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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

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