stability of differential equations

12 0 obj 54 0 obj << Consider the following example. 28 0 obj In addition that, we present definitions of stability and strict stability of fuzzy differential equations and also we have some theorems and comparison results. In general, systems of biological interest will not result in a set of linear ODEs, so don’t expect to get lucky too often. /A << /S /GoTo /D (subsection.4.2) >> /Subtype /Link /Type /Annot /Type /Annot >> endobj 37 0 obj >> endobj (2 Physical Stability) /Rect [71.004 459.825 175.716 470.673] %PDF-1.5 9. /Border[0 0 0]/H/I/C[1 0 0] 56 0 obj << 4 0 obj /Rect [85.948 326.903 248.699 335.814] 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). /Type /Annot Updates? 57 0 obj << 9 0 obj << /S /GoTo /D [42 0 R /FitH] >> In recent years, uncertain differential equations … Reference [1] J. H. Hubbard and B. H. West, Differential Equations: A Dynamical Systems Approach, New York: Springer, 1991. stream Example 2.5. >> endobj endobj /Subtype/Link/A<> The point x=3.7 is a semi-stable equilibrium of the differential equation. endobj 55 0 obj << Let's consider a predator-prey model with two variables: (1) density of prey and (2) density of predators. A given equation can have both stable and unstable solutions. (1 Introduction) 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. 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. 45 0 obj << The stability of a fixed point is found by determining the Floquet exponents (using Floquet theory):. 13 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. /Subtype/Link/A<> 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. investigation of the stability characteristics of a class of second-order differential equations and i = Ax + B(x) qx). In partial differential equations one may measure the distances between functions using Lp norms or th >> endobj /Subtype /Link /Border[0 0 0]/H/I/C[1 0 0] /Parent 63 0 R x��[[�۶~��Bp# &m��Nݧ69oI�CK��T"OH�>'��,�+x.�b{�D (3.1 Stability for Single-Step Methods) �%��~�!��]G��c*M&*u�3�j�߱�[l�!�J�o=��[��)�[9��`��PE3��*�S]Ahy��Y�8��.̿D��$' << /S /GoTo /D (section.3) >> endobj endobj endobj 48 0 obj << /Rect [85.948 305.81 267.296 316.658] 49 0 obj << By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. 67 0 obj << 5 0 obj However, the analysis of sets of linear ODEs is very useful when considering the stability of non -linear systems at equilibrium. Math. https://www.patreon.com/ProfessorLeonard Exploring Equilibrium Solutions and how critical points relate to increasing and decreasing populations. /Filter /FlateDecode 16 0 obj (4 The Simple Pendulum) >> endobj /Length 1018 Proof. (3.3 Choosing a Stable Step Size) 17, 322 – 341. /Rect [85.948 373.24 232.952 384.088] /A << /S /GoTo /D (subsection.4.1) >> /Rect [85.948 411.551 256.226 422.399] Electron J Qualit Th Diff Equat 63( 2011) 1-10. /Border[0 0 0]/H/I/C[0 1 1] << /S /GoTo /D (subsection.3.1) >> << /S /GoTo /D (subsection.3.3) >> 43 0 obj << LASALLE, J. P., An invariance principle in the theory of stability, differential equations and dynamical systems, "Proceedings of the International Symposium, Puerto Rico." [33] R. W. Ibrahim, Approximate solutions for fractional differential equation in the unit disk, Electron J Qualit Th Diff Equat 64 (2011) 1 … \[\frac{{dy}}{{dt}} = f\left( y \right)\] The only place that the independent variable, $t$ in this case, appears is in the derivative. Daletskii, M.G. The question of interest is whether the steady state is stable or unstable. Proof is given in MATB42. /D [42 0 R /XYZ 72 538.927 null] /Subtype /Link ��|��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��=��]ָ� stream /A << /S /GoTo /D (section.3) >> Now, let’s move on to the point of this section. Autonomous differential equations are differential equations that are of the form. >> endobj The logistics equation is an example of an autonomous differential equation. /A << /S /GoTo /D (section.2) >> /Resources 55 0 R 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 /Border[0 0 0]/H/I/C[1 0 0] Krein, "Stability of solutions of differential equations in Banach space" , Amer. 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) (1974) (Translated from Russian) [5] J. << /S /GoTo /D (section.4) >> �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~ << /S /GoTo /D (subsection.3.2) >> endobj 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. Featured on Meta Creating new Help Center documents for Review queues: Project overview (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. endobj For that reason, we will pursue this Corrections? /A << /S /GoTo /D (subsection.3.2) >> The point x=3.7 cannot be an equilibrium of the differential equation. 36 0 obj >> endobj /Subtype /Link 1 Linear stability analysis Equilibria are not always stable. %�� Press (1961) [6] /Type /Annot (3 Numerical Stability) 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. /Rect [71.004 430.706 186.12 441.555] /Subtype /Link 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. /Border[0 0 0]/H/I/C[1 0 0] /A << /S /GoTo /D (section.1) >> Hagstrom, T. and Keller, H. B. /Subtype /Link Navigate parenthood with the help of the Raising Curious Learners podcast. (4.1 Numerical Solution of the ODE) Edizioni "Oderisi," Gubbio, 1966, 95-106. Let us know if you have suggestions to improve this article (requires login). /Type /Annot La Salle, S. Lefschetz, "Stability by Lyapunov's direct method with applications" , Acad. 24 0 obj Hagstrom , T. and Lorenz , J. /A << /S /GoTo /D (section.4) >> Anal. << /S /GoTo /D (subsection.4.1) >> /Border[0 0 0]/H/I/C[0 1 1] Browse other questions tagged ordinary-differential-equations stability-theory or ask your own question. If the difference between the solutions approaches zero as x increases, the solution is called asymptotically stable. << /S /GoTo /D (subsection.4.2) >> Yu.L. /Type /Annot From the series: Differential Equations and Linear Algebra. >> 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. >> endobj Consider 32 0 obj Soc. 46 0 obj << endobj Suppose that we have a set of autonomous ordinary differential equations, written in vector form: x˙ =f(x): (1) endobj If a solution does not have either of these properties, it is called unstable. endobj endobj /A << /S /GoTo /D (subsection.4.3) >> It remains a classic guide, featuring material from original research papers, including the author's own studies. 'u��m�w�͕�k @]�YT /Filter /FlateDecode >> endobj >> endobj /Type /Page /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 ] The point x=3.7 is a stable equilibrium of the differential … 51 0 obj << endobj /Font << /F16 59 0 R /F8 60 0 R /F19 62 0 R >> /A << /S /GoTo /D (subsection.3.1) >> 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. 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. /Subtype /Link endobj F��4)1��M�z��N;�,#%�L:��KPG$��vcK��^�j{��"`%��kۄ�x"�}DR*��)�䒨�]��jM�(f҆�ތ&)�bs�7�|��I�:��ٝ/�|��|�\t缮�:�. Math. [19]. Omissions? Stability of models with several variables Detection of stability in these models is not that simple as in one-variable models. Our editors will review what you’ve submitted and determine whether to revise the article. endstream 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. /Border[0 0 0]/H/I/C[1 0 0] << /S /GoTo /D (section.2) >> >> endobj 25 0 obj /Type /Annot (4.3 Numerical Stability of the ODE Solvers) 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. 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. endobj After that, different types of stability of uncertain differential equations were explored, such as stability in moment [12] and almost sure stability [10]. (3.2 Stability for Multistep Methods) /Subtype /Link Featured on Meta Creating new Help Center documents for Review queues: Project overview >> endobj 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. /Rect [71.004 344.121 200.012 354.97] /Border[0 0 0]/H/I/C[1 0 0] The polynomial. /Type /Annot << /S /GoTo /D (subsection.4.3) >> 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. 17 0 obj endobj Professor Bellman then surveys important results concerning the boundedness, stability, and asymptotic behavior of second-order linear differential equations. ( 1995 ), ‘ All-time existence of smooth solutions to PDEs of mixed type and the invariant subspace of uniform states , Adv. To date exact formulas for the Lyapunov exponent, the criteria for the moment and almost sure endobj /Contents 56 0 R 8 0 obj 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 The point x=3.7 is an equilibrium of the differential equation, but you cannot determine its stability. 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. 41 0 obj >> endobj 42 0 obj << /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. 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. /Border[0 0 0]/H/I/C[1 0 0] 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. 47 0 obj << Dynamics of the model is described by the system of 2 differential equations: Here is a set of notes used by Paul Dawkins to teach his Differential Equations course at Lamar University. endobj /A << /S /GoTo /D (subsection.3.3) >> https://www.britannica.com/science/stability-solution-of-equations, Penn State IT Knowledge Base - Stability of Equilibrium Solutions. Strict Stability is a different stability definition and this stability type can give us an information about the rate of … endobj 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. /Rect [85.948 392.395 249.363 403.243] Stability, in mathematics, condition in which a slight disturbance in a system does not produce too disrupting an effect on that system. uncertain differential equation was presented by Liu [9], and some stability theorems were proved by Yao et al. /D [42 0 R /XYZ 71 721 null] /Rect [71.004 631.831 220.914 643.786] 1953 edition. 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 However, we will solve x_ = f(x) using some numerical method. /MediaBox [0 0 612 792] Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox. >> 52 0 obj << /Border[0 0 0]/H/I/C[1 0 0] endobj >> endobj 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. >> endobj << /S /GoTo /D (section.1) >> /Rect [71.004 490.88 151.106 499.791] 33 0 obj endobj 53 0 obj << /Rect [85.948 286.655 283.651 297.503] /Type /Annot 61 0 obj << 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. /Type /Annot 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. 29 0 obj Differential Equations and Linear Algebra, 3.2c: Two First Order Equations: Stability. 40 0 obj endobj (1986),‘ Exact boundary conditions at an artificial boundary for partial differential equations in cylinders ’, SIAM J. 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. /Type /Annot Differential Equations Book: Differential Equations for Engineers (Lebl) 8: Nonlinear Equations ... 8.2.2 Stability and classiﬁcation of isolated critical points. Gilbert Strang, Massachusetts Institute of Technology (MIT) A second order equation gives two first order equations for … endobj (4.2 Physical Stability for the Pendulum) In regard to the stability of nonlinear systems, results of the linear theory are used to drive the results of Poincaré and Liapounoff. ��/�yV�g^ϙ�ڀ��r>�1`��8�u�=�l�Z�H��Y� %��MG0c��/~��L#K��"�^�}��o�~��H�슾�� Stability Problems of Solutions of Differential Equations, "Proceedings of NATO Advanced Study Institute, Padua, Italy." ��s;��Sl�! /Rect [158.066 600.72 357.596 612.675] 44 0 obj << Browse other questions tagged quantum-mechanics differential-equations stability or ask your own question. 1 0 obj /Border[0 0 0]/H/I/C[1 0 0] /Length 3838 58 0 obj << /ProcSet [ /PDF /Text ] 20 0 obj 50 0 obj << 21 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. /Type /Annot /D [42 0 R /XYZ 72 683.138 null] �^\��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 >> This means that it is structurally able to provide a unique path to the fixed-point (the “steady- /Border[0 0 0]/H/I/C[1 0 0] /Subtype /Link >> endobj One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation.. Introduction to Differential Equations . [32] J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative.