stability of differential equations

>> >> endobj /Contents 56 0 R For that reason, we will pursue this In partial differential equations one may measure the distances between functions using Lp norms or th Example 2.5. /Type /Annot << /S /GoTo /D (subsection.3.1) >> The stability of a fixed point is found by determining the Floquet exponents (using Floquet theory):. >> endobj It remains a classic guide, featuring material from original research papers, including the author's own studies. /Subtype /Link Stability of models with several variables Detection of stability in these models is not that simple as in one-variable models. Proof is given in MATB42. From the series: Differential Equations and Linear Algebra. The solution y = cex of the equation y′ = y, on the other hand, is unstable, because the difference of any two solutions is (c1 - c2)ex, which increases without bound as x increases. >> endobj >> endobj 13 0 obj Daletskii, M.G. Math. Featured on Meta Creating new Help Center documents for Review queues: Project overview /Type /Annot FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS Theorem 2.4 If F and G are functions that are continuously diﬀerentiable throughout a simply connected region, then F dx+Gdy is exact if and only if ∂G/∂x = ∂F/∂y. Linear Stability Analysis for Systems of Ordinary Di erential Equations Consider the following two-dimensional system: x_ = f(x;y); y_ = g(x;y); and suppose that (x; y) is a steady state, that is, f(x ; y)=0 and g(x; y )=0. (1974) (Translated from Russian) [5] J. 3 Numerical Stability Physical stability of an equilibrium solution to a system of di erential equations addresses the behavior of solutions that start nearby the equilibrium solution. The point x=3.7 is a semi-stable equilibrium of the differential equation. /Subtype /Link 49 0 obj << endobj (3 Numerical Stability) 51 0 obj << The following was implemented in Maple by Marcus Davidsson (2009) davidsson_marcus@hotmail.com and is based upon the work by Shone (2003) Economic Dynamics: Phase Diagrams and their Economics Application and Dowling (1980) Shaums Outlines: An Introduction to Mathematical Economics If a solution does not have either of these properties, it is called unstable. 21 0 obj endobj << /S /GoTo /D (section.1) >> 25 0 obj /Rect [85.948 411.551 256.226 422.399] endobj << /S /GoTo /D (section.4) >> /Border[0 0 0]/H/I/C[1 0 0] /Subtype /Link By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. 48 0 obj << endobj This means that it is structurally able to provide a unique path to the fixed-point (the “steady- Corrections? Electron J Qualit Th Diff Equat 63( 2011) 1-10. (4.2 Physical Stability for the Pendulum) 1953 edition. << /S /GoTo /D (subsection.3.2) >> Navigate parenthood with the help of the Raising Curious Learners podcast. 1 Linear stability analysis Equilibria are not always stable. 9. /Type /Annot /Length 3838 Here is a set of notes used by Paul Dawkins to teach his Differential Equations course at Lamar University. Professor Bellman then surveys important results concerning the boundedness, stability, and asymptotic behavior of second-order linear differential equations. endobj 37 0 obj >> endobj (3.3 Choosing a Stable Step Size) Let us know if you have suggestions to improve this article (requires login). Therefore: a 2 × 2 system of differential equations can be studied as a mathematical object, and we may arrive at the conclusion that it possesses the saddle-path stability property. (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. (4.1 Numerical Solution of the ODE) La Salle, S. Lefschetz, "Stability by Lyapunov's direct method with applications" , Acad. Consider the following example. 46 0 obj << /Border[0 0 0]/H/I/C[1 0 0] �tm��-`/0�+�@P�h �#�Fͩ8�X(�kߚ��J`� XGDIP ��΅ۮ?3�.��N��C��9R%YO��/��|�4�qd9�j`�L��.�j�d�f�/�m�װ��"��V�Sx�Y5V�v�N~ ( 1995 ), ‘ All-time existence of smooth solutions to PDEs of mixed type and the invariant subspace of uniform states , Adv. endobj 'u��m�w�͕�k @]�YT 36 0 obj /A << /S /GoTo /D (subsection.3.2) >> Strict Stability is a different stability definition and this stability type can give us an information about the rate of … Krein, "Stability of solutions of differential equations in Banach space" , Amer. 55 0 obj << /D [42 0 R /XYZ 72 683.138 null] /Rect [85.948 373.24 232.952 384.088] /Border[0 0 0]/H/I/C[1 0 0] Stability OCW 18.03SC The reasoning which led to the above stability criterion for second-order equations applies to higher-order equations just as well. 61 0 obj << Featured on Meta Creating new Help Center documents for Review queues: Project overview 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. Browse other questions tagged quantum-mechanics differential-equations stability or ask your own question. endobj << /S /GoTo /D (section.2) >> Browse other questions tagged ordinary-differential-equations stability-theory or ask your own question. 45 0 obj << /Rect [85.948 392.395 249.363 403.243] https://www.britannica.com/science/stability-solution-of-equations, Penn State IT Knowledge Base - Stability of Equilibrium Solutions. 9 0 obj /Type /Annot The polynomial. (1 Introduction) >> endobj /Subtype/Link/A<> 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. stream 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). Math. >> The point x=3.7 is an equilibrium of the differential equation, but you cannot determine its stability. endobj The point x=3.7 is a stable equilibrium of the differential … /Type /Annot >> endobj After that, different types of stability of uncertain differential equations were explored, such as stability in moment [12] and almost sure stability [10]. 50 0 obj << ��/�yV�g^ϙ�ڀ��r>�1`��8�u�=�l�Z�H��Y� %��MG0c��/~��L#K��"�^�}��o�~��H�슾�� In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms.The precise definition of stability depends on the context. << /S /GoTo /D (section.3) >> /Type /Annot endobj In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. Gilbert Strang, Massachusetts Institute of Technology (MIT) A second order equation gives two first order equations for … << /S /GoTo /D (subsection.3.3) >> Soc. /Type /Page A given equation can have both stable and unstable solutions. Updates? (4 The Simple Pendulum) In recent years, uncertain differential equations … Our editors will review what you’ve submitted and determine whether to revise the article. x��[[�۶~��Bp# &m��Nݧ69oI�CK��T"OH�>'��,�+x.�b{�D /A << /S /GoTo /D (subsection.3.3) >> 1 0 obj (4.3 Numerical Stability of the ODE Solvers) %�� /Rect [71.004 344.121 200.012 354.97] https://www.patreon.com/ProfessorLeonard Exploring Equilibrium Solutions and how critical points relate to increasing and decreasing populations. /Subtype /Link 44 0 obj << /D [42 0 R /XYZ 71 721 null] In general, systems of biological interest will not result in a set of linear ODEs, so don’t expect to get lucky too often. 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 ] /Filter /FlateDecode /Subtype /Link 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. << /S /GoTo /D (subsection.4.3) >> /A << /S /GoTo /D (section.3) >> (3.1 Stability for Single-Step Methods) Reference [1] J. H. Hubbard and B. H. West, Differential Equations: A Dynamical Systems Approach, New York: Springer, 1991. /Type /Annot In addition that, we present definitions of stability and strict stability of fuzzy differential equations and also we have some theorems and comparison results. /Border[0 0 0]/H/I/C[1 0 0] �%��~�!��]G��c*M&*u�3�j�߱�[l�!�J�o=��[��)�[9��`��PE3��*�S]Ahy��Y�8��.̿D��$' (2 Physical Stability) Suitable for advanced undergraduates and graduate students, this was the first English-language text to offer detailed coverage of boundedness, stability, and asymptotic behavior of linear and nonlinear differential equations. 58 0 obj << Stability Problems of Solutions of Differential Equations, "Proceedings of NATO Advanced Study Institute, Padua, Italy." (3.2 Stability for Multistep Methods) endobj Stability, in mathematics, condition in which a slight disturbance in a system does not produce too disrupting an effect on that system. /Border[0 0 0]/H/I/C[1 0 0] In terms of the solution of a differential equation, a function f(x) is said to be stable if any other solution of the equation that starts out sufficiently close to it when x = 0 remains close to it for succeeding values of x. endobj 57 0 obj << 20 0 obj 17, 322 – 341. endobj Relatively slight errors in the initial population count, c, or in the breeding rate, a, will cause quite large errors in prediction, even if no disturbing influences occur. For example, the solution y = ce-x of the equation y′ = -y is asymptotically stable, because the difference of any two solutions c1e-x and c2e-x is (c1 - c2)e-x, which always approaches zero as x increases. F��4)1��M�z��N;�,#%�L:��KPG$��vcK��^�j{��"`%��kۄ�x"�}DR*��)�䒨�]��jM�(f҆�ތ&)�bs�7�|��I�:��ٝ/�|��|�\t缮�:�. 29 0 obj endstream 28 0 obj /A << /S /GoTo /D (section.1) >> Proof. [19]. Edizioni "Oderisi," Gubbio, 1966, 95-106. 40 0 obj 56 0 obj << endobj Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox. /Rect [71.004 631.831 220.914 643.786] 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. uncertain differential equation was presented by Liu [9], and some stability theorems were proved by Yao et al. 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. >> endobj 33 0 obj /Rect [85.948 286.655 283.651 297.503] [32] J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative. /Type /Annot Now, let’s move on to the point of this section. /D [42 0 R /XYZ 72 538.927 null] /ProcSet [ /PDF /Text ] Dynamics of the model is described by the system of 2 differential equations: endobj /Rect [85.948 305.81 267.296 316.658] 54 0 obj << 16 0 obj /Rect [71.004 430.706 186.12 441.555] /Subtype /Link Included are most of the standard topics in 1st and 2nd order differential equations, Laplace transforms, systems of differential eqauations, series solutions as well as a brief introduction to boundary value problems, Fourier series and partial differntial equations. << /S /GoTo /D [42 0 R /FitH] >> Stability, in mathematics, condition in which a slight disturbance in a system does not produce too disrupting an effect on that system.In terms of the solution of a differential equation, a function f(x) is said to be stable if any other solution of the equation that starts out sufficiently close to it when x = 0 remains close to it for succeeding values of x. If the difference between the solutions approaches zero as x increases, the solution is called asymptotically stable. /A << /S /GoTo /D (subsection.4.1) >> >> endobj /Filter /FlateDecode >> endobj /Rect [71.004 459.825 175.716 470.673] << /S /GoTo /D (subsection.4.1) >> >> endobj Hagstrom , T. and Lorenz , J. The logistics equation is an example of an autonomous differential equation. endobj 24 0 obj Differential Equations and Linear Algebra, 3.2c: Two First Order Equations: Stability. /Font << /F16 59 0 R /F8 60 0 R /F19 62 0 R >> 52 0 obj << 8 0 obj /Border[0 0 0]/H/I/C[1 0 0] 4 0 obj /Subtype /Link /Type /Annot /Length 1018 /A << /S /GoTo /D (subsection.4.2) >> 12 0 obj >> endobj [33] R. W. Ibrahim, Approximate solutions for fractional differential equation in the unit disk, Electron J Qualit Th Diff Equat 64 (2011) 1 … /A << /S /GoTo /D (section.2) >> /Type /Annot �^\��N��K�ݳ ��s~RJ/��3/�p��h�#A=�=m{��Euy{02�4ե �L��]�sz0f0�c$W��_�d&��ּ��.�?��{u��/�K�}��5�]Ix(��P�,Z��8�p+��@+a�6�BP��6��zx�{��$J`{�^�0��y��＄; ��z��.�8�uv�ނ0 ~��E�1gFnQ�{O�(�q8�+��r1�\��y��q7�'x��3r��4d�@f5��] ��Y�cΥ��q�4��_h�pg�a�{��b�Հ�H!I|��_G[v��N�߁L��r1�Q��L��`��:Y)I� � C4M��-5�c9íWa�u�`0,�3�Ex��54�~��W*�c��G��Xٳb��Z�]Qj��"*��@��K�=�u�]��s-��W��"��F��N�po�3 In regard to the stability of nonlinear systems, results of the linear theory are used to drive the results of Poincaré and Liapounoff. /Border[0 0 0]/H/I/C[1 0 0] >> endobj The question of interest is whether the steady state is stable or unstable. 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,$f�z%�@�jȡ��s_��r45UK0��,��X1ѥs�k��S�{dU�ڐli�)'��b�D�wCg�NlHC�f��h��D��j��Z�M��ǇR�~��U��4�]�W�Œ��SQ�yڱP��ߣ�q�C��I��m��P��Fw!Y�Π=��U^O!�9b.Dc.�>��N!��Na��^o:�IdN"�vh�6��^˛4͚5D�A�"�)g��ک��&j��#{ĥ��F_i��u=_릘�v0��>�D��^9z��]Ⱥs��%p�1��s+�ﮢl�Y�O&NL�i��6U�ӖA��QQݕr0�r�#�ܑ��Ydr2��!|D��^ݧ�;�i��iR�k�Á=��E�$��+ ��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 Autonomous differential equations are differential equations that are of the form. Thus, one of the difficulties in predicting population growth is the fact that it is governed by the equation y = axce, which is an unstable solution of the equation y′ = ay. 43 0 obj << Differential Equations Book: Differential Equations for Engineers (Lebl) 8: Nonlinear Equations ... 8.2.2 Stability and classiﬁcation of isolated critical points. /Subtype /Link investigation of the stability characteristics of a class of second-order differential equations and i = Ax + B(x) qx). endobj The end result is the same: Stability criterion for higher-order ODE’s — root form ODE (9) is stable ⇐⇒ all roots of (10) have negative real parts; (11) >> endobj Suppose that we have a set of autonomous ordinary differential equations, written in vector form: x˙ =f(x): (1) Let's consider a predator-prey model with two variables: (1) density of prey and (2) density of predators. 42 0 obj << 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. 41 0 obj /A << /S /GoTo /D (subsection.4.3) >> /Subtype /Link \[\frac{{dy}}{{dt}} = f\left( y \right)\] The only place that the independent variable, $t$ in this case, appears is in the derivative. Consider 32 0 obj Omissions? Anal. << /S /GoTo /D (subsection.4.2) >> for linear difference equations. 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. /Rect [71.004 490.88 151.106 499.791] /Resources 55 0 R 17 0 obj /Border[0 0 0]/H/I/C[0 1 1] /Parent 63 0 R /MediaBox [0 0 612 792] ��s;��Sl�! endobj ��|��튮�yA��7/�x�ԊI"�⫛�J�҂0�V7��k��2Ɠ��r#غ��ˮ-�r��?�xeV)IW�u��P��mxk+_7y��[�q��kf/l}{�p��o�]v�8ۡ�)s��C�6ܬ�ӻ�V�f�M��O��m^��m]��ޯ��~Ѣ�k[�5o��ͩh�~��z��^�z��VT�H�$(ꡪaJB= �q�)�l�2M�7Ǽ�O��Ϭv��9[)��?��o،��:��|W��mU�s��%j~�(y��v��p�N��F�j�Yke��sf_�� G�?`Y��ݢ�F�y�u�l�6�,�u�v��va��{pʻ �9��ܿ��a7��1\5ŀvV�c";+�O�[l/ U�@�b��R��G��^t�-Pzb�'�6/��Sg�7�a��2��jKa��Yws�4@B��"T% ?�0� HBYx�M�'�Fs�N��2BD7#§"T��*la�N��6[��}�<9I�MO�'��b�d�$5�_m.��{�H�:��(Mt'8��'��L��#Ae�ˈ�`��3�e�fA��Lµ3�Tz�y� ��Gx�ȓ\�I��j0�y�8A!��;��&�&��G,�ξ��~b��ik�ں%8�Mx��E��Q�QTvzF�@�(,ـ!C��EՒ��R��'&aWpt��G�B��q^��eo��H��wa�S��[�?_��Lch^O_�5��EͳD�N4_�oO�ٛ�%R�H�Hn,�1��#˘�ر�\]�i7`�0fQ�V�� v��{�r�Y"�?��r6��x*��-�5X�pP��F^S�.ޛ ��m�Ά��^p�\�Xƻ� JN��kO��=��]ָ� (1986),‘ Exact boundary conditions at an artificial boundary for partial differential equations in cylinders ’, SIAM J. The point x=3.7 cannot be an equilibrium of the differential equation. >> endobj However, we will solve x_ = f(x) using some numerical method. 47 0 obj << 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. If you want to learn differential equations, have a look at Differential Equations for Engineers If your interests are matrices and elementary linear algebra, try Matrix Algebra for Engineers If you want to learn vector calculus (also known as multivariable calculus, or calcu-lus three), you can sign up for Vector Calculus for Engineers /Subtype /Link One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation.. To date exact formulas for the Lyapunov exponent, the criteria for the moment and almost sure /Rect [85.948 326.903 248.699 335.814] /Border[0 0 0]/H/I/C[1 0 0] /Type /Annot /Border[0 0 0]/H/I/C[1 0 0] %PDF-1.5 endobj Introduction to Differential Equations . >> endobj Main theme of this volume is the stability of nonautonomous differential equations, with emphasis on the Lyapunov stability of solutions, the existence and smoothness of invariant manifolds, the construction and regularity of topological conjugacies, the study of center manifolds, as well as their reversibility and equivariance properties. /A << /S /GoTo /D (section.4) >> /Border[0 0 0]/H/I/C[0 1 1] Hagstrom, T. and Keller, H. B. 5 0 obj stream endobj LASALLE, J. P., An invariance principle in the theory of stability, differential equations and dynamical systems, "Proceedings of the International Symposium, Puerto Rico." >> endobj /Border[0 0 0]/H/I/C[1 0 0] However, the analysis of sets of linear ODEs is very useful when considering the stability of non -linear systems at equilibrium. /Rect [158.066 600.72 357.596 612.675] Press (1961) [6] /A << /S /GoTo /D (subsection.3.1) >> endobj /Type /Annot The paper discusses both p-th moment and almost sure exponential stability of solutions to stochastic functional differential equations with impulsive by using the Razumikhin-type technique.The main goal is to find some conditions that could be applied to control more easily than using the usual method with Lyapunov functionals. endobj /Subtype/Link/A<> 53 0 obj << Yu.L. 67 0 obj <<