# American Institute of Mathematical Sciences

February  2008, 22(1&2): 63-74. doi: 10.3934/dcds.2008.22.63

## Chaotic and nonchaotic mushrooms

 1 ABC Math Program and School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, United States

Received  March 2007 Revised  May 2008 Published  June 2008

In the paper [4] were introduced visual and simple classes of dynamical systems with divided phase space, i.e., with the coexistence of chaotic components of positive measure and islands of regular dynamics. These classes consist of mushroom billiards and their modifications. Here we generalize the construction of mushroom billiards and besides present another class of simple and visual billiards with divided phase space.
Citation: Leonid A. Bunimovich. Chaotic and nonchaotic mushrooms. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 63-74. doi: 10.3934/dcds.2008.22.63
 [1] Péter Bálint, Imre Péter Tóth. Hyperbolicity in multi-dimensional Hamiltonian systems with applications to soft billiards. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 37-59. doi: 10.3934/dcds.2006.15.37 [2] Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048 [3] Oskar Weinberger, Peter Ashwin. From coupled networks of systems to networks of states in phase space. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 2021-2041. doi: 10.3934/dcdsb.2018193 [4] Božidar Jovanović, Vladimir Jovanović. Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5163-5190. doi: 10.3934/dcds.2017224 [5] Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214 [6] Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form $H = H_1(x)+H_2(y)$. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 75-113. doi: 10.3934/dcds.2019004 [7] Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201 [8] Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855 [9] Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541 [10] Claudio Meneses. Linear phase space deformations with angular momentum symmetry. Journal of Geometric Mechanics, 2019, 11 (1) : 45-58. doi: 10.3934/jgm.2019003 [11] Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2019-2034. doi: 10.3934/dcdss.2019130 [12] Marcin Mazur, Jacek Tabor, Piotr Kościelniak. Semi-hyperbolicity and hyperbolicity. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 1029-1038. doi: 10.3934/dcds.2008.20.1029 [13] Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 [14] K. Tintarev. Critical values and minimal periods for autonomous Hamiltonian systems. Discrete & Continuous Dynamical Systems, 1995, 1 (3) : 389-400. doi: 10.3934/dcds.1995.1.389 [15] Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249 [16] Abed Bounemoura, Edouard Pennamen. Instability for a priori unstable Hamiltonian systems: A dynamical approach. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 753-793. doi: 10.3934/dcds.2012.32.753 [17] Carles Simó. Measuring the total amount of chaos in some Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5135-5164. doi: 10.3934/dcds.2014.34.5135 [18] Tianqing An, Zhi-Qiang Wang. Periodic solutions of Hamiltonian systems with anisotropic growth. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1069-1082. doi: 10.3934/cpaa.2010.9.1069 [19] Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623 [20] Lora Billings, Erik M. Bollt, David Morgan, Ira B. Schwartz. Stochastic global bifurcation in perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 123-132. doi: 10.3934/proc.2003.2003.123

2020 Impact Factor: 1.392