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The snapback repellers for chaos in multi-dimensional maps
| 1. | Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210, USA |
| 2. | Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan 300 |
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.
References:
| [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. 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. |
| [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.
|
| [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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. 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. |
show all references
References:
| [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. 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. |
| [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.
|
| [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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. 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. |
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