Finding a stable solution of a system of nonlinear
equations arising from dynamic systems

Carl. T. Kelley; Liqun  Qi; Xiaojiao Tong; Hongxia Yin

doi:10.3934/jimo.2011.7.497

Previous Article
JIMO Home
This Issue
Next Article
Optimality conditions for approximate solutions of vector optimization problems

April 2011, 7(2): 497-521. doi: 10.3934/jimo.2011.7.497

Finding a stable solution of a system of nonlinear equations arising from dynamic systems

Carl. T. Kelley ^1, , Liqun Qi ^2, , Xiaojiao Tong ^3, and Hongxia Yin ^4,

1.	Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, United States
2.	Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
3.	Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, 421002, China
4.	Department of Mathematics and Statistics, Minnesota State University Mankato, Mankato, MN 56001, United States

Received August 2009 Revised March 2011 Published April 2011

Abstract
Related Papers

In this paper, we present a new approach for finding a stable solution of a system of nonlinear equations arising from dynamical systems. We introduce the concept of stability functions and use this idea to construct stability solution models of several typical small signal stability problems in dynamical systems. Each model consists of a system of constrained semismooth equations. The advantage of the new models is twofold. Firstly, the stability requirement of dynamical systems is controlled by nonlinear inequalities. Secondly, the semismoothness property of the stability functions makes the models solvable by efficient numerical methods. We introduce smoothing functions for the stability functions and present a smoothing Newton method for solving the problems. Global and local quadratic convergence of the algorithm is established. Numerical examples from dynamical systems are also given to illustrate the efficiency of the new approach.

Keywords: stable solutions, saddle-node bifurcation, Hopf bifurcation, System of nonlinear equations, stability functions, smoothing Newton method..

Mathematics Subject Classification: Primary: 65H10, 65K10; Secondary: 90C3.

Citation: Carl. T. Kelley, Liqun Qi, Xiaojiao Tong, Hongxia Yin. Finding a stable solution of a system of nonlinear equations arising from dynamic systems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 497-521. doi: 10.3934/jimo.2011.7.497

References:

[1]	F. Alvarado, I. Dobson and Y. Hu, Computation of closest bifurcations in power systems,, IEEE Trans. Power System, 9 (1994), 918. doi: 10.1109/59.317655. Google Scholar
[2]	C. A. Cañizares, Calculating optimal system parameters to maximize the distance to saddle-node bifurcation points,, IEEE Trans. Circuits and System, 45 (1998), 225. doi: 10.1109/81.662696. Google Scholar
[3]	X. Chen, H. Qi, L. Qi and K.-L. Teo, Smooth convex approximation to the maximum eigenvalue function,, J. of Global Optimization, 30 (2004), 253. doi: 10.1007/s10898-004-8271-2. Google Scholar
[4]	H. D. Chiang, I. Dobson and R. J. Thomas, On voltage in electric power systems,, IEEE Trans. Power Systems, 5 (1990), 601. doi: 10.1109/59.54571. Google Scholar
[5]	T. Coffey, C. T. Kelley and D. E. Keyes, Pseudo-transient continuation and differential-algebraic equations,, SIAM J. Sci. Comp., 25 (2003), 553. doi: 10.1137/S106482750241044X. Google Scholar
[6]	F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley, (1983). Google Scholar
[7]	H. Dan, N. Yamashita and M. Fukushima, Convergence Properties of the Inexact Levenberg-Marquardt Method under Local Error Bound Conditions,, Optimization Methods and Software, 17 (2002), 605. doi: 10.1080/1055678021000049345. Google Scholar
[8]	G. Degla, An overview of semi-continuity results on the spectral radius and positivity,, J. Math. Anal. Appl., 338 (2008), 101. doi: 10.1016/j.jmaa.2007.05.011. Google Scholar
[9]	J. E. Dennis and R. B. Schnabel, "Numerical Methods for Unconstrained Optimization and Nonlinear Equations,", SIAM, (1996). Google Scholar
[10]	P. Deuflhard, Adaptive pseudo-transient continuation for nonlinear steady state problems, ZIP-Report02-12(March 2002),, in, 35 (2004). Google Scholar
[11]	I. Dobson, An iterative method to compute the closest saddle node or Hopf bifurcation in multidimensional parameter space,, in, (1992), 2513. Google Scholar
[12]	I. Dobson and L. Lu, Computing an optimum direction in control space to avoid saddle node bifurcation and voltage collapse in electric power systems,, IEEE Trans. Automatic Control, 37 (1992), 1616. doi: 10.1109/9.256397. Google Scholar
[13]	K. R. Fowler and C. T. Kelley, Pseudo-transient continuation for nonsmooth nonlinear equations,, SIAM J. Numer. Anal., 43 (2005), 1385. doi: 10.1137/S0036142903431298. Google Scholar
[14]	C. P. Gupta, R. K. Varma and S. C. Srivastava, A method to determine closest Hopf bifurcation in power systems considering exciter and load dynamics,, in, (1998), 293. Google Scholar
[15]	M. Hintermüller and M. Hinze, A SQP-semi-smooth Newton-type algorithm applied to control of the instationary Navier-Stokes system subject to control constraints,, SIAM J. Optim, 16 (2006), 1177. doi: 10.1137/030601259. Google Scholar
[16]	C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations,", in, 16 (1995). Google Scholar
[17]	C. T. Kelley, "Iterative Methods for Optimization,", SIAM, (1999). Google Scholar
[18]	C. T. Kelley, "Solving Nonlinear Equations with Newton's Method,", in, 1 (2003). Google Scholar
[19]	C. T. Kelley and D. E. Keyes, Convergence analysis of pseudo-transient continuation,, SIAM J. Numer. Anal., 35 (1998), 508. doi: 10.1137/S0036142996304796. Google Scholar
[20]	C. T. Kelley, Li-Zhi Liao, Liqun Qi, Moody T. Chu, J. P. Reese and C. Winton, Projected Pseudotransient continuation,, SIAM J. Numer. Nanl, 46 (2008), 3071. doi: 10.1137/07069866X. Google Scholar
[21]	P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose, C. Canizares, N. Hatziargyriou, D. Hill, A. Stankovic, C. Taylor, T. V. Cutsem and V. Vittal, Definition and classification of power system stability,, IEEE Transaction on Power Systems, 19 (2004), 1387. doi: 10.1109/TPWRS.2004.825981. Google Scholar
[22]	A. S. Lewis, Nonsmooth analysis of eigenvalues,, Math. Program., 84 (1999), 1. Google Scholar
[23]	A. S. Lewis and M. Overton, "Eigenvalue Optimization,", Acta Numerica, 5 (1996), 149. doi: 10.1017/S0962492900002646. Google Scholar
[24]	Y. Ma, H. Kawakami and C. K. Tse, Bifurcation analysis of switched dynamical systems with periodically moving borders,, IEEE Transactions on Circuits and Systems, 51 (2004), 1184. doi: 10.1109/TCSI.2004.829240. Google Scholar
[25]	Y. V. Makarov, Z. Y. Dong and D. J. Hill, A general method for small signal stability analysis,, IEEE Transaction on Power Systems, 13 (1998), 979. doi: 10.1109/59.709086. Google Scholar
[26]	R. Mifflin, Semismooth and semiconvex functions in constrained optimization,, SIAM J. Control Optim., 15 (1977), 957. doi: 10.1137/0315061. Google Scholar
[27]	J. M. Ortega and W. C. Rheinboldt, "Iterative Solutions of Nonlinear Equations in Several Variables,", Academic Press, (1970). Google Scholar
[28]	F. Oustry, A second-order bundle method to minimize the maximum eigenvalue function,, Math. Program, 89 (2000), 1. doi: 10.1007/PL00011388. Google Scholar
[29]	J. S. Pang, D. F. Sun and J. Sun, Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Cone Complementarity Problems,, Math. Oper. Res., 28 (2003), 39. doi: 10.1287/moor.28.1.39.14258. Google Scholar
[30]	H. Qi and L. Liao, A smoothing Newton method for extended vertical linear complementarity problems,, SIAM J. Matrix Anal. Appl., 21 (1999), 45. doi: 10.1137/S0895479897329837. Google Scholar
[31]	H. Qi and D. Sun, A quadratically convergent Newton method for computing the nearest correlation matrix,, SIAM J. Matrix Anal. Appl., 28 (2006), 360. doi: 10.1137/050624509. Google Scholar
[32]	H. Qi and X. Yang, Semismoothness of spectral functions,, SIAM J. Matrix Anal. Appl., 25 (2004), 766. doi: 10.1137/S0895479802417921. Google Scholar
[33]	L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations,, Math. Oper. Res., 18 (1993), 227. doi: 10.1287/moor.18.1.227. Google Scholar
[34]	L. Qi, D. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities,, Math. Program., 87 (2000), 1. Google Scholar
[35]	L. Qi and J. Sun, A nonsmooth version of Newton's method,, Math. Program., 58 (1993), 353. doi: 10.1007/BF01581275. Google Scholar
[36]	D. Sun and J. Sun, Semismooth matrix valued functions,, Math. Oper. Res., 27 (2002), 150. doi: 10.1287/moor.27.1.150.342. Google Scholar
[37]	D. Sun and J. Sun, Strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems,, SIAM J. Numer Anal., 40 (2002), 2352. doi: 10.1137/S0036142901393814. Google Scholar
[38]	M. D. Smooke and R. M. Mattheij, On the solution of nonlinear two-point boundary value problem on successively refined grids,, Appl. Numer. Math., 1 (1985), 463. doi: i:10.1016/0168-9274(85)90032-7. Google Scholar
[39]	A. Shestakov and J. Milovich, "Applications of Pseudo-Transient Continuation and Newton-Krylov Methods of the Poisson-Boltzmann and Radiation Diffusion Equation,", Tech. report UCRL-JC-139339, (2000). Google Scholar
[40]	X. J. Tong and S. Zhou, A smoothing projected Newton-type method for semismooth equations with bound constraints,, Journal of Industrial Management Optimization, 1 (2005), 235. Google Scholar
[41]	V. Venkatasubramanian, H. Schattler and J. Zaborsky, Dynamics of large constrained nonlinear systems - a taxonomy theory,, Proc. IEEE, 83 (1995), 1530. doi: 10.1109/5.481633. Google Scholar
[42]	M. Ulbrich, Semismooth Newton Methods for operator equations in function spaces,, SIAM J. Optim., 13 (2002), 805. doi: 10.1137/S1052623400371569. Google Scholar
[43]	X. Wu, C. K. Tse, O. Dranga and J. Lu, Fast-scale instability of single-stage power-factor-correction of power supplies,, IEEE Transactions on Circuits and Systems, 53 (2006), 204. doi: 10.1109/TCSI.2005.854293. Google Scholar

show all references

References:

[1]	F. Alvarado, I. Dobson and Y. Hu, Computation of closest bifurcations in power systems,, IEEE Trans. Power System, 9 (1994), 918. doi: 10.1109/59.317655. Google Scholar
[2]	C. A. Cañizares, Calculating optimal system parameters to maximize the distance to saddle-node bifurcation points,, IEEE Trans. Circuits and System, 45 (1998), 225. doi: 10.1109/81.662696. Google Scholar
[3]	X. Chen, H. Qi, L. Qi and K.-L. Teo, Smooth convex approximation to the maximum eigenvalue function,, J. of Global Optimization, 30 (2004), 253. doi: 10.1007/s10898-004-8271-2. Google Scholar
[4]	H. D. Chiang, I. Dobson and R. J. Thomas, On voltage in electric power systems,, IEEE Trans. Power Systems, 5 (1990), 601. doi: 10.1109/59.54571. Google Scholar
[5]	T. Coffey, C. T. Kelley and D. E. Keyes, Pseudo-transient continuation and differential-algebraic equations,, SIAM J. Sci. Comp., 25 (2003), 553. doi: 10.1137/S106482750241044X. Google Scholar
[6]	F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley, (1983). Google Scholar
[7]	H. Dan, N. Yamashita and M. Fukushima, Convergence Properties of the Inexact Levenberg-Marquardt Method under Local Error Bound Conditions,, Optimization Methods and Software, 17 (2002), 605. doi: 10.1080/1055678021000049345. Google Scholar
[8]	G. Degla, An overview of semi-continuity results on the spectral radius and positivity,, J. Math. Anal. Appl., 338 (2008), 101. doi: 10.1016/j.jmaa.2007.05.011. Google Scholar
[9]	J. E. Dennis and R. B. Schnabel, "Numerical Methods for Unconstrained Optimization and Nonlinear Equations,", SIAM, (1996). Google Scholar
[10]	P. Deuflhard, Adaptive pseudo-transient continuation for nonlinear steady state problems, ZIP-Report02-12(March 2002),, in, 35 (2004). Google Scholar
[11]	I. Dobson, An iterative method to compute the closest saddle node or Hopf bifurcation in multidimensional parameter space,, in, (1992), 2513. Google Scholar
[12]	I. Dobson and L. Lu, Computing an optimum direction in control space to avoid saddle node bifurcation and voltage collapse in electric power systems,, IEEE Trans. Automatic Control, 37 (1992), 1616. doi: 10.1109/9.256397. Google Scholar
[13]	K. R. Fowler and C. T. Kelley, Pseudo-transient continuation for nonsmooth nonlinear equations,, SIAM J. Numer. Anal., 43 (2005), 1385. doi: 10.1137/S0036142903431298. Google Scholar
[14]	C. P. Gupta, R. K. Varma and S. C. Srivastava, A method to determine closest Hopf bifurcation in power systems considering exciter and load dynamics,, in, (1998), 293. Google Scholar
[15]	M. Hintermüller and M. Hinze, A SQP-semi-smooth Newton-type algorithm applied to control of the instationary Navier-Stokes system subject to control constraints,, SIAM J. Optim, 16 (2006), 1177. doi: 10.1137/030601259. Google Scholar
[16]	C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations,", in, 16 (1995). Google Scholar
[17]	C. T. Kelley, "Iterative Methods for Optimization,", SIAM, (1999). Google Scholar
[18]	C. T. Kelley, "Solving Nonlinear Equations with Newton's Method,", in, 1 (2003). Google Scholar
[19]	C. T. Kelley and D. E. Keyes, Convergence analysis of pseudo-transient continuation,, SIAM J. Numer. Anal., 35 (1998), 508. doi: 10.1137/S0036142996304796. Google Scholar
[20]	C. T. Kelley, Li-Zhi Liao, Liqun Qi, Moody T. Chu, J. P. Reese and C. Winton, Projected Pseudotransient continuation,, SIAM J. Numer. Nanl, 46 (2008), 3071. doi: 10.1137/07069866X. Google Scholar
[21]	P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose, C. Canizares, N. Hatziargyriou, D. Hill, A. Stankovic, C. Taylor, T. V. Cutsem and V. Vittal, Definition and classification of power system stability,, IEEE Transaction on Power Systems, 19 (2004), 1387. doi: 10.1109/TPWRS.2004.825981. Google Scholar
[22]	A. S. Lewis, Nonsmooth analysis of eigenvalues,, Math. Program., 84 (1999), 1. Google Scholar
[23]	A. S. Lewis and M. Overton, "Eigenvalue Optimization,", Acta Numerica, 5 (1996), 149. doi: 10.1017/S0962492900002646. Google Scholar
[24]	Y. Ma, H. Kawakami and C. K. Tse, Bifurcation analysis of switched dynamical systems with periodically moving borders,, IEEE Transactions on Circuits and Systems, 51 (2004), 1184. doi: 10.1109/TCSI.2004.829240. Google Scholar
[25]	Y. V. Makarov, Z. Y. Dong and D. J. Hill, A general method for small signal stability analysis,, IEEE Transaction on Power Systems, 13 (1998), 979. doi: 10.1109/59.709086. Google Scholar
[26]	R. Mifflin, Semismooth and semiconvex functions in constrained optimization,, SIAM J. Control Optim., 15 (1977), 957. doi: 10.1137/0315061. Google Scholar
[27]	J. M. Ortega and W. C. Rheinboldt, "Iterative Solutions of Nonlinear Equations in Several Variables,", Academic Press, (1970). Google Scholar
[28]	F. Oustry, A second-order bundle method to minimize the maximum eigenvalue function,, Math. Program, 89 (2000), 1. doi: 10.1007/PL00011388. Google Scholar
[29]	J. S. Pang, D. F. Sun and J. Sun, Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Cone Complementarity Problems,, Math. Oper. Res., 28 (2003), 39. doi: 10.1287/moor.28.1.39.14258. Google Scholar
[30]	H. Qi and L. Liao, A smoothing Newton method for extended vertical linear complementarity problems,, SIAM J. Matrix Anal. Appl., 21 (1999), 45. doi: 10.1137/S0895479897329837. Google Scholar
[31]	H. Qi and D. Sun, A quadratically convergent Newton method for computing the nearest correlation matrix,, SIAM J. Matrix Anal. Appl., 28 (2006), 360. doi: 10.1137/050624509. Google Scholar
[32]	H. Qi and X. Yang, Semismoothness of spectral functions,, SIAM J. Matrix Anal. Appl., 25 (2004), 766. doi: 10.1137/S0895479802417921. Google Scholar
[33]	L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations,, Math. Oper. Res., 18 (1993), 227. doi: 10.1287/moor.18.1.227. Google Scholar
[34]	L. Qi, D. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities,, Math. Program., 87 (2000), 1. Google Scholar
[35]	L. Qi and J. Sun, A nonsmooth version of Newton's method,, Math. Program., 58 (1993), 353. doi: 10.1007/BF01581275. Google Scholar
[36]	D. Sun and J. Sun, Semismooth matrix valued functions,, Math. Oper. Res., 27 (2002), 150. doi: 10.1287/moor.27.1.150.342. Google Scholar
[37]	D. Sun and J. Sun, Strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems,, SIAM J. Numer Anal., 40 (2002), 2352. doi: 10.1137/S0036142901393814. Google Scholar
[38]	M. D. Smooke and R. M. Mattheij, On the solution of nonlinear two-point boundary value problem on successively refined grids,, Appl. Numer. Math., 1 (1985), 463. doi: i:10.1016/0168-9274(85)90032-7. Google Scholar
[39]	A. Shestakov and J. Milovich, "Applications of Pseudo-Transient Continuation and Newton-Krylov Methods of the Poisson-Boltzmann and Radiation Diffusion Equation,", Tech. report UCRL-JC-139339, (2000). Google Scholar
[40]	X. J. Tong and S. Zhou, A smoothing projected Newton-type method for semismooth equations with bound constraints,, Journal of Industrial Management Optimization, 1 (2005), 235. Google Scholar
[41]	V. Venkatasubramanian, H. Schattler and J. Zaborsky, Dynamics of large constrained nonlinear systems - a taxonomy theory,, Proc. IEEE, 83 (1995), 1530. doi: 10.1109/5.481633. Google Scholar
[42]	M. Ulbrich, Semismooth Newton Methods for operator equations in function spaces,, SIAM J. Optim., 13 (2002), 805. doi: 10.1137/S1052623400371569. Google Scholar
[43]	X. Wu, C. K. Tse, O. Dranga and J. Lu, Fast-scale instability of single-stage power-factor-correction of power supplies,, IEEE Transactions on Circuits and Systems, 53 (2006), 204. doi: 10.1109/TCSI.2005.854293. Google Scholar

[1]	Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
[2]	Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[3]	Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027
[4]	Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450
[5]	Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185
[6]	Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[7]	Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
[8]	Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149
[9]	Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[10]	Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184
[11]	Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021066
[12]	Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021005
[13]	Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[14]	Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[15]	Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026
[16]	Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206
[17]	Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203
[18]	Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448
[19]	Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021068
[20]	Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018

2019 Impact Factor: 1.366

American Institute of Mathematical Sciences

Finding a stable solution of a system of nonlinear equations arising from dynamic systems

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Finding a stable solution of a system of nonlinear equations arising from dynamic systems

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors