# American Institute of Mathematical Sciences

April  2007, 17(2): 349-364. doi: 10.3934/dcds.2007.17.349

## Geometric properties of the integral curves of an implicit differential equation

 1 Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, United Kingdom

Received  December 2005 Revised  September 2006 Published  November 2006

We study geometric properties of the integral curves of an implicit differential equation in a neighbourhood of a codimension $\le 1$ singularity. We also deal with the way these singularities bifurcate in generic families of equations and the changes in the associated geometry. The main tool used here is the Legendre transformation.
Citation: Farid Tari. Geometric properties of the integral curves of an implicit differential equation. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 349-364. doi: 10.3934/dcds.2007.17.349
 [1] Oğul Esen, Partha Guha. On the geometry of the Schmidt-Legendre transformation. Journal of Geometric Mechanics, 2018, 10 (3) : 251-291. doi: 10.3934/jgm.2018010 [2] Zhong-Qing Wang, Li-Lian Wang. A Legendre-Gauss collocation method for nonlinear delay differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 685-708. doi: 10.3934/dcdsb.2010.13.685 [3] Farid Tari. Two-parameter families of implicit differential equations. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 139-162. doi: 10.3934/dcds.2005.13.139 [4] Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87 [5] Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299 [6] Shan Li, Shi-Mi Yan, Zhong-Qing Wang. Efficient Legendre dual-Petrov-Galerkin methods for odd-order differential equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1543-1563. doi: 10.3934/dcdsb.2019239 [7] Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47 [8] Fahd Jarad, Sugumaran Harikrishnan, Kamal Shah, Kuppusamy Kanagarajan. Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 723-739. doi: 10.3934/dcdss.2020040 [9] Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433 [10] Zalman Balanov, Meymanat Farzamirad, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree. part II: Symmetric Hopf bifurcations of functional differential equations. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 923-960. doi: 10.3934/dcds.2006.16.923 [11] Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038 [12] Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 [13] Jie Shen, Li-Lian Wang. Laguerre and composite Legendre-Laguerre Dual-Petrov-Galerkin methods for third-order equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1381-1402. doi: 10.3934/dcdsb.2006.6.1381 [14] Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291 [15] Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829 [16] Dariusz Idczak. A global implicit function theorem and its applications to functional equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2549-2556. doi: 10.3934/dcdsb.2014.19.2549 [17] A. Pedas, G. Vainikko. Smoothing transformation and piecewise polynomial projection methods for weakly singular Fredholm integral equations. Communications on Pure and Applied Analysis, 2006, 5 (2) : 395-413. doi: 10.3934/cpaa.2006.5.395 [18] Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control and Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31 [19] Huijie Qiao, Jiang-Lun Wu. On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1449-1467. doi: 10.3934/dcdsb.2018215 [20] Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123

2020 Impact Factor: 1.392