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The snapback repellers for chaos in multi-dimensional maps

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  • 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.

    Mathematics Subject Classification: Primary: 37C29, 37C70, 34C28.

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  • [1] G. Alefeld and J. Herzberger, Introduction to Interval Computations, in Academic Press, NY, 1983.
    [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.
    [3] Z. AraiW. KaliesH. KokubuK. MischaikowH. 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.
    [4] J. R. BeddingtonC. A. Free and J. H. Lawton, Dynamic complexity in predator-prey models framed in difference equations, Nature, 255 (1975), 58-60. 
    [5] G. ChenS.-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.
    [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.
    [7] L. Gardini and F. Tramontana, Snapback repellers and chaotic attractors, Physical Rev. E, 81 (2010), 046202, 5pp.  doi: 10.1103/PhysRevE.81.046202.
    [8] L. GardiniI. SushkoV. 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.
    [9] Z. JingZ. Jia and Y. Chang, Chaos behavior in the discrete FitzHugh nerve system, China Set. A-Math, 44 (2001), 1571-1578.  doi: 10.1007/BF02880796.
    [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.
    [11] M.-C. LiM.-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.
    [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.
    [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.
    [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.
    [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.
    [16] R. E. Moore and F. Bierbaum, Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.
    [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.
    [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.
    [19] J. Sugie, Nonexistence of periodic solutions for the FitzHugh nerve system, Quart. Appl. Math., 49 (1991), 543-554.  doi: 10.1090/qam/1121685.
    [20] Y. ZhangQ. ZhangL. 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.
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