American Institute of Mathematical Sciences

Advanced
June  2016, 3(2): 113-137. doi: 10.3934/jcd.2016006

Rigorous bounds for polynomial Julia sets

Luiz Henrique de Figueiredo 1, , Diego Nehab 1, , Jorge Stolfi 2, and  João Batista S. de Oliveira 3,

1. 

IMPA, Rio de Janeiro, Brazil, Brazil
2. 

Instituto de Computacão, UNICAMP, Campinas, Brazil
3. 

Faculdade de Informática, PUC-RS, Porto Alegre, Brazil

Received  March 2016 Revised  August 2016 Published  November 2016

We present an algorithm for computing images of polynomial Julia sets that are reliable in the sense that they carry mathematical guarantees against sampling artifacts and rounding errors in floating-point arithmetic. We combine cell mapping based on interval arithmetic with label propagation in graphs to avoid function iteration and rounding errors. As a result, our algorithm avoids point sampling and can reliably classify entire rectangles in the complex plane as being on either side of the Julia set. The union of the rectangles that cannot be so classified is guaranteed to contain the Julia set. Our algorithm computes a refinable quadtree decomposition of the complex plane adapted to the Julia set which can be used for rendering and for approximating geometric properties such as the area of the filled Julia set and the fractal dimension of the Julia set.
Keywords: cell mapping, computer-assisted proofs., robust rendering, Julia sets, fractals, adaptive refinement.
Mathematics Subject Classification: 37-04, 37F50, 37F10, 37M99, 65P9.
Citation: Luiz Henrique de Figueiredo, Diego Nehab, Jorge Stolfi, João Batista S. de Oliveira. Rigorous bounds for polynomial Julia sets. Journal of Computational Dynamics, 2016, 3 (2) : 113-137. doi: 10.3934/jcd.2016006
References:
[1]

Z. Arai, On hyperbolic plateaus of the Hénon map, Experimental Mathematics, 16 (2007), 181-188. doi: 10.1080/10586458.2007.10128992.  Google Scholar

[2]

P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bulletin of the American Mathematical Society, 11 (1984), 85-141. doi: 10.1090/S0273-0979-1984-15240-6.  Google Scholar

[3]

B. Branner, The Mandelbrot set, in Amer. Math. Soc., 39 (1989), 75-105. doi: 10.1090/psapm/039/1010237.  Google Scholar

[4]

M. Braverman, Hyperbolic Julia sets are poly-time computable, Electronic Notes in Theoretical Computer Science, 120 (2005), 17-30. doi: 10.1016/j.entcs.2004.06.031.  Google Scholar

[5]

M. Braverman and M. Yampolsky, Computability of Julia Sets, Springer-Verlag, 2009.  Google Scholar

[6]

R. Carniel, A quasi-cell mapping approach to the global dynamical analysis of Newton's root-finding algorithm, Applied Numerical Mathematics, 15 (1994), 133-152. doi: 10.1016/0168-9274(94)00016-6.  Google Scholar

[7]

L. H. de Figueiredo and J. Stolfi, Affine arithmetic: concepts and applications, Numerical Algorithms, 37 (2004), 147-158. doi: 10.1023/B:NUMA.0000049462.70970.b6.  Google Scholar

[8]

M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numerische Mathematik, 75 (1997), 293-317. doi: 10.1007/s002110050240.  Google Scholar

[9]

M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, in Handbook of dynamical systems, North-Holland, 2 (2002), 221-264. doi: 10.1016/S1874-575X(02)80026-1.  Google Scholar

[10]

R. L. Devaney and L. Keen (eds.), Chaos and Fractals: The Mathematics behind the Computer Graphics, Proceedings of Symposia in Applied Mathematics 39, AMS, 1989.  Google Scholar

[11]

A. Douady, Does a Julia set depend continuously on the polynomial?, in Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets (ed. R. L. Devaney), Proceedings of Symposia in Applied Mathematics, AMS, 49 (1994), 91-138. doi: 10.1090/psapm/049/1315535.  Google Scholar

[12]

M. B. Durkin, The accuracy of computer algorithms in dynamical systems, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 1 (1991), 625-639. doi: 10.1142/S0218127491000452.  Google Scholar

[13]

Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic, Nonlinearity, 15 (2002), 1759-1779. doi: 10.1088/0951-7715/15/6/304.  Google Scholar

[14]

E. Grassmann and J. Rokne, The range of values of a circular complex polynomial over a circular complex interval, Computing, 23 (1979), 139-169. doi: 10.1007/BF02252094.  Google Scholar

[15]

T. H. Gronwall, Some remarks on conformal representation,, Annals of Mathematics, 16 (): 72.  doi: 10.2307/1968044.  Google Scholar

[16]

S. L. Hruska, Constructing an expanding metric for dynamical systems in one complex variable, Nonlinearity, 18 (2005), 81-100. doi: 10.1088/0951-7715/18/1/005.  Google Scholar

[17]

S. L. Hruska, Rigorous numerical studies of the dynamics of polynomial skew products of $C^2$, in Complex dynamics, vol. 396 of Contemp. Math., AMS, 2006, 85-100. doi: 10.1090/conm/396/07395.  Google Scholar

[18]

C. S. Hsu, Cell-to-cell Mapping: A Method of Global Analysis for Nonlinear Systems, Springer-Verlag, 1987. doi: 10.1007/978-1-4757-3892-6.  Google Scholar

[19]

C. S. Hsu, Global analysis by cell mapping, International Journal of Bifurcations and Chaos, 2 (1992), 727-771. doi: 10.1142/S0218127492000422.  Google Scholar

[20]

O. Junge, Rigorous discretization of subdivision techniques, in International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., 2000, 916-918.  Google Scholar

[21]

W. D. Kalies, K. Mischaikow and R. C. A. M. VanderVorst, An algorithmic approach to chain recurrence, Foundations of Computational Mathematics, 5 (2005), 409-449. doi: 10.1007/s10208-004-0163-9.  Google Scholar

[22]

L. Keen, Julia sets, in Proc. Sympos. Appl. Math., 39 (1989), 57-74. doi: 10.1090/psapm/039/1010236.  Google Scholar

[23]

V. Kreinovich, Interval software,, , ().   Google Scholar

[24]

D. Michelucci and S. Foufou, Interval-based tracing of strange attractors, International Journal of Computational Geometry & Applications, 16 (2006), 27-39. doi: 10.1142/S0218195906001914.  Google Scholar

[25]

J. Milnor, Remarks on iterated cubic maps, Experimental Mathematics, 1 (1992), 5-24.  Google Scholar

[26]

J. Milnor, Dynamics in One Complex Variable, vol. 160 of Annals of Mathematics Studies, 3rd edition, Princeton University Press, 2006.  Google Scholar

[27]

R. E. Moore, Interval Analysis, Prentice-Hall, 1966.  Google Scholar

[28]

R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM, 2009. doi: 10.1137/1.9780898717716.  Google Scholar

[29]

R. P. Munafo, Roundoff error, http://mrob.com/pub/muency/roundofferror.html, 1996, Accessed: 2015-12-09. Google Scholar

[30]

D. Nehab and H. Hoppe, A fresh look at generalized sampling, Foundations and Trends in Computer Graphics and Vision, 8 (2014), 1-84. doi: 10.1561/0600000053.  Google Scholar

[31]

G. Osipenko, Dynamical Systems, Graphs, and Algorithms, vol. 1889 of Lecture Notes in Mathematics, Springer-Verlag, 2007.  Google Scholar

[32]

A. Paiva, L. H. de Figueiredo and J. Stolfi, Robust visualization of strange attractors using affine arithmetic, Computers & Graphics, 30 (2006), 1020-1026. doi: 10.1016/j.cag.2006.08.016.  Google Scholar

[33]

H.-O. Peitgen and P. H. Richter, The Beauty of Fractals: Images of Complex Dynamical Systems, Springer-Verlag, 1986. doi: 10.1007/978-3-642-61717-1.  Google Scholar

[34]

H.-O. Peitgen and D. Saupe (eds.), The Science of Fractal Images, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-3784-6.  Google Scholar

[35]

M. S. Petković and L. D. Petković, Complex Interval Arithmetic and Its Applications, Wiley-VCH Verlag, 1998.  Google Scholar

[36]

R. Rettinger, A fast algorithm for {Julia} sets of hyperbolic rational functions, Electronic Notes in Theoretical Computer Science, 120 (2005), 145-157. doi: 10.1016/j.entcs.2004.06.041.  Google Scholar

[37]

R. Rettinger and K. Weihrauch, The computational complexity of some Julia sets, in Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC 2003), ACM, 2003, 177-185. doi: 10.1145/780542.780570.  Google Scholar

[38]

J. Rokne, The range of values of a complex polynomial over a complex interval, Computing, 22 (1979), 153-169. doi: 10.1007/BF02253127.  Google Scholar

[39]

D. Ruelle, Repellers for real analytic maps, Ergodic Theory and Dynamical Systems, 2 (1982), 99-107. doi: 10.1017/S0143385700009603.  Google Scholar

[40]

S. M. Rump and M. Kashiwagi, Implementation and improvements of affine arithmetic, Nonlinear Theory and Its Applications, IEICE, 6 (2015), 341-359. Google Scholar

[41]

H. Samet, The quadtree and related hierarchical data structures, Computing Surveys, 16 (1984), 187-260. doi: 10.1145/356924.356930.  Google Scholar

[42]

D. Saupe, Efficient computation of Julia sets and their fractal dimension, Physica D, 28 (1987), 358-370. doi: 10.1016/0167-2789(87)90024-8.  Google Scholar

[43]

N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, Walter de Gruyter & Co., 1993. doi: 10.1515/9783110889314.  Google Scholar

[44]

C. M. Stroh, Julia Sets of Complex Polynomials and Their Implementation on the Computer, Master's thesis, University of Linz, 1997. Google Scholar

[45]

R. Tarjan, Depth-first search and linear graph algorithms, SIAM Journal on Computing, 1 (1972), 146-160. doi: 10.1137/0201010.  Google Scholar

[46]

W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1197-1202. doi: 10.1016/S0764-4442(99)80439-X.  Google Scholar

[47]

J. Tupper, Reliable two-dimensional graphing methods for mathematical formulae with two free variables, in Proceedings of SIGGRAPH '01, ACM, 2001, 77-86. doi: 10.1145/383259.383267.  Google Scholar

show all references

References:
[1]

Z. Arai, On hyperbolic plateaus of the Hénon map, Experimental Mathematics, 16 (2007), 181-188. doi: 10.1080/10586458.2007.10128992.  Google Scholar

[2]

P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bulletin of the American Mathematical Society, 11 (1984), 85-141. doi: 10.1090/S0273-0979-1984-15240-6.  Google Scholar

[3]

B. Branner, The Mandelbrot set, in Amer. Math. Soc., 39 (1989), 75-105. doi: 10.1090/psapm/039/1010237.  Google Scholar

[4]

M. Braverman, Hyperbolic Julia sets are poly-time computable, Electronic Notes in Theoretical Computer Science, 120 (2005), 17-30. doi: 10.1016/j.entcs.2004.06.031.  Google Scholar

[5]

M. Braverman and M. Yampolsky, Computability of Julia Sets, Springer-Verlag, 2009.  Google Scholar

[6]

R. Carniel, A quasi-cell mapping approach to the global dynamical analysis of Newton's root-finding algorithm, Applied Numerical Mathematics, 15 (1994), 133-152. doi: 10.1016/0168-9274(94)00016-6.  Google Scholar

[7]

L. H. de Figueiredo and J. Stolfi, Affine arithmetic: concepts and applications, Numerical Algorithms, 37 (2004), 147-158. doi: 10.1023/B:NUMA.0000049462.70970.b6.  Google Scholar

[8]

M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numerische Mathematik, 75 (1997), 293-317. doi: 10.1007/s002110050240.  Google Scholar

[9]

M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, in Handbook of dynamical systems, North-Holland, 2 (2002), 221-264. doi: 10.1016/S1874-575X(02)80026-1.  Google Scholar

[10]

R. L. Devaney and L. Keen (eds.), Chaos and Fractals: The Mathematics behind the Computer Graphics, Proceedings of Symposia in Applied Mathematics 39, AMS, 1989.  Google Scholar

[11]

A. Douady, Does a Julia set depend continuously on the polynomial?, in Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets (ed. R. L. Devaney), Proceedings of Symposia in Applied Mathematics, AMS, 49 (1994), 91-138. doi: 10.1090/psapm/049/1315535.  Google Scholar

[12]

M. B. Durkin, The accuracy of computer algorithms in dynamical systems, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 1 (1991), 625-639. doi: 10.1142/S0218127491000452.  Google Scholar

[13]

Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic, Nonlinearity, 15 (2002), 1759-1779. doi: 10.1088/0951-7715/15/6/304.  Google Scholar

[14]

E. Grassmann and J. Rokne, The range of values of a circular complex polynomial over a circular complex interval, Computing, 23 (1979), 139-169. doi: 10.1007/BF02252094.  Google Scholar

[15]

T. H. Gronwall, Some remarks on conformal representation,, Annals of Mathematics, 16 (): 72.  doi: 10.2307/1968044.  Google Scholar

[16]

S. L. Hruska, Constructing an expanding metric for dynamical systems in one complex variable, Nonlinearity, 18 (2005), 81-100. doi: 10.1088/0951-7715/18/1/005.  Google Scholar

[17]

S. L. Hruska, Rigorous numerical studies of the dynamics of polynomial skew products of $C^2$, in Complex dynamics, vol. 396 of Contemp. Math., AMS, 2006, 85-100. doi: 10.1090/conm/396/07395.  Google Scholar

[18]

C. S. Hsu, Cell-to-cell Mapping: A Method of Global Analysis for Nonlinear Systems, Springer-Verlag, 1987. doi: 10.1007/978-1-4757-3892-6.  Google Scholar

[19]

C. S. Hsu, Global analysis by cell mapping, International Journal of Bifurcations and Chaos, 2 (1992), 727-771. doi: 10.1142/S0218127492000422.  Google Scholar

[20]

O. Junge, Rigorous discretization of subdivision techniques, in International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., 2000, 916-918.  Google Scholar

[21]

W. D. Kalies, K. Mischaikow and R. C. A. M. VanderVorst, An algorithmic approach to chain recurrence, Foundations of Computational Mathematics, 5 (2005), 409-449. doi: 10.1007/s10208-004-0163-9.  Google Scholar

[22]

L. Keen, Julia sets, in Proc. Sympos. Appl. Math., 39 (1989), 57-74. doi: 10.1090/psapm/039/1010236.  Google Scholar

[23]

V. Kreinovich, Interval software,, , ().   Google Scholar

[24]

D. Michelucci and S. Foufou, Interval-based tracing of strange attractors, International Journal of Computational Geometry & Applications, 16 (2006), 27-39. doi: 10.1142/S0218195906001914.  Google Scholar

[25]

J. Milnor, Remarks on iterated cubic maps, Experimental Mathematics, 1 (1992), 5-24.  Google Scholar

[26]

J. Milnor, Dynamics in One Complex Variable, vol. 160 of Annals of Mathematics Studies, 3rd edition, Princeton University Press, 2006.  Google Scholar

[27]

R. E. Moore, Interval Analysis, Prentice-Hall, 1966.  Google Scholar

[28]

R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM, 2009. doi: 10.1137/1.9780898717716.  Google Scholar

[29]

R. P. Munafo, Roundoff error, http://mrob.com/pub/muency/roundofferror.html, 1996, Accessed: 2015-12-09. Google Scholar

[30]

D. Nehab and H. Hoppe, A fresh look at generalized sampling, Foundations and Trends in Computer Graphics and Vision, 8 (2014), 1-84. doi: 10.1561/0600000053.  Google Scholar

[31]

G. Osipenko, Dynamical Systems, Graphs, and Algorithms, vol. 1889 of Lecture Notes in Mathematics, Springer-Verlag, 2007.  Google Scholar

[32]

A. Paiva, L. H. de Figueiredo and J. Stolfi, Robust visualization of strange attractors using affine arithmetic, Computers & Graphics, 30 (2006), 1020-1026. doi: 10.1016/j.cag.2006.08.016.  Google Scholar

[33]

H.-O. Peitgen and P. H. Richter, The Beauty of Fractals: Images of Complex Dynamical Systems, Springer-Verlag, 1986. doi: 10.1007/978-3-642-61717-1.  Google Scholar

[34]

H.-O. Peitgen and D. Saupe (eds.), The Science of Fractal Images, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-3784-6.  Google Scholar

[35]

M. S. Petković and L. D. Petković, Complex Interval Arithmetic and Its Applications, Wiley-VCH Verlag, 1998.  Google Scholar

[36]

R. Rettinger, A fast algorithm for {Julia} sets of hyperbolic rational functions, Electronic Notes in Theoretical Computer Science, 120 (2005), 145-157. doi: 10.1016/j.entcs.2004.06.041.  Google Scholar

[37]

R. Rettinger and K. Weihrauch, The computational complexity of some Julia sets, in Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC 2003), ACM, 2003, 177-185. doi: 10.1145/780542.780570.  Google Scholar

[38]

J. Rokne, The range of values of a complex polynomial over a complex interval, Computing, 22 (1979), 153-169. doi: 10.1007/BF02253127.  Google Scholar

[39]

D. Ruelle, Repellers for real analytic maps, Ergodic Theory and Dynamical Systems, 2 (1982), 99-107. doi: 10.1017/S0143385700009603.  Google Scholar

[40]

S. M. Rump and M. Kashiwagi, Implementation and improvements of affine arithmetic, Nonlinear Theory and Its Applications, IEICE, 6 (2015), 341-359. Google Scholar

[41]

H. Samet, The quadtree and related hierarchical data structures, Computing Surveys, 16 (1984), 187-260. doi: 10.1145/356924.356930.  Google Scholar

[42]

D. Saupe, Efficient computation of Julia sets and their fractal dimension, Physica D, 28 (1987), 358-370. doi: 10.1016/0167-2789(87)90024-8.  Google Scholar

[43]

N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, Walter de Gruyter & Co., 1993. doi: 10.1515/9783110889314.  Google Scholar

[44]

C. M. Stroh, Julia Sets of Complex Polynomials and Their Implementation on the Computer, Master's thesis, University of Linz, 1997. Google Scholar

[45]

R. Tarjan, Depth-first search and linear graph algorithms, SIAM Journal on Computing, 1 (1972), 146-160. doi: 10.1137/0201010.  Google Scholar

[46]

W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1197-1202. doi: 10.1016/S0764-4442(99)80439-X.  Google Scholar

[47]

J. Tupper, Reliable two-dimensional graphing methods for mathematical formulae with two free variables, in Proceedings of SIGGRAPH '01, ACM, 2001, 77-86. doi: 10.1145/383259.383267.  Google Scholar

[1]

Maxime Breden, Jean-Philippe Lessard. Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2825-2858. doi: 10.3934/dcdsb.2018164
[2]

István Balázs, Jan Bouwe van den Berg, Julien Courtois, János Dudás, Jean-Philippe Lessard, Anett Vörös-Kiss, JF Williams, Xi Yuan Yin. Computer-assisted proofs for radially symmetric solutions of PDEs. Journal of Computational Dynamics, 2018, 5 (1&2) : 61-80. doi: 10.3934/jcd.2018003
[3]

Chiara Caracciolo, Ugo Locatelli. Computer-assisted estimates for Birkhoff normal forms. Journal of Computational Dynamics, 2020, 7 (2) : 425-460. doi: 10.3934/jcd.2020017
[4]

Maciej J. Capiński, Emmanuel Fleurantin, J. D. Mireles James. Computer assisted proofs of two-dimensional attracting invariant tori for ODEs. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6681-6707. doi: 10.3934/dcds.2020162
[5]

Thomas Wanner. Computer-assisted equilibrium validation for the diblock copolymer model. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 1075-1107. doi: 10.3934/dcds.2017045
[6]

A. Aschwanden, A. Schulze-Halberg, D. Stoffer. Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 721-736. doi: 10.3934/dcds.2006.14.721
[7]

Nahid Banihashemi, C. Yalçın Kaya. Inexact restoration and adaptive mesh refinement for optimal control. Journal of Industrial & Management Optimization, 2014, 10 (2) : 521-542. doi: 10.3934/jimo.2014.10.521
[8]

Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95
[9]

Koh Katagata. Quartic Julia sets including any two copies of quadratic Julia sets. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2103-2112. doi: 10.3934/dcds.2016.36.2103
[10]

Robert L. Devaney, Daniel M. Look. Buried Sierpinski curve Julia sets. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 1035-1046. doi: 10.3934/dcds.2005.13.1035
[11]

Danilo Antonio Caprio. A class of adding machines and Julia sets. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5951-5970. doi: 10.3934/dcds.2016061
[12]

Nathaniel D. Emerson. Dynamics of polynomials with disconnected Julia sets. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 801-834. doi: 10.3934/dcds.2003.9.801
[13]

Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553
[14]

Xuemei Li, Rafael de la Llave. Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 623-641. doi: 10.3934/dcdss.2010.3.623
[15]

Tien-Cuong Dinh, Nessim Sibony. Rigidity of Julia sets for Hénon type maps. Journal of Modern Dynamics, 2014, 8 (3&4) : 499-548. doi: 10.3934/jmd.2014.8.499
[16]

Tarik Aougab, Stella Chuyue Dong, Robert S. Strichartz. Laplacians on a family of quadratic Julia sets II. Communications on Pure & Applied Analysis, 2013, 12 (1) : 1-58. doi: 10.3934/cpaa.2013.12.1
[17]

Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293
[18]

Ali Messaoudi, Rafael Asmat Uceda. Stochastic adding machine and $2$-dimensional Julia sets. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5247-5269. doi: 10.3934/dcds.2014.34.5247
[19]

Ranjit Bhattacharjee, Robert L. Devaney, R.E. Lee Deville, Krešimir Josić, Monica Moreno-Rocha. Accessible points in the Julia sets of stable exponentials. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 299-318. doi: 10.3934/dcdsb.2001.1.299
[20]

Kazuhisa Ichikawa. Synergistic effect of blocking cancer cell invasion revealed by computer simulations. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1189-1202. doi: 10.3934/mbe.2015.12.1189

 Impact Factor: 

Tools

Metrics

  • PDF downloads (107)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy