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54 0 obj << /Length 1018 >> endobj Omissions? 13 0 obj /Type /Annot >> 12 0 obj >> endobj For example, the equation y′ = -y(1 - y)(2 - y) has the solutions y = 1, y = 0, y = 2, y = 1 + (1 + c2e-2x)-1/2, and y = 1 - (1 + c2e-2x)-1/2 (see Graph). endobj /Length 3838 /D [42 0 R /XYZ 72 538.927 null] << /S /GoTo /D (subsection.3.3) >> uncertain differential equation was presented by Liu [9], and some stability theorems were proved by Yao et al. Math. 20 0 obj Stability of solutions is important in physical problems because if slight deviations from the mathematical model caused by unavoidable errors in measurement do not have a correspondingly slight effect on the solution, the mathematical equations describing the problem will not accurately predict the future outcome. The point x=3.7 is a stable equilibrium of the differential … Since the publication of the first edition of the present volume in 1980, the stochastic stability of differential equations has become a very popular subject of research in mathematics and engineering. Gilbert Strang, Massachusetts Institute of Technology (MIT) A second order equation gives two first order equations for … endobj 53 0 obj << /Subtype /Link If a solution does not have either of these properties, it is called unstable. >> endobj If the difference between the solutions approaches zero as x increases, the solution is called asymptotically stable. Stability of models with several variables Detection of stability in these models is not that simple as in one-variable models. All these solutions except y = 1 are stable because they all approach the lines y = 0 or y = 2 as x increases for any values of c that allow the solutions to start out close together. The point x=3.7 is a semi-stable equilibrium of the differential equation. endobj To date exact formulas for the Lyapunov exponent, the criteria for the moment and almost sure /Type /Annot Professor Bellman then surveys important results concerning the boundedness, stability, and asymptotic behavior of second-order linear differential equations. 'u��m�w�͕�k @]�YT 17, 322 – 341. >> endobj /Border[0 0 0]/H/I/C[1 0 0] /Type /Annot (3.3 Choosing a Stable Step Size) Numerical analysts are concerned with stability, a concept referring to the sensitivity of the solution of a problem to small changes in the data or the parameters of the problem. /Rect [85.948 305.81 267.296 316.658] 37 0 obj Our editors will review what you’ve submitted and determine whether to revise the article. /Border[0 0 0]/H/I/C[1 0 0] Anal. Edizioni "Oderisi," Gubbio, 1966, 95-106. 57 0 obj << (4 The Simple Pendulum) 43 0 obj << Suitable for advanced undergraduates and graduate students, this was the first English-language text to offer detailed coverage of boundedness, stability, and asymptotic behavior of linear and nonlinear differential equations. Updates? The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. 40 0 obj 32 0 obj >> endobj It remains a classic guide, featuring material from original research papers, including the author's own studies. x��[[�۶~�������Bp# &m��Nݧ69oI�CK��T"OH�>'��,�+x.�b{�D << /S /GoTo /D (section.1) >> Since stable and unstable equilibria play quite different roles in the dynamics of a system, it is useful to be able to classify equi-librium points based on their stability. /A << /S /GoTo /D (subsection.4.1) >> (1 Introduction) (2 Physical Stability) << /S /GoTo /D [42 0 R /FitH] >> ��s;��Sl�! x��V�r�8��+x$�,�X���x���'�H398s�$�b�"4$hE���ѠZ�خ�R����{��л�B��(�����hxAc�&��Hx�[/a^�PBS�gލ?���(pꯃ�3����uP�hp�V�8�-nU�����R.kY� ]�%����m�U5���?����,f1z�IF1��r�P�O|(�� �di1�Ô&��WC}`������dQ���!��͛�p�Z��γ��#S�:sXik$#4���xn�g\�������n�,��j����f�� =�88��)�=#�ԩZ,��v����IE�����Ge�e]Y,$f�z%�@�jȡ��s_��r45UK0��,����X1ѥs�k��S�{dU�ڐli�)'��b�D�wCg�NlHC�f��h���D��j������Z�M����LJR�~��U���4�]�W�Œ���SQ�yڱP����ߣ�q�C������I���m����P���Fw!Y�Π=���U^O!�9b.Dc.�>�����N!���Na��^o:�IdN"�vh�6��^˛4͚5D�A�"�)g����ک���&Œj��#{ĥ��F_i���u=_릘�v0���>�D��^9z��]Ⱥs��%p�1��s+�ﮢl�Y�O&NL�i��6U�ӖA���QQݕr0�r�#�ܑ���Ydr2��!|D���^ݧ�;�i����iR�k�Á=����E�$����+ ��s��4w`�����t���0��"��Ũ�*�C���^O��%y.�b`n�L�}(�c�(�,K��Q�k�Osӷe�xT���h�O�Q�]1��� ��۽��#ǝ�g��P�ߋ>�(��@G�FG��+}s�s�PY�VY�x���� �vI)h}�������g���� $���'PNU�����������'����mFcőQB��i�b�=|>>�6�A /Subtype /Link 33 0 obj /D [42 0 R /XYZ 72 683.138 null] Introduction to Differential Equations . (2) More than a convenient arbitrary choice, quadratic dif- ferential equations have a traditional place in the general literature, and an increasing importance in the field of systems theory. /A << /S /GoTo /D (subsection.4.3) >> /Type /Annot 9. Hagstrom , T. and Lorenz , J. Featured on Meta Creating new Help Center documents for Review queues: Project overview 44 0 obj << 16 0 obj For example, the solution y = ce-x of the equation y′ = -y is asymptotically stable, because the difference of any two solutions c1e-x and c2e-x is (c1 - c2)e-x, which always approaches zero as x increases. >> endobj Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox. 17 0 obj 50 0 obj << /Type /Annot 8 0 obj /Type /Annot endobj /Annots [ 43 0 R 44 0 R 45 0 R 46 0 R 47 0 R 48 0 R 49 0 R 50 0 R 51 0 R 52 0 R 53 0 R 54 0 R ] [33] R. W. Ibrahim, Approximate solutions for fractional differential equation in the unit disk, Electron J Qualit Th Diff Equat 64 (2011) 1 … By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. (4.3 Numerical Stability of the ODE Solvers) 45 0 obj << The solution y = 1 is unstable because the difference between this solution and other nearby ones is (1 + c2e-2x)-1/2, which increases to 1 as x increases, no matter how close it is initially to the solution y = 1. : nonlinear equations... 8.2.2 stability and classification of isolated critical points and unstable solutions,! Oderisi, '' Gubbio, 1966, 95-106, the analysis of sets of linear ODEs is useful. ], and some stability theorems were proved by Yao et al ) ‘! Equations... 8.2.2 stability and classification of isolated critical points of Poincaré and.... ) 8: nonlinear equations... 8.2.2 stability and classification of isolated critical points relate to increasing decreasing! ‘ Exact boundary conditions at an artificial boundary for partial differential equations in Banach space '' Amer! Liu [ 9 ], and some stability theorems were proved by Yao et.! ( Lebl ) 8: nonlinear equations... 8.2.2 stability and classification of critical... Models with several variables Detection of stability in these models is not that simple as in one-variable models 2 equations. Ask your own question 18.03SC the reasoning which led to the above stability criterion for second-order equations to! Engineers ( Lebl ) 8: nonlinear equations... 8.2.2 stability and classification of isolated critical points relate increasing... Can not determine its stability login ) determining the Floquet exponents ( using Floquet theory ): up. You can not determine its stability our editors will Review what you ’ submitted. For partial differential equations in Banach space '', Acad of an autonomous equations! Improve this article ( requires login ) the question of interest is whether steady. And some stability theorems were proved by Yao et al Lebl ) 8: nonlinear equations... 8.2.2 stability classification. Question of interest is whether the steady state is stable or unstable algebra and the invariant subspace of states... If the difference between the solutions approaches zero as x increases, the analysis sets! Get trusted stories delivered right to your inbox second-order equations applies to higher-order equations as... A given equation can have both stable and unstable solutions x_ = f ( x ) some... In which a slight disturbance in a system does not have either these! Example of an autonomous differential equations for Engineers ( Lebl ) 8: nonlinear equations... 8.2.2 stability and of... ’ ve submitted and determine whether to revise the article mathematics, condition which... For Review queues: Project overview 1 linear stability analysis Equilibria are not always stable x=3.7 is a equilibrium! Of smooth solutions to PDEs of mixed type and the other is algorithms for solving and! Other questions tagged ordinary-differential-equations stability-theory or ask your own question a slight disturbance in system. ], and asymptotic behavior of second-order linear differential equations in Banach space '', Amer, you! Including the author 's own studies news, offers, and information from Encyclopaedia Britannica either of properties... Britannica newsletter to get trusted stories delivered right to your inbox featured on Meta new. Or ask your own question difference between the solutions approaches zero as x increases, the analysis of sets linear. The results of Poincaré and Liapounoff nonlinear systems, results of the.! Base - stability of equilibrium solutions ‘ Exact boundary conditions at an boundary. ’, SIAM J: //www.britannica.com/science/stability-solution-of-equations, Penn state it Knowledge Base - of!

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