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April  2012, 32(4): 1287-1307. doi: 10.3934/dcds.2012.32.1287

Box dimension and bifurcations of one-dimensional discrete dynamical systems

1. 

Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia

Received  March 2010 Revised  September 2011 Published  October 2011

This paper is devoted to study the box dimension of the orbits of one-dimensional discrete dynamical systems and their bifurcations in nonhyperbolic fixed points. It is already known that there is a connection between some bifurcations in a nonhyperbolic fixed point of one-dimensional maps, and the box dimension of the orbits near that point. The main purpose of this paper is to generalize that result to the one-dimensional maps of class $C^{k}$ and apply it to one and two-parameter bifurcations of maps with the generalized nondegeneracy conditions. These results show that the value of the box dimension changes at the bifurcation point, and depends only on the order of the nondegeneracy condition. Furthermore, we obtain the reverse result, that is, the order of the nondegeneracy of a map in a nonhyperbolic fixed point can be obtained from the box dimension of a orbit near that point. This reverse result can be applied to the continuous planar dynamical systems by using the Poincaré map, in order to get the multiplicity of a weak focus or nonhyperbolic limit cycle. We also apply the main result to the bifurcations of nonhyperbolic periodic orbits in the plane.
Citation: Lana Horvat Dmitrović. Box dimension and bifurcations of one-dimensional discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1287-1307. doi: 10.3934/dcds.2012.32.1287
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 Hopf-Neimark-Sacker bifurcation theorem, J. Math. Anal. Appl., 237 (1999), 93-105. 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), 405-419. doi: 10.1016/S0362-546X(01)00908-7.

[4]

N. Elezović, V. Županović and D. Žubrinić, Box dimension of trajectories of some discrete dynamical systems, Chaos Solitons Fractals, 34 (2007), 244-252. 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," 2nd edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998.

[7]

M. L. Lapidus and C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums, Proc. London Math. Soc. (3), 66 (1993), 41-69.

[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, Minkowski-Bouligand dimension of solutions of the one-dimensional $p$-Laplacian, J. Differential Equations, 190 (2003), 268-305. doi: 10.1016/S0022-0396(02)00149-3.

[11]

M. Pašić, D. Žubrinić and V. Županović, Oscillatory and phase dimensions of solutions of some second-order differential equations, Bull. Sci. Math., 133 (2009), 859-874.

[12]

L. Perko, "Differential Equations and Dynamical Systems," 2nd edition, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 1996.

[13]

C. Tricot, "Curves and Fractal Dimension," With a foreword by Michel Mendès France, Springer-Verlag, New York, 1995.

[14]

S. Wiggins, "Introduction to Applied Non-linear Dynamical Systems and Chaos," 2nd edition, Texts in Applied Mathematics, 2, Springer-Verlag, 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), 394-402.

[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), 947-960.

[18]

D. Žubrinić and V. Županović, Fractal analysis of spiral trajectories of some planar vector fields, Bull. Sci. Math., 129 (2005), 457-485.

[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), 959-963.

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 Hopf-Neimark-Sacker bifurcation theorem, J. Math. Anal. Appl., 237 (1999), 93-105. 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), 405-419. doi: 10.1016/S0362-546X(01)00908-7.

[4]

N. Elezović, V. Županović and D. Žubrinić, Box dimension of trajectories of some discrete dynamical systems, Chaos Solitons Fractals, 34 (2007), 244-252. 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," 2nd edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998.

[7]

M. L. Lapidus and C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums, Proc. London Math. Soc. (3), 66 (1993), 41-69.

[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, Minkowski-Bouligand dimension of solutions of the one-dimensional $p$-Laplacian, J. Differential Equations, 190 (2003), 268-305. doi: 10.1016/S0022-0396(02)00149-3.

[11]

M. Pašić, D. Žubrinić and V. Županović, Oscillatory and phase dimensions of solutions of some second-order differential equations, Bull. Sci. Math., 133 (2009), 859-874.

[12]

L. Perko, "Differential Equations and Dynamical Systems," 2nd edition, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 1996.

[13]

C. Tricot, "Curves and Fractal Dimension," With a foreword by Michel Mendès France, Springer-Verlag, New York, 1995.

[14]

S. Wiggins, "Introduction to Applied Non-linear Dynamical Systems and Chaos," 2nd edition, Texts in Applied Mathematics, 2, Springer-Verlag, 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), 394-402.

[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), 947-960.

[18]

D. Žubrinić and V. Županović, Fractal analysis of spiral trajectories of some planar vector fields, Bull. Sci. Math., 129 (2005), 457-485.

[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), 959-963.

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