# American Institute of Mathematical Sciences

• Previous Article
Computation of whiskered invariant tori and their associated manifolds: New fast algorithms
• DCDS Home
• This Issue
• Next Article
Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one
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 & Continuous Dynamical Systems - A, 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, (1990).   Google Scholar [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.  doi: 10.1006/jmaa.1999.6460.  Google Scholar [3] F. Balibrea and J. C. Valverde, Cusp and generalized flip bifurcations under higher degree conditions,, Nonlinear Anal., 52 (2003), 405.  doi: 10.1016/S0362-546X(01)00908-7.  Google Scholar [4] N. Elezović, V. Županović and D. Žubrinić, Box dimension of trajectories of some discrete dynamical systems,, Chaos Solitons Fractals, 34 (2007), 244.  doi: 10.1016/j.chaos.2006.03.060.  Google Scholar [5] K. Falconer, "Fractal Geometry: Mathematical Foundations and Applications,", John Wiley & Sons, (1990).   Google Scholar [6] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", 2nd edition, 112 (1998).   Google Scholar [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.   Google Scholar [8] P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability,", Cambridge Studies in Advanced Mathematics, 44 (1995).   Google Scholar [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 (1993).   Google Scholar [10] M. Pašić, Minkowski-Bouligand dimension of solutions of the one-dimensional $p$-Laplacian,, J. Differential Equations, 190 (2003), 268.  doi: 10.1016/S0022-0396(02)00149-3.  Google Scholar [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.   Google Scholar [12] L. Perko, "Differential Equations and Dynamical Systems,", 2nd edition, 7 (1996).   Google Scholar [13] C. Tricot, "Curves and Fractal Dimension,", With a foreword by Michel Mendès France, (1995).   Google Scholar [14] S. Wiggins, "Introduction to Applied Non-linear Dynamical Systems and Chaos,", 2nd edition, 2 (2003).   Google Scholar [15] D. Žubrinić, Analysis of Minkowski content of fractal sets and applications,, Real Anal. Exchange, 31 (): 315.   Google Scholar [16] D. Žubrinić and V. Županović, Fractal dimension in dynamics,, in, (2006), 394.   Google Scholar [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.   Google Scholar [18] D. Žubrinić and V. Županović, Fractal analysis of spiral trajectories of some planar vector fields,, Bull. Sci. Math., 129 (2005), 457.   Google Scholar [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.   Google Scholar

show all references

##### References:
 [1] D. K. Arrowsmith and C. M. Place, "An Introduction to Dynamical Systems,", Cambridge University Press, (1990).   Google Scholar [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.  doi: 10.1006/jmaa.1999.6460.  Google Scholar [3] F. Balibrea and J. C. Valverde, Cusp and generalized flip bifurcations under higher degree conditions,, Nonlinear Anal., 52 (2003), 405.  doi: 10.1016/S0362-546X(01)00908-7.  Google Scholar [4] N. Elezović, V. Županović and D. Žubrinić, Box dimension of trajectories of some discrete dynamical systems,, Chaos Solitons Fractals, 34 (2007), 244.  doi: 10.1016/j.chaos.2006.03.060.  Google Scholar [5] K. Falconer, "Fractal Geometry: Mathematical Foundations and Applications,", John Wiley & Sons, (1990).   Google Scholar [6] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", 2nd edition, 112 (1998).   Google Scholar [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.   Google Scholar [8] P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability,", Cambridge Studies in Advanced Mathematics, 44 (1995).   Google Scholar [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 (1993).   Google Scholar [10] M. Pašić, Minkowski-Bouligand dimension of solutions of the one-dimensional $p$-Laplacian,, J. Differential Equations, 190 (2003), 268.  doi: 10.1016/S0022-0396(02)00149-3.  Google Scholar [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.   Google Scholar [12] L. Perko, "Differential Equations and Dynamical Systems,", 2nd edition, 7 (1996).   Google Scholar [13] C. Tricot, "Curves and Fractal Dimension,", With a foreword by Michel Mendès France, (1995).   Google Scholar [14] S. Wiggins, "Introduction to Applied Non-linear Dynamical Systems and Chaos,", 2nd edition, 2 (2003).   Google Scholar [15] D. Žubrinić, Analysis of Minkowski content of fractal sets and applications,, Real Anal. Exchange, 31 (): 315.   Google Scholar [16] D. Žubrinić and V. Županović, Fractal dimension in dynamics,, in, (2006), 394.   Google Scholar [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.   Google Scholar [18] D. Žubrinić and V. Županović, Fractal analysis of spiral trajectories of some planar vector fields,, Bull. Sci. Math., 129 (2005), 457.   Google Scholar [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.   Google Scholar
 [1] Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257 [2] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [3] Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344 [4] Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 299-319. doi: 10.3934/dcdss.2020322 [5] Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073 [6] João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 [7] Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 [8] Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342 [9] Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 [10] Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366 [11] Lei Liu, Li Wu. Multiplicity of closed characteristics on $P$-symmetric compact convex hypersurfaces in $\mathbb{R}^{2n}$. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378 [12] Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460 [13] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [14] Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331 [15] Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325 [16] Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $\beta$-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267 [17] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 [18] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [19] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [20] Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268

2019 Impact Factor: 1.338