• Previous Article
    Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one
  • DCDS Home
  • This Issue
  • Next Article
    Computation of whiskered invariant tori and their associated manifolds: New fast algorithms
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]

Zhiqin Qiao, Deming Zhu, Qiuying Lu. Bifurcation of a heterodimensional cycle with weak inclination flip. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1009-1025. doi: 10.3934/dcdsb.2012.17.1009

[2]

Alexey A. Petrov, Sergei Yu. Pilyugin. Shadowing near nonhyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3761-3772. doi: 10.3934/dcds.2014.34.3761

[3]

Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061

[4]

Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147

[5]

Eric Benoît. Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 911-929. doi: 10.3934/dcdss.2009.2.911

[6]

Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846

[7]

Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129

[8]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297

[9]

Fabio Scalco Dias, Luis Fernando Mello. The center--focus problem and small amplitude limit cycles in rigid systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1627-1637. doi: 10.3934/dcds.2012.32.1627

[10]

Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439

[11]

Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447

[12]

Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017

[13]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709

[14]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692

[15]

Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979

[16]

Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381

[17]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805

[18]

Alejandro Allendes, Alexander Quaas. Multiplicity results for extremal operators through bifurcation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 51-65. doi: 10.3934/dcds.2011.29.51

[19]

Freddy Dumortier, Robert Roussarie. Bifurcation of relaxation oscillations in dimension two. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 631-674. doi: 10.3934/dcds.2007.19.631

[20]

Wacław Marzantowicz, Piotr Maciej Przygodzki. Finding periodic points of a map by use of a k-adic expansion. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 495-514. doi: 10.3934/dcds.1999.5.495

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]