de Ciencias de Zaragoza. 25: 229 24, (24). Families of symmetric periodic orbits in the three body problem and the figure eight F. J. MuñozAlmaraz, J. Galán and E. Freire
More informationSymmetric planar non collinear relative equilibria for the Lennard Jones potential 3 body problem with two equal masses
Monografías de la Real Academia de Ciencias de Zaragoza. 25: 93 114, (2004). Symmetric planar non collinear relative equilibria for the Lennard Jones potential 3 body problem with two equal masses M. Corbera,
More informationElements of Abstract Group Theory
Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for
More informationA Googlelike Model of Road Network Dynamics and its Application to Regulation and Control
A Googlelike Model of Road Network Dynamics and its Application to Regulation and Control Emanuele Crisostomi, Steve Kirkland, Robert Shorten August, 2010 Abstract Inspired by the ability of Markov chains
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationBoundary value problems in complex analysis I
Boletín de la Asociación Matemática Venezolana, Vol. XII, No. (2005) 65 Boundary value problems in complex analysis I Heinrich Begehr Abstract A systematic investigation of basic boundary value problems
More informationRECENTLY, there has been a great deal of interest in
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 1, JANUARY 1999 187 An Affine Scaling Methodology for Best Basis Selection Bhaskar D. Rao, Senior Member, IEEE, Kenneth KreutzDelgado, Senior Member,
More informationThe Effect of Network Topology on the Spread of Epidemics
1 The Effect of Network Topology on the Spread of Epidemics A. Ganesh, L. Massoulié, D. Towsley Microsoft Research Dept. of Computer Science 7 J.J. Thomson Avenue University of Massachusetts CB3 0FB Cambridge,
More informationWMR Control Via Dynamic Feedback Linearization: Design, Implementation, and Experimental Validation
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 6, NOVEMBER 2002 835 WMR Control Via Dynamic Feedback Linearization: Design, Implementation, and Experimental Validation Giuseppe Oriolo, Member,
More informationSUMMING CURIOUS, SLOWLY CONVERGENT, HARMONIC SUBSERIES
SUMMING CURIOUS, SLOWLY CONVERGENT, HARMONIC SUBSERIES THOMAS SCHMELZER AND ROBERT BAILLIE 1 Abstract The harmonic series diverges But if we delete from it all terms whose denominators contain any string
More informationWiener s test for superbrownian motion and the Brownian snake
Probab. Theory Relat. Fields 18, 13 129 (1997 Wiener s test for superbrownian motion and the Brownian snake JeanStephane Dhersin and JeanFrancois Le Gall Laboratoire de Probabilites, Universite Paris
More informationPolitical Uncertainty and the Peso Problem*
año 2 vols. 3 y 4 Revista de Economía Política de Buenos Aires 33 Political Uncertainty and the Peso Problem* Javier GarcíaFronti Universidad de Buenos Aires and Lei Zhang University of Warwick Abstract
More informationCalderón problem. Mikko Salo
Calderón problem Lecture notes, Spring 2008 Mikko Salo Department of Mathematics and Statistics University of Helsinki Contents Chapter 1. Introduction 1 Chapter 2. Multiple Fourier series 5 2.1. Fourier
More informationANALYTIC NUMBER THEORY LECTURE NOTES BASED ON DAVENPORT S BOOK
ANALYTIC NUMBER THEORY LECTURE NOTES BASED ON DAVENPORT S BOOK ANDREAS STRÖMBERGSSON These lecture notes follow to a large extent Davenport s book [5], but with things reordered and often expanded. The
More informationReinitialization Free Level Set Evolution via Reaction Diffusion
1 Reinitialization Free Level Set Evolution via Reaction Diffusion Kaihua Zhang, Lei Zhang, Member, IEEE, Huihui Song and David Zhang, Fellow, IEEE Abstract This paper presents a novel reactiondiffusion
More informationThe Backpropagation Algorithm
7 The Backpropagation Algorithm 7. Learning as gradient descent We saw in the last chapter that multilayered networks are capable of computing a wider range of Boolean functions than networks with a single
More informationAn existence result for a nonconvex variational problem via regularity
An existence result for a nonconvex variational problem via regularity Irene Fonseca, Nicola Fusco, Paolo Marcellini Abstract Local Lipschitz continuity of minimizers of certain integrals of the Calculus
More informationCLASSIFICATORY PROBLEMS IN AFFINE GEOMETRY APPROACHED BY DIFFERENTIAL EQUATIONS METHODS INTRODUCTION
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 47, Número 2, 2006, Páginas 95 107 CLASSIFICATORY PROBLEMS IN AFFINE GEOMETRY APPROACHED BY DIFFERENTIAL EQUATIONS METHODS SALVADOR GIGENA Abstract. We
More informationTHE CONGRUENT NUMBER PROBLEM
THE CONGRUENT NUMBER PROBLEM KEITH CONRAD 1. Introduction A right triangle is called rational when its legs and hypotenuse are all rational numbers. Examples of rational right triangles include Pythagorean
More informationA Perfect Example for The BFGS Method
A Perfect Example for The BFGS Method YuHong Dai Abstract Consider the BFGS quasinewton method applied to a general nonconvex function that has continuous second derivatives. This paper aims to construct
More informationInstabilities in the SunJupiterAsteroid Three Body Problem
Noname manuscript No. (will be inserted by the editor) Instabilities in the SunJupiterAsteroid Three Body Problem John C. Urschel Joseph R. Galante Received: date / Accepted: date Abstract We consider
More information