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Global invariant manifolds near a Shilnikov homoclinic bifurcation
The computation of convex invariant sets via Newton's method
1. | Chair of Applied Mathematics, University of Bayreuth, 95440 Bayreuth, Germany |
2. | Chair of Applied Mathematics, University of Paderborn, 33098 Paderborn, Germany, Germany, Germany |
3. | Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, United States |
References:
[1] |
J.-P. Aubin, Mutational and Morphological Analysis. Tools for Shape Evolution and Morphogenesis,, Systems & Control: Foundations & Applications, (1999).
doi: 10.1007/978-1-4612-1576-9. |
[2] |
R. Baier, C. Büskens, I. A. Chahma and M. Gerdts, Approximation of reachable sets by direct solution methods of optimal control problems,, Optim. Meth. Softw., 22 (2007), 433.
doi: 10.1080/10556780600604999. |
[3] |
R. Baier and E. M. Farkhi, Directed derivatives of convex compact-valued mappings,, in Advances in Convex Analysis and Global Optimization: Honoring the Memory of C. Caratheodory (1873-1950) (eds. N. Hadjisavvas and P. M. Pardalos), (2001), 1873.
doi: 10.1007/978-1-4613-0279-7_32. |
[4] |
R. Baier and E. M. Farkhi, The directed subdifferential of DC functions,, in Nonlinear Analysis and Optimization II: Optimization. A Conference in Celebration of Alex Ioffe's 70th and Simeon Reich's 60th Birthdays, (2008), 18.
doi: 10.1090/conm/514/10098. |
[5] |
R. Baier and M. Hessel-von Molo, Newton's method and secant method for set-valued mappings,, in Proceedings on the 8th International Conference on, (2011), 6.
doi: 10.1007/978-3-642-29843-1_9. |
[6] |
R. Baier and G. Perria, Set-valued Hermite interpolation,, J. Approx. Theory, 163 (2011), 1349.
doi: 10.1016/j.jat.2010.11.004. |
[7] |
R. Baier and E. M. Farkhi, Differences of convex compact sets in the space of directed sets. I. The space of directed sets,, Set-Valued Anal., 9 (2001), 217.
doi: 10.1023/A:1012046027626. |
[8] |
R. Baier and E. M. Farkhi, Differences of convex compact sets in the space of directed sets. II. Visualization of directed sets,, Set-Valued Anal., 9 (2001), 247.
doi: 10.1023/A:1012009529492. |
[9] |
W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems,, IMA J. Numer. Anal., 10 (1990), 379.
doi: 10.1093/imanum/10.3.379. |
[10] |
F. Blanchini, Set invariance in control,, Automatica, 35 (1999), 1747.
doi: 10.1016/S0005-1098(99)00113-2. |
[11] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - set oriented numerical methods for dynamical systems,, in Ergodic Theory, (2001), 145.
|
[12] |
M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors,, Numer. Math., 75 (1997), 293.
doi: 10.1007/s002110050240. |
[13] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM J. Numer. Anal., 36 (1999), 491.
doi: 10.1137/S0036142996313002. |
[14] |
V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis,, vol. 7 of Approximation and Optimization, (1995).
|
[15] |
L. Dieci, J. Lorenz and R. D. Russell, Numerical calculation of invariant tori,, SIAM J. Sci. Statist. Comput., 12 (1991), 607.
doi: 10.1137/0912033. |
[16] |
N. S. Dimitrova and S. M. Markov, Interval methods of Newton type for nonlinear equations,, Pliska Stud. Math. Bulgar., 5 (1983), 105.
|
[17] |
T. Donchev and E. M. Farkhi, Fixed set iterations for relaxed Lipschitz multimaps,, Nonlinear Anal., 53 (2003), 997.
doi: 10.1016/S0362-546X(03)00036-1. |
[18] |
H. Hadwiger, Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt,, Math. Z., 53 (1950), 210.
doi: 10.1007/BF01175656. |
[19] |
E. Hansen, A multidimensional interval Newton method,, Reliab. Comput., 12 (2006), 253.
doi: 10.1007/s11155-006-9000-y. |
[20] |
A. J. Homburg, H. M. Osinga and G. Vegter, On the computation of invariant manifolds of fixed points,, ZAMM Z. Angew. Math. Phys., 46 (1995), 171.
doi: 10.1007/BF00944751. |
[21] |
C. S. Hsu, Global analysis by cell mapping,, Internat. J. Bifurc. Chaos Appl. Sci. Engrg., 2 (1992), 727.
doi: 10.1142/S0218127492000422. |
[22] |
L. V. Kantorovich, On Newton's method for functional equations,, Dokl. Akad. Nauk SSSR, 59 (1948), 1237.
|
[23] |
L. V. Kantorovich, The majorant principle and Newton's method,, Dokl. Akad. Nauk SSSR, 76 (1951), 17.
|
[24] |
L. V. Kantorovich and G. P. Akilov, Functional Analysis,, 2nd edition, (1982).
|
[25] |
I. G. Kevrekidis, R. Aris, L. D. Schmidt and S. Pelikan, Numerical computation of invariant circles of maps,, Phys. D, 16 (1985), 243.
doi: 10.1016/0167-2789(85)90061-2. |
[26] |
I. G. Kevrekidis, C. W. Gear and G. Hummer, Equation-free: The computer-aided analysis of complex multiscale systems,, AIChE Journ., 50 (2004), 1346.
doi: 10.1002/aic.10106. |
[27] |
I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis,, Commun. Math. Sci., 1 (2003), 715.
doi: 10.4310/CMS.2003.v1.n4.a5. |
[28] |
D. Klatte and B. Kummer, Nonsmooth equations in optimization. Regularity, calculus, methods and applications,, vol. 60 of Nonconvex Optimization and its Applications, (2002).
|
[29] |
B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763.
doi: 10.1142/S0218127405012533. |
[30] |
R. Krawczyk, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken,, Computing (Arch. Elektron. Rechnen), 4 (1969), 187.
|
[31] |
R. Krawczyk, Einschlieş ung von Nullstellen mit Hilfe einer Intervallarithmetik,, Computing (Arch. Elektron. Rechnen), 5 (1970), 356.
|
[32] |
E. Kreuzer, Analysis of chaotic systems using the cell mapping approach,, Arch. Appl. Mech., 55 (1985), 285.
doi: 10.1007/BF00538223. |
[33] |
S. M. Markov, Some applications of extended interval arithmetic to interval iterations,, Comput. Suppl., 2 (1980), 69.
|
[34] |
K. Nickel, Das Auflösungsverhalten von nichtlinearen Fixmengen-Systemen,, in Iterative solution of nonlinear systems of equations (Oberwolfach, (1982), 106.
|
[35] |
D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets, vol. 548 of Mathematics and Its Applications,, Kluwer Academic Publishers, (2002).
|
[36] |
G. Perria, Set-valued Interpolation, vol. 79 of Bayreuth. Math. Schr.,, Department of Mathematics, (2007).
|
[37] |
B. T. Polyak, Convexity of quadratic transformations and its use in control and optimization,, J. Optim. Theory Appl., 99 (1998), 553.
doi: 10.1023/A:1021798932766. |
[38] |
B. T. Polyak, Convexity of nonlinear image of a small ball with applications to optimization,, Set-Valued Anal., 9 (2001), 159.
doi: 10.1023/A:1011287523150. |
[39] |
L. S. Pontryagin, Linear differential games. II,, Sov. Math., 8 (1967), 910. Google Scholar |
[40] |
R. T. Rockafellar, Convex Analysis,, vol. 28 of Princeton Mathematical Series, (1970).
|
[41] |
A. M. Rubinov and I. S. Akhundov, Difference of compact sets in the sense of Demyanov and its application to non-smooth analysis,, Optimization, 23 (1992), 179.
doi: 10.1080/02331939208843757. |
[42] |
P. Saint-Pierre, Newton and other continuation methods for multivalued inclusions,, Set-Valued Anal., 3 (1995), 143.
doi: 10.1007/BF01038596. |
[43] |
C. I. Siettos, C. C. Pantelides and I. G. Kevrekidis, Enabling dynamic process simulators to perform alternative tasks: A time-stepper-based toolkit for computer-aided analysis,, Ind. Engrg. Chem. Res., 42 (2003), 6795.
doi: 10.1021/ie021062w. |
[44] |
C. Theodoropoulos, Y.-H. Qian and I. G. Kevrekidis, "Coarse'' stability and bifurcation analysis using time-steppers: A reaction-diffusion example,, Proc. Natl. Acad. Sci., 97 (2000), 9840.
doi: 10.1073/pnas.97.18.9840. |
[45] |
L. S. Tuckerman and D. Barkley, Bifurcation analysis for timesteppers,, in Numerical methods for bifurcation problems and large-scale dynamical systems (Minneapolis, (2000), 453.
doi: 10.1007/978-1-4612-1208-9_20. |
[46] |
X. Wang, Convergence of Newton's method and inverse function theorem in Banach space,, Math. Comp., 68 (1999), 169.
doi: 10.1090/S0025-5718-99-00999-0. |
[47] |
T. Yamamoto, A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions,, Numer. Math., 49 (1986), 203.
doi: 10.1007/BF01389624. |
show all references
References:
[1] |
J.-P. Aubin, Mutational and Morphological Analysis. Tools for Shape Evolution and Morphogenesis,, Systems & Control: Foundations & Applications, (1999).
doi: 10.1007/978-1-4612-1576-9. |
[2] |
R. Baier, C. Büskens, I. A. Chahma and M. Gerdts, Approximation of reachable sets by direct solution methods of optimal control problems,, Optim. Meth. Softw., 22 (2007), 433.
doi: 10.1080/10556780600604999. |
[3] |
R. Baier and E. M. Farkhi, Directed derivatives of convex compact-valued mappings,, in Advances in Convex Analysis and Global Optimization: Honoring the Memory of C. Caratheodory (1873-1950) (eds. N. Hadjisavvas and P. M. Pardalos), (2001), 1873.
doi: 10.1007/978-1-4613-0279-7_32. |
[4] |
R. Baier and E. M. Farkhi, The directed subdifferential of DC functions,, in Nonlinear Analysis and Optimization II: Optimization. A Conference in Celebration of Alex Ioffe's 70th and Simeon Reich's 60th Birthdays, (2008), 18.
doi: 10.1090/conm/514/10098. |
[5] |
R. Baier and M. Hessel-von Molo, Newton's method and secant method for set-valued mappings,, in Proceedings on the 8th International Conference on, (2011), 6.
doi: 10.1007/978-3-642-29843-1_9. |
[6] |
R. Baier and G. Perria, Set-valued Hermite interpolation,, J. Approx. Theory, 163 (2011), 1349.
doi: 10.1016/j.jat.2010.11.004. |
[7] |
R. Baier and E. M. Farkhi, Differences of convex compact sets in the space of directed sets. I. The space of directed sets,, Set-Valued Anal., 9 (2001), 217.
doi: 10.1023/A:1012046027626. |
[8] |
R. Baier and E. M. Farkhi, Differences of convex compact sets in the space of directed sets. II. Visualization of directed sets,, Set-Valued Anal., 9 (2001), 247.
doi: 10.1023/A:1012009529492. |
[9] |
W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems,, IMA J. Numer. Anal., 10 (1990), 379.
doi: 10.1093/imanum/10.3.379. |
[10] |
F. Blanchini, Set invariance in control,, Automatica, 35 (1999), 1747.
doi: 10.1016/S0005-1098(99)00113-2. |
[11] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - set oriented numerical methods for dynamical systems,, in Ergodic Theory, (2001), 145.
|
[12] |
M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors,, Numer. Math., 75 (1997), 293.
doi: 10.1007/s002110050240. |
[13] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM J. Numer. Anal., 36 (1999), 491.
doi: 10.1137/S0036142996313002. |
[14] |
V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis,, vol. 7 of Approximation and Optimization, (1995).
|
[15] |
L. Dieci, J. Lorenz and R. D. Russell, Numerical calculation of invariant tori,, SIAM J. Sci. Statist. Comput., 12 (1991), 607.
doi: 10.1137/0912033. |
[16] |
N. S. Dimitrova and S. M. Markov, Interval methods of Newton type for nonlinear equations,, Pliska Stud. Math. Bulgar., 5 (1983), 105.
|
[17] |
T. Donchev and E. M. Farkhi, Fixed set iterations for relaxed Lipschitz multimaps,, Nonlinear Anal., 53 (2003), 997.
doi: 10.1016/S0362-546X(03)00036-1. |
[18] |
H. Hadwiger, Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt,, Math. Z., 53 (1950), 210.
doi: 10.1007/BF01175656. |
[19] |
E. Hansen, A multidimensional interval Newton method,, Reliab. Comput., 12 (2006), 253.
doi: 10.1007/s11155-006-9000-y. |
[20] |
A. J. Homburg, H. M. Osinga and G. Vegter, On the computation of invariant manifolds of fixed points,, ZAMM Z. Angew. Math. Phys., 46 (1995), 171.
doi: 10.1007/BF00944751. |
[21] |
C. S. Hsu, Global analysis by cell mapping,, Internat. J. Bifurc. Chaos Appl. Sci. Engrg., 2 (1992), 727.
doi: 10.1142/S0218127492000422. |
[22] |
L. V. Kantorovich, On Newton's method for functional equations,, Dokl. Akad. Nauk SSSR, 59 (1948), 1237.
|
[23] |
L. V. Kantorovich, The majorant principle and Newton's method,, Dokl. Akad. Nauk SSSR, 76 (1951), 17.
|
[24] |
L. V. Kantorovich and G. P. Akilov, Functional Analysis,, 2nd edition, (1982).
|
[25] |
I. G. Kevrekidis, R. Aris, L. D. Schmidt and S. Pelikan, Numerical computation of invariant circles of maps,, Phys. D, 16 (1985), 243.
doi: 10.1016/0167-2789(85)90061-2. |
[26] |
I. G. Kevrekidis, C. W. Gear and G. Hummer, Equation-free: The computer-aided analysis of complex multiscale systems,, AIChE Journ., 50 (2004), 1346.
doi: 10.1002/aic.10106. |
[27] |
I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis,, Commun. Math. Sci., 1 (2003), 715.
doi: 10.4310/CMS.2003.v1.n4.a5. |
[28] |
D. Klatte and B. Kummer, Nonsmooth equations in optimization. Regularity, calculus, methods and applications,, vol. 60 of Nonconvex Optimization and its Applications, (2002).
|
[29] |
B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763.
doi: 10.1142/S0218127405012533. |
[30] |
R. Krawczyk, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken,, Computing (Arch. Elektron. Rechnen), 4 (1969), 187.
|
[31] |
R. Krawczyk, Einschlieş ung von Nullstellen mit Hilfe einer Intervallarithmetik,, Computing (Arch. Elektron. Rechnen), 5 (1970), 356.
|
[32] |
E. Kreuzer, Analysis of chaotic systems using the cell mapping approach,, Arch. Appl. Mech., 55 (1985), 285.
doi: 10.1007/BF00538223. |
[33] |
S. M. Markov, Some applications of extended interval arithmetic to interval iterations,, Comput. Suppl., 2 (1980), 69.
|
[34] |
K. Nickel, Das Auflösungsverhalten von nichtlinearen Fixmengen-Systemen,, in Iterative solution of nonlinear systems of equations (Oberwolfach, (1982), 106.
|
[35] |
D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets, vol. 548 of Mathematics and Its Applications,, Kluwer Academic Publishers, (2002).
|
[36] |
G. Perria, Set-valued Interpolation, vol. 79 of Bayreuth. Math. Schr.,, Department of Mathematics, (2007).
|
[37] |
B. T. Polyak, Convexity of quadratic transformations and its use in control and optimization,, J. Optim. Theory Appl., 99 (1998), 553.
doi: 10.1023/A:1021798932766. |
[38] |
B. T. Polyak, Convexity of nonlinear image of a small ball with applications to optimization,, Set-Valued Anal., 9 (2001), 159.
doi: 10.1023/A:1011287523150. |
[39] |
L. S. Pontryagin, Linear differential games. II,, Sov. Math., 8 (1967), 910. Google Scholar |
[40] |
R. T. Rockafellar, Convex Analysis,, vol. 28 of Princeton Mathematical Series, (1970).
|
[41] |
A. M. Rubinov and I. S. Akhundov, Difference of compact sets in the sense of Demyanov and its application to non-smooth analysis,, Optimization, 23 (1992), 179.
doi: 10.1080/02331939208843757. |
[42] |
P. Saint-Pierre, Newton and other continuation methods for multivalued inclusions,, Set-Valued Anal., 3 (1995), 143.
doi: 10.1007/BF01038596. |
[43] |
C. I. Siettos, C. C. Pantelides and I. G. Kevrekidis, Enabling dynamic process simulators to perform alternative tasks: A time-stepper-based toolkit for computer-aided analysis,, Ind. Engrg. Chem. Res., 42 (2003), 6795.
doi: 10.1021/ie021062w. |
[44] |
C. Theodoropoulos, Y.-H. Qian and I. G. Kevrekidis, "Coarse'' stability and bifurcation analysis using time-steppers: A reaction-diffusion example,, Proc. Natl. Acad. Sci., 97 (2000), 9840.
doi: 10.1073/pnas.97.18.9840. |
[45] |
L. S. Tuckerman and D. Barkley, Bifurcation analysis for timesteppers,, in Numerical methods for bifurcation problems and large-scale dynamical systems (Minneapolis, (2000), 453.
doi: 10.1007/978-1-4612-1208-9_20. |
[46] |
X. Wang, Convergence of Newton's method and inverse function theorem in Banach space,, Math. Comp., 68 (1999), 169.
doi: 10.1090/S0025-5718-99-00999-0. |
[47] |
T. Yamamoto, A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions,, Numer. Math., 49 (1986), 203.
doi: 10.1007/BF01389624. |
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