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An Adaptive Homotopy Method for Computing Bifurcations of Nonlinear Parametric Systems

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Abstract

In this paper, we present an adaptive step-size homotopy tracking method for computing bifurcation points of nonlinear systems. There are four components in this new method: (1) an adaptive tracking technique is developed near bifurcation points; (2) an inflation technique is backed up when the adaptive tracking fails; (3) Puiseux series interpolation is used to compute bifurcation points; and (4) the tangent cone structure of the bifurcation point is approximated numerically to compute solutions on different branches. Various numerical examples of nonlinear systems are given to illustrate the efficiency of this new approach. This new adaptive homotopy tracking method is also applied to a system of nonlinear PDEs and shows robustness and efficiency for large-scale nonlinear discretized systems.

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References

  1. Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods, vol. 45. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  2. Bates, D., Hauenstein, J., Sommese, A.: Efficient path tracking methods. Numer. Algorithms 58(4), 451–459 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bates, D., Hauenstein, J., Sommese, A., Wampler, C.: Bertini: Software for numerical algebraic geometry (2006)

  4. Bates, D., Hauenstein, J., Sommese, A., Wampler, C.: Adaptive multiprecision path tracking. SIAM J. Numer. Anal. 46(2), 722–746 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bates, D., Hauenstein, J., Sommese, A., Wampler, C.: Numerically Solving Polynomial Systems with Bertini, vol. 25. SIAM, Philadelphia (2013)

    Book  MATH  Google Scholar 

  6. Bates, J., Hauenstein, D., Sommese, A.: A parallel endgame. Contemp. Math. 556, 25–35 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Buffoni, B., Toland, J., Toland, J.F.: Analytic Theory of Global Bifurcation: An Introduction. Princeton University Press, Princeton (2003)

    Book  MATH  Google Scholar 

  8. Chen, X., Hambrock, R., Lou, Y.: Evolution of conditional dispersal: a reaction–diffusion–advection model. J. Math. Biol. 57(3), 361–386 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chicone, C.: Lyapunov–Schmidt reduction and melnikov integrals for bifurcation of periodic solutions in coupled oscillators. J. Differ. Equ. 112(2), 407–447 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Choi, S., Harney, D., Book, N.: A robust path tracking algorithm for homotopy continuation. Comput. Chem. Eng. 20(6), 647–655 (1996)

    Article  Google Scholar 

  11. Dayton, B., Zeng, Z.: Computing the multiplicity structure in solving polynomial systems. In: Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation. ACM, pp. 116–123 (2005)

  12. Deuflhard, P.: Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, vol. 35. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  13. Fischer, G.: Plane Algebraic Curves, vol. 15. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  14. Friedman, A., Hao, W.: A mathematical model of atherosclerosis with reverse cholesterol transport and associated risk factors. Bull. Math. Biol. 77(5), 758–781 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Friedman, A., Hu, B.: Bifurcation from stability to instability for a free boundary problem arising in a tumor model. Arch. Ration. Mech. Anal. 180(2), 293–330 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Friedman, A., Hu, B.: Bifurcation for a free boundary problem modeling tumor growth by stokes equation. SIAM J. Math. Anal. 39(1), 174–194 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Haber, R., Unbehauen, H.: Structure identification of nonlinear dynamic systems survey on input/output approaches. Automatica 26(4), 651–677 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hambrock, R., Lou, Y.: The evolution of conditional dispersal strategies in spatially heterogeneous habitats. Bull. Math. Biol. 71(8), 1793–1817 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hao, W., Crouser, E., Friedman, A.: Mathematical model of sarcoidosis. Proc. Natl. Acad. Sci. 111(45), 16065–16070 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hao, W., Friedman, A.: The ldl–hdl profile determines the risk of atherosclerosis: a mathematical model. PLoS ONE 9(3), e90497 (2014)

    Article  Google Scholar 

  21. Hao, W., Hauenstein, J., Hu, B., Liu, Y., Sommese, A., Zhang, Y.-T.: Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core. Nonlinear Anal. Real World Appl. 13(2), 694–709 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hao, W., Hauenstein, J., Hu, B., Sommese, A.: A three-dimensional steady-state tumor system. Appl. Math. Comput. 218(6), 2661–2669 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Hao, W., Hauenstein, J., Shu, C.-W., Sommese, A., Xu, Z., Zhang, Y.-T.: A homotopy method based on weno schemes for solving steady state problems of hyperbolic conservation laws. J. Comput. Phys. 250(5), 332–346 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hao, W., Lam, K.Y., Lou, Y.: Concentration phenomena in an integro-pde model for evolution of conditional dispersal. Indiana Univ. Math. J. (2017, to appear)

  25. Hao, W., Nepomechie, R., Sommese, A.: Completeness of solutions of Bethe’s equations. Phys. Rev. E 88(5), 052113 (2013)

    Article  Google Scholar 

  26. Hao, W., Nepomechie, R., Sommese, A.: Singular solutions, repeated roots and completeness for higher-spin chains. J. Stat. Mech. Theory Exp. 2014(3), P03024 (2014)

    Article  MathSciNet  Google Scholar 

  27. Hauenstein, J., Wampler, C.: Isosingular sets and deflation. Found. Comput. Math. 13(3), 371–403 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hou, T., Lowengrub, J., Shelley, M.: Boundary integral methods for multicomponent fluids and multiphase materials. J. Comput. Phys. 169(2), 302–362 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Huber, B., Verschelde, J.: Polyhedral end games for polynomial continuation. Numer. Algorithms 18(1), 91–108 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Khalil, H.: Nonlinear Systems. Prentice Hall, Upper Saddle River (2002)

    MATH  Google Scholar 

  31. Lam, K.Y., Lou, Y.: Evolution of conditional dispersal: evolutionarily stable strategies in spatial models. J. Math. Biol. 68(4), 851–877 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lee, T.-L., Li, T.-Y., Tsai, C.-H.: Hom4ps-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing 83(2), 109–133 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lee, Y., Wu, J., Xu, J., Zikatanov, L.: Robust subspace correction methods for nearly singular systems. Math. Models Methods Appl. Sci. 17(11), 1937–1963 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Leykin, A., Verschelde, J., Zhao, A.: Newton’s method with deflation for isolated singularities of polynomial systems. Theor. Comput. Sci. 359(1–3), 111–122 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, T.-Y., Sauer, T., Yorke, J.: The cheater’s homotopy: an efficient procedure for solving systems of polynomial equations. SIAM J. Numer. Anal. 26(5), 1241–1251 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, T.-Y., Zeng, Z.: Homotopy-determinant algorithm for solving nonsymmetric eigenvalue problems. Math. Comput. 59(200), 483–502 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  37. Morgan, A., Sommese, A.: Computing all solutions to polynomial systems using homotopy continuation. Appl. Math. Comput. 24(2), 115–138 (1987)

    MathSciNet  MATH  Google Scholar 

  38. Morgan, A., Sommese, A.: A homotopy for solving general polynomial systems that respects m-homogeneous structures. Appl. Math. Comput. 24(2), 101–113 (1987)

    MathSciNet  MATH  Google Scholar 

  39. Morgan, A., Sommese, A., Wampler, C.: A power series method for computing singular solutions to nonlinear analytic systems. Numer. Math. 63(1), 391–409 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rabier, P., Rheinboldt, W.: On a computational method for the second fundamental tensor and its application to bifurcation problems. Numer. Math. 57(1), 681–694 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  41. Rheinboldt, W.: Numerical methods for a class of finite dimensional bifurcation problems. SIAM J. Numer. Anal. 15(1), 1–11 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sommese, A., Wampler, C.: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, vol. 99. World Scientific, Singapore (2005)

    Book  MATH  Google Scholar 

  43. Strogatz, S.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, New York (2014)

    MATH  Google Scholar 

  44. Verschelde, J.: Algorithm 795: Phcpack: a general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25(2), 251–276 (1999)

    Article  MATH  Google Scholar 

  45. Wang, X., Golubitsky, M.: Singularity theory of fitness functions under dimorphism equivalence. J. Math. Biol. 73(3), 525–573 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  46. Watson, L., Billups, S., Morgan, A.: Algorithm 652: Hompack: a suite of codes for globally convergent homotopy algorithms. ACM Trans. Math. Softw. 13(3), 281–310 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  47. Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  48. Xu, J., Chen, L., Nochetto, R.: Optimal multilevel methods for h (grad), h (curl), and h (div) systems on graded and unstructured grids. Multiscale Nonlinear Adapt. Approx. 1(1), 599–659 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zeng, Z.: Algorithm 835: Multroot—a matlab package for computing polynomial roots and multiplicities. ACM Trans. Math. Softw. 30(2), 218–236 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  50. Zeng, Z.: Computing multiple roots of inexact polynomials. Math. Comput. 74(250), 869–903 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This study was supported by the National Science Foundation (Grant No. DMS-1818769).

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  1. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Wenrui Hao & Chunyue Zheng

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  1. Wenrui Hao
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  2. Chunyue Zheng
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Correspondence to Wenrui Hao.

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Hao, W., Zheng, C. An Adaptive Homotopy Method for Computing Bifurcations of Nonlinear Parametric Systems. J Sci Comput 82, 53 (2020). https://doi.org/10.1007/s10915-020-01160-w

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