American Institute of Mathematical Sciences

Advanced
January  2021, 8(1): 59-97. doi: 10.3934/jcd.2021004

A general framework for validated continuation of periodic orbits in systems of polynomial ODEs

Jan Bouwe van den Berg 1, and  Elena Queirolo 2,

1. 

VU Amsterdam, Department of Mathematics, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
2. 

Rutgers University, Department of Mathematics, 110 Frelinghusen Rd, Piscataway, NJ 08854-8019, USA

Received  February 2019 Revised  November 2019 Published  October 2020

Fund Project: The authors are partially supported by NWO-VICI grant 639033109

In this paper a parametrized Newton-Kantorovich approach is applied to continuation of periodic orbits in arbitrary polynomial vector fields. This allows us to rigorously validate numerically computed branches of periodic solutions. We derive the estimates in full generality and present sample continuation proofs obtained using an implementation in Matlab. The presented approach is applicable to any polynomial vector field of any order and dimension. A variety of examples is presented to illustrate the efficacy of the method.

Keywords: Validated numerics, periodic orbits, continuation, solution branch, polynomial ODEs.
Mathematics Subject Classification: Primary: 37C27, 34A12; Secondary: 65G40.
Citation: Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004
References:
[1]

G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Ration. Mech. Anal., 197 (2010), 1033-1051.  doi: 10.1007/s00205-010-0309-7.  Google Scholar

[2]

J. B. van den Berg, M. Breden, J.-P. Lessard and M. Murray, Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof, J. Differential Equations, 264 (2018), 3086-3130.  doi: 10.1016/j.jde.2017.11.011.  Google Scholar

[3]

J. B. van den Berg, C. Groothedde and J.-P. Lessard, A general method for computer-assisted proofs of periodic solutions in delay differential problems, 2018, Preprint. Google Scholar

[4]

J. B. van den Berg and J.-P. Lessard, Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 7 (2008), 988-1031.  doi: 10.1137/070709128.  Google Scholar

[5]

J. B. van den Berg, J.-P. Lessard and K. Mischaikow, Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), 1565-1584.  doi: 10.1090/S0025-5718-10-02325-2.  Google Scholar

[6]

J. B. van den Berg, and J.-P. Lessard and E. Queirolo, Rigorous verification of Hopf bifurcations in ODEs, 2020, Preprint Google Scholar

[7]

J. B. van den Berg and E. Queirolo, MATLABcode for "A general framework for validated continuation of periodic orbits in systems of polynomial ODEs", 2018, https://www.math.vu.nl/ janbouwe/code/continuation/. Google Scholar

[8]

J. B. van den Berg and R. Sheombarsing, Validated computations for connecting orbits in polynomial vector fields, Indag. Math. (N.S.), 31 (2020), 310-373. doi: 10.1016/j.indag.2020.01.007.  Google Scholar

[9]

J. B. van den Berg and J. F. Williams, Validation of the bifurcation diagram in the 2D Ohta-Kawasaki problem, Nonlinearity, 30 (2017), 1584-1638.  doi: 10.1088/1361-6544/aa60e8.  Google Scholar

[10]

M. Breden and R. Castelli, Existence and instability of steady states for a triangular cross-diffusion system: A computer-assisted proof, Journal of Differential Equations, 264 (2018), 6418-6458.  doi: 10.1016/j.jde.2018.01.033.  Google Scholar

[11]

M. Breden, J.-P. Lessard and M. Vanicat, Global bifurcation diagrams of steady states of systems of pdes via rigorous numerics: A 3-component reaction-diffusion system, Acta applicandae mathematicae, 128 (2013), 113-152.  doi: 10.1007/s10440-013-9823-6.  Google Scholar

[12]

R. Castelli, Rigorous computation of non-uniform patterns for the 2-dimensional gray-scott reaction-diffusion equation, Acta Applicandae Mathematicae, 151 (2017), 27-52.  doi: 10.1007/s10440-017-0101-x.  Google Scholar

[13]

A. R. Champneys and B. Sandstede, Numerical computation of coherent structures, In Numerical continuation methods for dynamical systems, Underst. Complex Syst., Springer, Dordrecht, 2007, pages 331-358. doi: 10.1007/978-1-4020-6356-5_11.  Google Scholar

[14]

R. H. Clewley, W. E. Sherwood, M. D. LaMar and J. M. Guckenheimer, PyDSTool, a software environment for dynamical systems modeling, 2007, https://sourceforge.net/projects/pydstool/. Google Scholar

[15]

A. Correc and J.-P. Lessard, Coexistence of nontrivial solutions of the one-dimensional Ginzburg-Landau equation: A computer-assisted proof, European J. Appl. Math., 26 (2015), 33-60.  doi: 10.1017/S0956792514000308.  Google Scholar

[16]

H. Dankowicz and F. Schilder, Recipes for continuation, volume 11 of Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. https://sourceforge.net/projects/cocotools/. doi: 10.1137/1.9781611972573.  Google Scholar

[17]

S. Day, J.-P. Lessard and K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM J. Numer. Anal., 45 (2007), 1398-1424.  doi: 10.1137/050645968.  Google Scholar

[18]

A. Dhooge, W. Govaerts, Yu. A. Kuznetsov, H. G. E. Meijer and B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14 (2008), 147-175. https://sourceforge.net/projects/matcont/. doi: 10.1080/13873950701742754.  Google Scholar

[19]

E. J. Doedel, B. E. Oldeman, A. R. Champneys, F. Dercole, T. F. Fairgrieve, Y. Kuznetsov, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. H. Zhang, AUTO-07p : Continuation and bifurcation software for ordinary differential equations, 2012, http://sourceforge.net/projects/auto-07p/. Google Scholar

[20]

B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, volume 14 of Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. http://www.math.pitt.edu/ bard/xpp/xpp.html. doi: 10.1137/1.9780898718195.  Google Scholar

[21]

J.-Ll. Figueras A. Haro and A. Luque, Rigorous computer-assisted application of KAM theory: A modern approach, Found. Comput. Math., 17 (2017), 1123-1193.  doi: 10.1007/s10208-016-9339-3.  Google Scholar

[22]

M. Gameiro, J.-P. Lessard and A. Pugliese, Computation of smooth manifolds via rigorous multi-parameter continuation in infinite dimensions, Found. Comput. Math., 16 (2016), 531-575.  doi: 10.1007/s10208-015-9259-7.  Google Scholar

[23]

A. Gasull, H. Giacomini and M. Grau, Effective construction of Poincaré-Bendixson regions, J. Appl. Anal. Comput., 7 (2017), 1549-1569.  doi: 10.11948/2017094.  Google Scholar

[24]

A. Hungria, J.-P. Lessard and J. D Mireles James, Rigorous numerics for analytic solutions of differential equations: The radii polynomial approach, Mathematics of Computation, 85 (2016), 1427-1459.  doi: 10.1090/mcom/3046.  Google Scholar

[25]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, volume 112 of Applied Mathematical Sciences, Springer-Verlag, New York, third edition, 2004. doi: 10.1007/978-1-4757-3978-7.  Google Scholar

[26]

J.-P. Lessard, Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation, J. Differential Equations, 248 (2010), 992-1016.  doi: 10.1016/j.jde.2009.11.008.  Google Scholar

[27]

J.-P. Lessard and J. D. Mireles James, Computer assisted Fourier analysis in sequence spaces of varying regularity, SIAM J. Math. Anal., 49 (2017), 530-561.  doi: 10.1137/16M1056006.  Google Scholar

[28]

J.-P. Lessard, J. D. Mireles James and J. Ransford, Automatic differentiation for Fourier series and the radii polynomial approach, Phys. D, 334 (2016), 174-186.  doi: 10.1016/j.physd.2016.02.007.  Google Scholar

[29]

K. Makino and M. Berz, Rigorous integration of flows and odes using taylor models, In Proceedings of the 2009 conference on Symbolic Numeric Computation, pages 79-84, 2009. doi: 10.1145/1577190.1577206.  Google Scholar

[30]

S. M Rump, Intlab—interval laboratory, In Developments in reliable computing, pages 77-104. Springer, 1999. doi: 10.1007/978-94-017-1247-7_7.  Google Scholar

[31]

G. S. Rychkov, The maximum number of limit cycles of polynomial liénard systems of degree five is equal to two, Differential Equations, 11 (1975), 301-302.   Google Scholar

[32]

T. Wanner, Computer-assisted bifurcation diagram validation and applications in materials science, In Rigorous numerics in dynamics, volume 74 of Proc. Sympos. Appl. Math., pages 123-174. Amer. Math. Soc., Providence, RI, 2018.  Google Scholar

[33]

P. Zgliczyński, Steady state bifurcations for the Kuramoto-Sivashinsky equation: a computer assisted proof, J. Comput. Dyn., 2 (2015), 95-142.  doi: 10.3934/jcd.2015.2.95.  Google Scholar

[34]

CAPD: Computer assisted proofs in dynamics, a package for rigorous numerics, http://capd.ii.uj.edu.pl/. Google Scholar

show all references

References:
[1]

G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Ration. Mech. Anal., 197 (2010), 1033-1051.  doi: 10.1007/s00205-010-0309-7.  Google Scholar

[2]

J. B. van den Berg, M. Breden, J.-P. Lessard and M. Murray, Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof, J. Differential Equations, 264 (2018), 3086-3130.  doi: 10.1016/j.jde.2017.11.011.  Google Scholar

[3]

J. B. van den Berg, C. Groothedde and J.-P. Lessard, A general method for computer-assisted proofs of periodic solutions in delay differential problems, 2018, Preprint. Google Scholar

[4]

J. B. van den Berg and J.-P. Lessard, Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 7 (2008), 988-1031.  doi: 10.1137/070709128.  Google Scholar

[5]

J. B. van den Berg, J.-P. Lessard and K. Mischaikow, Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), 1565-1584.  doi: 10.1090/S0025-5718-10-02325-2.  Google Scholar

[6]

J. B. van den Berg, and J.-P. Lessard and E. Queirolo, Rigorous verification of Hopf bifurcations in ODEs, 2020, Preprint Google Scholar

[7]

J. B. van den Berg and E. Queirolo, MATLABcode for "A general framework for validated continuation of periodic orbits in systems of polynomial ODEs", 2018, https://www.math.vu.nl/ janbouwe/code/continuation/. Google Scholar

[8]

J. B. van den Berg and R. Sheombarsing, Validated computations for connecting orbits in polynomial vector fields, Indag. Math. (N.S.), 31 (2020), 310-373. doi: 10.1016/j.indag.2020.01.007.  Google Scholar

[9]

J. B. van den Berg and J. F. Williams, Validation of the bifurcation diagram in the 2D Ohta-Kawasaki problem, Nonlinearity, 30 (2017), 1584-1638.  doi: 10.1088/1361-6544/aa60e8.  Google Scholar

[10]

M. Breden and R. Castelli, Existence and instability of steady states for a triangular cross-diffusion system: A computer-assisted proof, Journal of Differential Equations, 264 (2018), 6418-6458.  doi: 10.1016/j.jde.2018.01.033.  Google Scholar

[11]

M. Breden, J.-P. Lessard and M. Vanicat, Global bifurcation diagrams of steady states of systems of pdes via rigorous numerics: A 3-component reaction-diffusion system, Acta applicandae mathematicae, 128 (2013), 113-152.  doi: 10.1007/s10440-013-9823-6.  Google Scholar

[12]

R. Castelli, Rigorous computation of non-uniform patterns for the 2-dimensional gray-scott reaction-diffusion equation, Acta Applicandae Mathematicae, 151 (2017), 27-52.  doi: 10.1007/s10440-017-0101-x.  Google Scholar

[13]

A. R. Champneys and B. Sandstede, Numerical computation of coherent structures, In Numerical continuation methods for dynamical systems, Underst. Complex Syst., Springer, Dordrecht, 2007, pages 331-358. doi: 10.1007/978-1-4020-6356-5_11.  Google Scholar

[14]

R. H. Clewley, W. E. Sherwood, M. D. LaMar and J. M. Guckenheimer, PyDSTool, a software environment for dynamical systems modeling, 2007, https://sourceforge.net/projects/pydstool/. Google Scholar

[15]

A. Correc and J.-P. Lessard, Coexistence of nontrivial solutions of the one-dimensional Ginzburg-Landau equation: A computer-assisted proof, European J. Appl. Math., 26 (2015), 33-60.  doi: 10.1017/S0956792514000308.  Google Scholar

[16]

H. Dankowicz and F. Schilder, Recipes for continuation, volume 11 of Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. https://sourceforge.net/projects/cocotools/. doi: 10.1137/1.9781611972573.  Google Scholar

[17]

S. Day, J.-P. Lessard and K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM J. Numer. Anal., 45 (2007), 1398-1424.  doi: 10.1137/050645968.  Google Scholar

[18]

A. Dhooge, W. Govaerts, Yu. A. Kuznetsov, H. G. E. Meijer and B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14 (2008), 147-175. https://sourceforge.net/projects/matcont/. doi: 10.1080/13873950701742754.  Google Scholar

[19]

E. J. Doedel, B. E. Oldeman, A. R. Champneys, F. Dercole, T. F. Fairgrieve, Y. Kuznetsov, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. H. Zhang, AUTO-07p : Continuation and bifurcation software for ordinary differential equations, 2012, http://sourceforge.net/projects/auto-07p/. Google Scholar

[20]

B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, volume 14 of Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. http://www.math.pitt.edu/ bard/xpp/xpp.html. doi: 10.1137/1.9780898718195.  Google Scholar

[21]

J.-Ll. Figueras A. Haro and A. Luque, Rigorous computer-assisted application of KAM theory: A modern approach, Found. Comput. Math., 17 (2017), 1123-1193.  doi: 10.1007/s10208-016-9339-3.  Google Scholar

[22]

M. Gameiro, J.-P. Lessard and A. Pugliese, Computation of smooth manifolds via rigorous multi-parameter continuation in infinite dimensions, Found. Comput. Math., 16 (2016), 531-575.  doi: 10.1007/s10208-015-9259-7.  Google Scholar

[23]

A. Gasull, H. Giacomini and M. Grau, Effective construction of Poincaré-Bendixson regions, J. Appl. Anal. Comput., 7 (2017), 1549-1569.  doi: 10.11948/2017094.  Google Scholar

[24]

A. Hungria, J.-P. Lessard and J. D Mireles James, Rigorous numerics for analytic solutions of differential equations: The radii polynomial approach, Mathematics of Computation, 85 (2016), 1427-1459.  doi: 10.1090/mcom/3046.  Google Scholar

[25]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, volume 112 of Applied Mathematical Sciences, Springer-Verlag, New York, third edition, 2004. doi: 10.1007/978-1-4757-3978-7.  Google Scholar

[26]

J.-P. Lessard, Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation, J. Differential Equations, 248 (2010), 992-1016.  doi: 10.1016/j.jde.2009.11.008.  Google Scholar

[27]

J.-P. Lessard and J. D. Mireles James, Computer assisted Fourier analysis in sequence spaces of varying regularity, SIAM J. Math. Anal., 49 (2017), 530-561.  doi: 10.1137/16M1056006.  Google Scholar

[28]

J.-P. Lessard, J. D. Mireles James and J. Ransford, Automatic differentiation for Fourier series and the radii polynomial approach, Phys. D, 334 (2016), 174-186.  doi: 10.1016/j.physd.2016.02.007.  Google Scholar

[29]

K. Makino and M. Berz, Rigorous integration of flows and odes using taylor models, In Proceedings of the 2009 conference on Symbolic Numeric Computation, pages 79-84, 2009. doi: 10.1145/1577190.1577206.  Google Scholar

[30]

S. M Rump, Intlab—interval laboratory, In Developments in reliable computing, pages 77-104. Springer, 1999. doi: 10.1007/978-94-017-1247-7_7.  Google Scholar

[31]

G. S. Rychkov, The maximum number of limit cycles of polynomial liénard systems of degree five is equal to two, Differential Equations, 11 (1975), 301-302.   Google Scholar

[32]

T. Wanner, Computer-assisted bifurcation diagram validation and applications in materials science, In Rigorous numerics in dynamics, volume 74 of Proc. Sympos. Appl. Math., pages 123-174. Amer. Math. Soc., Providence, RI, 2018.  Google Scholar

[33]

P. Zgliczyński, Steady state bifurcations for the Kuramoto-Sivashinsky equation: a computer assisted proof, J. Comput. Dyn., 2 (2015), 95-142.  doi: 10.3934/jcd.2015.2.95.  Google Scholar

[34]

CAPD: Computer assisted proofs in dynamics, a package for rigorous numerics, http://capd.ii.uj.edu.pl/. Google Scholar

Figure 1.  Representation of the matrix introduced in Equation (25)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The shape of the operator $ Q $, as defined in Lemma 5.1. The vertical columns represent the vectors $ y^{[k]} $, where $ k $ denotes the column index
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The boundaries of the validation interval $ (r_{\min},r_{\max}) $ versus the number of modes $ K $ for various values of $ \nu $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  On the right: the minimal number of modes $ K $ necessary for the validation of a periodic orbit of (52) depending on $ \mu $. The stepsize in $ \mu $ is not chosen uniformly, because the problem is more sensitive to changes in $ \mu $ for $ \mu $ bigger, therefore requesting a steeper change in the number of modes. One the left: several of the validated periodic orbits
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The computation time versus the number of modes $ K $. The reference line has a slope corresponding to $ K^3 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The computation time versus the dimension $ N $ of the system (53). The reference lines have slopes corresponding to quadratic and cubic dependence on $ N $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  The computation time versus the order $ D $ of the system (54). The reference lines have slopes corresponding to quadratic and cubic dependence on $ D $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Validation of a periodic solution for the Lorenz system (55) with the classical parameter values $ \sigma = 10 $, $ \beta = 8/3 $ and $ \rho = 28 $. The solution has a period close to 25. This solution was validated with $ \nu = 1+10^{-6} $ and $ K = 800 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Step size versus $ \mu $. Some oscillation is noticeable in the step size. This is due to the fact that the stepsize is decreased when the number of modes is changed, see Section 9. During this validation, the number of modes has been increased automatically from 4 to about 400
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  Validated continuation for the Lorenz system (55) from $ \rho = 28 $ to $ \rho = 14.8 $. In the left graph, the lower orbit still has two distinct swirls in the left lobe, but they are too close to each other to be seen in this graph
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  Validated continuation for $ 10 $ coupled Lorenz systems (77) from $ \epsilon = 0.14205 $ to $ \epsilon = 0.14252 $. In the left graph we have depicted in blue the orbit of the first three component, in green the orbit of the last three component, with the biggest validated $ \epsilon $, in the right graph we depicted the values of $ \epsilon $, where we can notice the adaptation of the stepsize
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 12.  Continuation through the fold of the Rychkov system (78). The validation started at the top, where a small initial stepsize was chosen. One can see how the stepsize was automatically adjusted along the validated curve. As can be seen from the values along the horizontal axis, we only depict a validated continuation for parameter values very close to saddle-node
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176
[2]

Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020371
[3]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
[4]

Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010
[5]

Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, 2021, 20 (2) : 547-558. doi: 10.3934/cpaa.2020280
[6]

Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A SOCP relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 151-168. doi: 10.3934/jimo.2019104
[7]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021005
[8]

Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021026
[9]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002
[10]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115
[11]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
[12]

Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020172
[13]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267
[14]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[15]

Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system. Electronic Research Archive, 2021, 29 (1) : 1691-1708. doi: 10.3934/era.2020087
[16]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037
[17]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033
[18]

Tinghua Hu, Yang Yang, Zhengchun Zhou. Golay complementary sets with large zero odd-periodic correlation zones. Advances in Mathematics of Communications, 2021, 15 (1) : 23-33. doi: 10.3934/amc.2020040
[19]

Yicheng Liu, Yipeng Chen, Jun Wu, Xiao Wang. Periodic consensus in network systems with general distributed processing delays. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2021002
[20]

Zhihua Liu, Yayun Wu, Xiangming Zhang. Existence of periodic wave trains for an age-structured model with diffusion. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021009

 Impact Factor: 

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences