American Institute of Mathematical Sciences

Advanced
June & December  2018, 5(1&2): 81-92. doi: 10.3934/jcd.2018004

The snapback repellers for chaos in multi-dimensional maps

Kang-Ling Liao 1, , Chih-Wen Shih 2, and  Chi-Jer Yu 2,

1. 

Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210, USA
2. 

Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan 300

Published  November 2018

The key of Marotto's theorem on chaos for multi-dimensional maps is the existence of snapback repeller. For practical application of the theory, locating a computable repelling neighborhood of the repelling fixed point has thus become the key issue. For some multi-dimensional maps $F$, basic information of $F$ is not sufficient to indicate the existence of snapback repeller for $F$. In this investigation, for a repeller $\bar{\bf z}$ of $F$, we start from estimating the repelling neighborhood of $\bar{\bf z}$ under $F^{k}$ for some $k ≥ 2$, by a theory built on the first or second derivative of $F^k$. By employing the Interval Arithmetic computation, we locate a snapback point ${\bf z}_0$ in this repelling neighborhood and examine the nonzero determinant condition for the Jacobian of $F$ along the orbit through ${\bf z}_0$. With this new approach, we are able to conclude the existence of snapback repellers under the valid definition, hence chaotic behaviors, in a discrete-time predator-prey model, a population model, and the FitzHugh nerve model.

Keywords: Snapback repeller, homoclinic orbit, chaos, Interval arithmetic.
Mathematics Subject Classification: Primary: 37C29, 37C70, 34C28.
Citation: Kang-Ling Liao, Chih-Wen Shih, Chi-Jer Yu. The snapback repellers for chaos in multi-dimensional maps. Journal of Computational Dynamics, 2018, 5 (1&2) : 81-92. doi: 10.3934/jcd.2018004
References:
[1]

G. Alefeld and J. Herzberger, Introduction to Interval Computations, in Academic Press, NY, 1983.  Google Scholar

[2]

G. Alefeld, Inclusion methods for systems of nonlinear equations-the interval Newton method and modifications, Topics in Validated Computations, Elsevier, Amsterdam, 5 (1994), 7-26.  Google Scholar

[3]

Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka and P. Pilarczyk, A database schema for the analysis of global dynamics of multi parameter systems, SIAM J. Appl. Dyn. Syst., 8 (2009), 757-789.  doi: 10.1137/080734935.  Google Scholar

[4]

J. R. Beddington, C. A. Free and J. H. Lawton, Dynamic complexity in predator-prey models framed in difference equations, Nature, 255 (1975), 58-60.   Google Scholar

[5]

G. Chen, S.-B. Hsu and J. Zhou, Snapback repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the span, J. Math. Phys., 39 (1998), 6459-6489.  doi: 10.1063/1.532670.  Google Scholar

[6]

S. S. Chen and C. W. Shih, Transversal homoclinic orbits in a transiently chaotic neural network, Chaos, 12 (2002), 654-671.  doi: 10.1063/1.1488895.  Google Scholar

[7]

L. Gardini and F. Tramontana, Snapback repellers and chaotic attractors, Physical Rev. E, 81 (2010), 046202, 5pp.  doi: 10.1103/PhysRevE.81.046202.  Google Scholar

[8]

L. Gardini, I. Sushko, V. Avrutin and M. Schanz, Critical homoclinic orbits lead to snapback repellers, Chaos, Solitons, and Fractals, 44 (2011), 433-449.  doi: 10.1016/j.chaos.2011.03.004.  Google Scholar

[9]

Z. Jing, Z. Jia and Y. Chang, Chaos behavior in the discrete FitzHugh nerve system, China Set. A-Math, 44 (2001), 1571-1578.  doi: 10.1007/BF02880796.  Google Scholar

[10]

C. Li and G. Chen, An improved version of the Marotto theorem, Chaos, Solitons, and Fractals, 18 (2003), 69-77.  doi: 10.1016/S0960-0779(02)00605-7.  Google Scholar

[11]

M.-C. Li, M.-J. Lyu and P. Zgliczyński, Topological entropy for multidimensional perturbations of snapback repellers and one-dimensional maps, Nonlinearity, 21 (2008), 2555-2567.  doi: 10.1088/0951-7715/21/11/005.  Google Scholar

[12]

T.-Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.  doi: 10.1080/00029890.1975.11994008.  Google Scholar

[13]

K.-L. Liao and C.-W. Shih, Snapback repellers and homoclinic orbits for multi-dimensional maps, J. Math. Anal. Appl., 386 (2012), 387-400.  doi: 10.1016/j.jmaa.2011.08.011.  Google Scholar

[14]

F. R. Marotto, Snapback repellers imply chaos in $\mathbb{R}^{n}$, J. Math. Anal. Appl., 63 (1978), 199-223.  doi: 10.1016/0022-247X(78)90115-4.  Google Scholar

[15]

F. R. Marotto, On redefining a snapback repeller, Chaos, Solitons, and Fractals, 25 (2005), 25-28.  doi: 10.1016/j.chaos.2004.10.003.  Google Scholar

[16]

R. E. Moore and F. Bierbaum, Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.  Google Scholar

[17]

C.-C. Peng, Numerical computation of orbits and rigorous verification of existence of snapback repellers, Chaos, 17 (2007), 013107, 8pp.  doi: 10.1063/1.2430907.  Google Scholar

[18]

Y. Shi and P. Yu, Chaos induced by regular snapback repellers, J. Math. Anal. Appl., 337 (2008), 1480-1494.  doi: 10.1016/j.jmaa.2007.05.005.  Google Scholar

[19]

J. Sugie, Nonexistence of periodic solutions for the FitzHugh nerve system, Quart. Appl. Math., 49 (1991), 543-554.  doi: 10.1090/qam/1121685.  Google Scholar

[20]

Y. Zhang, Q. Zhang, L. Zhao and C. Yang, Dynamical behaviors and chaos control in a discrete functional response model, Chaos, Solitons, and Fractals, 34 (2007), 1318-1327.  doi: 10.1016/j.chaos.2006.04.032.  Google Scholar

show all references

References:
[1]

G. Alefeld and J. Herzberger, Introduction to Interval Computations, in Academic Press, NY, 1983.  Google Scholar

[2]

G. Alefeld, Inclusion methods for systems of nonlinear equations-the interval Newton method and modifications, Topics in Validated Computations, Elsevier, Amsterdam, 5 (1994), 7-26.  Google Scholar

[3]

Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka and P. Pilarczyk, A database schema for the analysis of global dynamics of multi parameter systems, SIAM J. Appl. Dyn. Syst., 8 (2009), 757-789.  doi: 10.1137/080734935.  Google Scholar

[4]

J. R. Beddington, C. A. Free and J. H. Lawton, Dynamic complexity in predator-prey models framed in difference equations, Nature, 255 (1975), 58-60.   Google Scholar

[5]

G. Chen, S.-B. Hsu and J. Zhou, Snapback repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the span, J. Math. Phys., 39 (1998), 6459-6489.  doi: 10.1063/1.532670.  Google Scholar

[6]

S. S. Chen and C. W. Shih, Transversal homoclinic orbits in a transiently chaotic neural network, Chaos, 12 (2002), 654-671.  doi: 10.1063/1.1488895.  Google Scholar

[7]

L. Gardini and F. Tramontana, Snapback repellers and chaotic attractors, Physical Rev. E, 81 (2010), 046202, 5pp.  doi: 10.1103/PhysRevE.81.046202.  Google Scholar

[8]

L. Gardini, I. Sushko, V. Avrutin and M. Schanz, Critical homoclinic orbits lead to snapback repellers, Chaos, Solitons, and Fractals, 44 (2011), 433-449.  doi: 10.1016/j.chaos.2011.03.004.  Google Scholar

[9]

Z. Jing, Z. Jia and Y. Chang, Chaos behavior in the discrete FitzHugh nerve system, China Set. A-Math, 44 (2001), 1571-1578.  doi: 10.1007/BF02880796.  Google Scholar

[10]

C. Li and G. Chen, An improved version of the Marotto theorem, Chaos, Solitons, and Fractals, 18 (2003), 69-77.  doi: 10.1016/S0960-0779(02)00605-7.  Google Scholar

[11]

M.-C. Li, M.-J. Lyu and P. Zgliczyński, Topological entropy for multidimensional perturbations of snapback repellers and one-dimensional maps, Nonlinearity, 21 (2008), 2555-2567.  doi: 10.1088/0951-7715/21/11/005.  Google Scholar

[12]

T.-Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.  doi: 10.1080/00029890.1975.11994008.  Google Scholar

[13]

K.-L. Liao and C.-W. Shih, Snapback repellers and homoclinic orbits for multi-dimensional maps, J. Math. Anal. Appl., 386 (2012), 387-400.  doi: 10.1016/j.jmaa.2011.08.011.  Google Scholar

[14]

F. R. Marotto, Snapback repellers imply chaos in $\mathbb{R}^{n}$, J. Math. Anal. Appl., 63 (1978), 199-223.  doi: 10.1016/0022-247X(78)90115-4.  Google Scholar

[15]

F. R. Marotto, On redefining a snapback repeller, Chaos, Solitons, and Fractals, 25 (2005), 25-28.  doi: 10.1016/j.chaos.2004.10.003.  Google Scholar

[16]

R. E. Moore and F. Bierbaum, Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.  Google Scholar

[17]

C.-C. Peng, Numerical computation of orbits and rigorous verification of existence of snapback repellers, Chaos, 17 (2007), 013107, 8pp.  doi: 10.1063/1.2430907.  Google Scholar

[18]

Y. Shi and P. Yu, Chaos induced by regular snapback repellers, J. Math. Anal. Appl., 337 (2008), 1480-1494.  doi: 10.1016/j.jmaa.2007.05.005.  Google Scholar

[19]

J. Sugie, Nonexistence of periodic solutions for the FitzHugh nerve system, Quart. Appl. Math., 49 (1991), 543-554.  doi: 10.1090/qam/1121685.  Google Scholar

[20]

Y. Zhang, Q. Zhang, L. Zhao and C. Yang, Dynamical behaviors and chaos control in a discrete functional response model, Chaos, Solitons, and Fractals, 34 (2007), 1318-1327.  doi: 10.1016/j.chaos.2006.04.032.  Google Scholar

[1]

Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019210
[2]

Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883
[3]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485
[4]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293
[5]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351
[6]

Bao Qing Hu, Song Wang. A novel approach in uncertain programming part I: new arithmetic and order relation for interval numbers. Journal of Industrial & Management Optimization, 2006, 2 (4) : 351-371. doi: 10.3934/jimo.2006.2.351
[7]

Marta Štefánková. Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3435-3443. doi: 10.3934/dcds.2016.36.3435
[8]

Martin Wechselberger, Warren Weckesser. Homoclinic clusters and chaos associated with a folded node in a stellate cell model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 829-850. doi: 10.3934/dcdss.2009.2.829
[9]

Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295
[10]

Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015
[11]

Tanja Eisner, Rainer Nagel. Arithmetic progressions -- an operator theoretic view. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 657-667. doi: 10.3934/dcdss.2013.6.657
[12]

Mehdi Pourbarat. On the arithmetic difference of middle Cantor sets. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4259-4278. doi: 10.3934/dcds.2018186
[13]

Vladimir Dragović, Milena Radnović. Pseudo-integrable billiards and arithmetic dynamics. Journal of Modern Dynamics, 2014, 8 (1) : 109-132. doi: 10.3934/jmd.2014.8.109
[14]

Gerhard Frey. Relations between arithmetic geometry and public key cryptography. Advances in Mathematics of Communications, 2010, 4 (2) : 281-305. doi: 10.3934/amc.2010.4.281
[15]

Wei Lin, Jianhong Wu, Guanrong Chen. Generalized snap-back repeller and semi-conjugacy to shift operators of piecewise continuous transformations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 103-119. doi: 10.3934/dcds.2007.19.103
[16]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91
[17]

Stefano Galatolo. Orbit complexity and data compression. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 477-486. doi: 10.3934/dcds.2001.7.477
[18]

Shiqiu Liu, Frédérique Oggier. On applications of orbit codes to storage. Advances in Mathematics of Communications, 2016, 10 (1) : 113-130. doi: 10.3934/amc.2016.10.113
[19]

Peng Sun. Minimality and gluing orbit property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4041-4056. doi: 10.3934/dcds.2019162
[20]

Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757

 Impact Factor: 

Tools

Metrics

  • PDF downloads (56)
  • HTML views (764)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences