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Slow motion of particle systems as a limit of a reactiondiffusion equation with halfLaplacian in dimension one
Box dimension and bifurcations of onedimensional discrete dynamical systems
1.  Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia 
References:
[1] 
D. K. Arrowsmith and C. M. Place, "An Introduction to Dynamical Systems," Cambridge University Press, Cambridge, 1990. 
[2] 
F. Balibrea and J. C. Valverde, Bifurcations under nondegenerated conditions of higher degree and a new simple proof of the HopfNeimarkSacker bifurcation theorem, J. Math. Anal. Appl., 237 (1999), 93105. doi: 10.1006/jmaa.1999.6460. 
[3] 
F. Balibrea and J. C. Valverde, Cusp and generalized flip bifurcations under higher degree conditions, Nonlinear Anal., 52 (2003), 405419. doi: 10.1016/S0362546X(01)009087. 
[4] 
N. Elezović, V. Županović and D. Žubrinić, Box dimension of trajectories of some discrete dynamical systems, Chaos Solitons Fractals, 34 (2007), 244252. doi: 10.1016/j.chaos.2006.03.060. 
[5] 
K. Falconer, "Fractal Geometry: Mathematical Foundations and Applications," John Wiley & Sons, Ltd., Chichester, 1990. 
[6] 
Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," 2^{nd} edition, Applied Mathematical Sciences, 112, SpringerVerlag, New York, 1998. 
[7] 
M. L. Lapidus and C. Pomerance, The Riemann zetafunction and the onedimensional WeylBerry conjecture for fractal drums, Proc. London Math. Soc. (3), 66 (1993), 4169. 
[8] 
P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995. 
[9] 
J. Palis and F. Takens, "Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors," Cambridge Studies in Advanced Mathematics, 35, Cambridge University Press, Cambridge, 1993. 
[10] 
M. Pašić, MinkowskiBouligand dimension of solutions of the onedimensional $p$Laplacian, J. Differential Equations, 190 (2003), 268305. doi: 10.1016/S00220396(02)001493. 
[11] 
M. Pašić, D. Žubrinić and V. Županović, Oscillatory and phase dimensions of solutions of some secondorder differential equations, Bull. Sci. Math., 133 (2009), 859874. 
[12] 
L. Perko, "Differential Equations and Dynamical Systems," 2^{nd} edition, Texts in Applied Mathematics, 7, SpringerVerlag, New York, 1996. 
[13] 
C. Tricot, "Curves and Fractal Dimension," With a foreword by Michel Mendès France, SpringerVerlag, New York, 1995. 
[14] 
S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," 2^{nd} edition, Texts in Applied Mathematics, 2, SpringerVerlag, New York, 2003. 
[15] 
D. Žubrinić, Analysis of Minkowski content of fractal sets and applications,, Real Anal. Exchange, 31 (): 315. 
[16] 
D. Žubrinić and V. Županović, Fractal dimension in dynamics, in "Encyclopedia of Math. Physics" (eds. J.P. Françoise, G. L. Naber and S. T. Tsou), Academic Press/Elsevier Science, Oxford, (2006), 394402. 
[17] 
D. Žubrinić and V. Županović, Poincaré map in fractal analysis of spiral trajectories of planar vector fields, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 947960. 
[18] 
D. Žubrinić and V. Županović, Fractal analysis of spiral trajectories of some planar vector fields, Bull. Sci. Math., 129 (2005), 457485. 
[19] 
D. Žubrinić and V. Županović, Fractal analysis of spiral trajectories of some vector fields in $\mathbbR^3$, C. R. Math. Acad. Sci. Paris, 342 (2006), 959963. 
show all references
References:
[1] 
D. K. Arrowsmith and C. M. Place, "An Introduction to Dynamical Systems," Cambridge University Press, Cambridge, 1990. 
[2] 
F. Balibrea and J. C. Valverde, Bifurcations under nondegenerated conditions of higher degree and a new simple proof of the HopfNeimarkSacker bifurcation theorem, J. Math. Anal. Appl., 237 (1999), 93105. doi: 10.1006/jmaa.1999.6460. 
[3] 
F. Balibrea and J. C. Valverde, Cusp and generalized flip bifurcations under higher degree conditions, Nonlinear Anal., 52 (2003), 405419. doi: 10.1016/S0362546X(01)009087. 
[4] 
N. Elezović, V. Županović and D. Žubrinić, Box dimension of trajectories of some discrete dynamical systems, Chaos Solitons Fractals, 34 (2007), 244252. doi: 10.1016/j.chaos.2006.03.060. 
[5] 
K. Falconer, "Fractal Geometry: Mathematical Foundations and Applications," John Wiley & Sons, Ltd., Chichester, 1990. 
[6] 
Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," 2^{nd} edition, Applied Mathematical Sciences, 112, SpringerVerlag, New York, 1998. 
[7] 
M. L. Lapidus and C. Pomerance, The Riemann zetafunction and the onedimensional WeylBerry conjecture for fractal drums, Proc. London Math. Soc. (3), 66 (1993), 4169. 
[8] 
P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995. 
[9] 
J. Palis and F. Takens, "Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors," Cambridge Studies in Advanced Mathematics, 35, Cambridge University Press, Cambridge, 1993. 
[10] 
M. Pašić, MinkowskiBouligand dimension of solutions of the onedimensional $p$Laplacian, J. Differential Equations, 190 (2003), 268305. doi: 10.1016/S00220396(02)001493. 
[11] 
M. Pašić, D. Žubrinić and V. Županović, Oscillatory and phase dimensions of solutions of some secondorder differential equations, Bull. Sci. Math., 133 (2009), 859874. 
[12] 
L. Perko, "Differential Equations and Dynamical Systems," 2^{nd} edition, Texts in Applied Mathematics, 7, SpringerVerlag, New York, 1996. 
[13] 
C. Tricot, "Curves and Fractal Dimension," With a foreword by Michel Mendès France, SpringerVerlag, New York, 1995. 
[14] 
S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," 2^{nd} edition, Texts in Applied Mathematics, 2, SpringerVerlag, New York, 2003. 
[15] 
D. Žubrinić, Analysis of Minkowski content of fractal sets and applications,, Real Anal. Exchange, 31 (): 315. 
[16] 
D. Žubrinić and V. Županović, Fractal dimension in dynamics, in "Encyclopedia of Math. Physics" (eds. J.P. Françoise, G. L. Naber and S. T. Tsou), Academic Press/Elsevier Science, Oxford, (2006), 394402. 
[17] 
D. Žubrinić and V. Županović, Poincaré map in fractal analysis of spiral trajectories of planar vector fields, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 947960. 
[18] 
D. Žubrinić and V. Županović, Fractal analysis of spiral trajectories of some planar vector fields, Bull. Sci. Math., 129 (2005), 457485. 
[19] 
D. Žubrinić and V. Županović, Fractal analysis of spiral trajectories of some vector fields in $\mathbbR^3$, C. R. Math. Acad. Sci. Paris, 342 (2006), 959963. 
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