American Institute of Mathematical Sciences

Advanced
May  2004, 4(2): 391-406. doi: 10.3934/dcdsb.2004.4.391

On the parametric dependences of a class of non-linear singular maps

T. Gilbert 1, and  J. R. Dorfman 1,

1. 

Department of Physics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, United States, United States

Received  October 2002 Revised  June 2003 Published  February 2004

We discuss a two-parameter family of maps that generalize piecewise linear, expanding maps of the circle. One parameter measures the effect of a non-linearity which bends the branches of the linear map. The second parameter rotates points by a fixed angle. For small values of the nonlinearity parameter, we compute the invariant measure and show that it has a singular density to first order in the nonlinearity parameter. Its Fourier modes have forms similar to the Weierstrass function. We discuss the consequences of this singularity on the Lyapunov exponents and on the transport properties of the corresponding multibaker map. For larger non-linearities, the map becomes non-hyperbolic and exhibits a series of period-adding bifurcations.
Keywords: invariant measures, Singular maps, weierstrass function, period-adding bifurcations..
Mathematics Subject Classification: 37D5.
Citation: T. Gilbert, J. R. Dorfman. On the parametric dependences of a class of non-linear singular maps. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 391-406. doi: 10.3934/dcdsb.2004.4.391
[1]

Thorsten Hüls. A model function for non-autonomous bifurcations of maps. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 351-363. doi: 10.3934/dcdsb.2007.7.351
[2]

Arno Berger, Roland Zweimüller. Invariant measures for general induced maps and towers. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3885-3901. doi: 10.3934/dcds.2013.33.3885
[3]

Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795
[4]

Kaijen Cheng, Kenneth Palmer, Yuh-Jenn Wu. Period 3 and chaos for unimodal maps. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1933-1949. doi: 10.3934/dcds.2014.34.1933
[5]

Oliver Jenkinson. Optimization and majorization of invariant measures. Electronic Research Announcements, 2007, 13: 1-12.

[6]

Siniša Slijepčević. Stability of invariant measures. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1345-1363. doi: 10.3934/dcds.2009.24.1345
[7]

Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185
[8]

Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811
[9]

Marcus Pivato. Invariant measures for bipermutative cellular automata. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 723-736. doi: 10.3934/dcds.2005.12.723
[10]

Yakov Pesin, Samuel Senti. Equilibrium measures for maps with inducing schemes. Journal of Modern Dynamics, 2008, 2 (3) : 397-430. doi: 10.3934/jmd.2008.2.397
[11]

Vítor Araújo, Ali Tahzibi. Physical measures at the boundary of hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 849-876. doi: 10.3934/dcds.2008.20.849
[12]

Leonardo Mora. Homoclinic bifurcations, fat attractors and invariant curves. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1133-1148. doi: 10.3934/dcds.2003.9.1133
[13]

Alexander O. Brown, Christopher S. Tang. The impact of alternative performance measures on single-period inventory policy. Journal of Industrial & Management Optimization, 2006, 2 (3) : 297-318. doi: 10.3934/jimo.2006.2.297
[14]

Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597
[15]

Amir Mohammadi. Measures invariant under horospherical subgroups in positive characteristic. Journal of Modern Dynamics, 2011, 5 (2) : 237-254. doi: 10.3934/jmd.2011.5.237
[16]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017
[17]

Vasso Anagnostopoulou. Stochastic dominance for shift-invariant measures. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 667-682. doi: 10.3934/dcds.2019027
[18]

Gamaliel Blé. External arguments and invariant measures for the quadratic family. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 241-260. doi: 10.3934/dcds.2004.11.241
[19]

Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94.

[20]

Huichi Huang. Fourier coefficients of $\times p$-invariant measures. Journal of Modern Dynamics, 2017, 11: 551-562. doi: 10.3934/jmd.2017021

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences