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Properties of basins of attraction for planar discrete cooperative maps
Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881-0816, USA |
It is shown that locally asymptotically stable equilibria of planar cooperative or competitive maps have basin of attraction $ \mathcal{B} $ with relatively simple geometry: the boundary of each component of $ \mathcal{B} $ consists of the union of two unordered curves, and the components of $ \mathcal{B} $ are not comparable as sets. The boundary curves are Lipschitz if the map is of class $ C^1 $. Further, if a periodic point is in $ \partial \mathcal{B} $, then $ \partial\mathcal{B} $ is tangential to the line through the point with direction given by the eigenvector associated with the smaller characteristic value of the map at the point. Examples are given.
References:
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A. Berger and A. Duh,
Global saddle-type dynamics for convex second-order difference equations, J. Difference Equ. Appl., 23 (2017), 1807-1823.
doi: 10.1080/10236198.2017.1367390. |
[2] |
A. Brett and M. R. S. Kulenović,
Basins of attraction of equilibrium points of monotone difference equations, Sarajevo J. Math., 5 (2009), 211-233.
|
[3] |
A. Cima, A. Gasull and V. Mañosa,
Basin of attraction of triangular maps with applications, J. Difference Equ. Appl., 20 (2014), 423-437.
doi: 10.1080/10236198.2013.852187. |
[4] |
S. Elaydi, E. Kwessi and G. Livadiotis,
Hierarchical competition models with the Allee effect III: Multispecies, J. Biol. Dyn., 12 (2018), 271-287.
doi: 10.1080/17513758.2018.1439537. |
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S. Elaydi, An Introduction to Difference Equations, Third edition, Undergraduate Texts in Mathematics, Springer, New York, 2005. |
[6] |
M. Garić-Demirović, M. R. S. Kulenović and M. Nurkanović,
Basins of attraction of certain homogeneous second order quadratic fractional difference equation, J. Concr. Appl. Math., 13 (2015), 35-50.
|
[7] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1990. |
[8] |
P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991. |
[9] |
M. W. Hirsch and H. Smith,
Monotone dynamical systems, Handbook of Differential Equations: Ordinary Differential Equations, Elsevier B. V., Amsterdam, 2 (2005), 239-357.
|
[10] |
M. W. Hirsch and H. Smith,
Monotone maps: A review, J. Difference Equ. Appl., 11 (2005), 379-398.
doi: 10.1080/10236190412331335445. |
[11] |
G. L. Karakostas,
The dynamics of a cooperative difference system with coefficient a Metzler matrix, J. Difference Equ. Appl., 20 (2014), 685-693.
doi: 10.1080/10236198.2013.799153. |
[12] |
M. R. S. Kulenović and O. Merino,
Global bifurcations for competitive systems in the plane, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 133-149.
doi: 10.3934/dcdsb.2009.12.133. |
[13] |
M. R. S. Kulenović and O. Merino,
Invariant manifolds for planar competitive and cooperative maps, J. Difference Equ. Appl., 24 (2018), 898-915.
doi: 10.1080/10236198.2018.1438418. |
[14] |
M. R. S. Kulenović, O. Merino and M. Nurkanović,
Global dynamics of certain competitive system in the plane, J. Difference Equ. Appl., 18 (2012), 1951-1966.
doi: 10.1080/10236198.2011.605357. |
[15] |
M. R. S. Kulenović, S. Moranjkić and Z. Nurkanović,
Global dynamics and bifurcation of a perturbed sigmoid Beverton-Holt difference equation, Math. Methods Appl. Sci., 39 (2016), 2696-2715.
doi: 10.1002/mma.3722. |
[16] |
G. Livadiotis, L. Assas, S. Elaydi, E. Kwessi and D. Ribble,
Competition models with the Allee effect, J. Difference Equ. Appl., 20 (2015), 1127-1151.
doi: 10.1080/10236198.2014.897341. |
[17] |
S. W. McDonald, C. Grebogi, E. Ott and J. A. Yorke,
Fractal basin boundaries, Phys. D, 17 (1985), 125-153.
doi: 10.1016/0167-2789(85)90001-6. |
[18] |
J. W. Milnor, Attractor, Scholarpedia, 1 (2006), 1815. Google Scholar |
[19] |
H. E. Nusse and J. A. Yorke,
Basins of attraction, Science, 271 (1996), 1376-1380.
doi: 10.1126/science.271.5254.1376. |
[20] |
H. E. Nusse and J. A. Yorke,
The structure of basins of attraction and their trapping regions, Ergodic Theory Dynam. Systems, 17 (1997), 463-481.
doi: 10.1017/S0143385797069782. |
[21] |
H. E. Nusse and J. A. Yorke,
Characterizing the basins with the most entangled boundaries, Ergodic Theory Dynam. Systems, 23 (2003), 895-906.
doi: 10.1017/S0143385702001360. |
[22] |
H. E. Nusse and J. A. Yorke,
Bifurcations of basins of attraction from the view point of prime ends, Topology Appl., 154 (2007), 2567-2579.
doi: 10.1016/j.topol.2006.07.019. |
[23] |
H. L. Smith,
Planar competitive and cooperative difference equations, J. Difference Equ. Appl., 3 (1998), 335-357.
doi: 10.1080/10236199708808108. |
show all references
References:
[1] |
A. Berger and A. Duh,
Global saddle-type dynamics for convex second-order difference equations, J. Difference Equ. Appl., 23 (2017), 1807-1823.
doi: 10.1080/10236198.2017.1367390. |
[2] |
A. Brett and M. R. S. Kulenović,
Basins of attraction of equilibrium points of monotone difference equations, Sarajevo J. Math., 5 (2009), 211-233.
|
[3] |
A. Cima, A. Gasull and V. Mañosa,
Basin of attraction of triangular maps with applications, J. Difference Equ. Appl., 20 (2014), 423-437.
doi: 10.1080/10236198.2013.852187. |
[4] |
S. Elaydi, E. Kwessi and G. Livadiotis,
Hierarchical competition models with the Allee effect III: Multispecies, J. Biol. Dyn., 12 (2018), 271-287.
doi: 10.1080/17513758.2018.1439537. |
[5] |
S. Elaydi, An Introduction to Difference Equations, Third edition, Undergraduate Texts in Mathematics, Springer, New York, 2005. |
[6] |
M. Garić-Demirović, M. R. S. Kulenović and M. Nurkanović,
Basins of attraction of certain homogeneous second order quadratic fractional difference equation, J. Concr. Appl. Math., 13 (2015), 35-50.
|
[7] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1990. |
[8] |
P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991. |
[9] |
M. W. Hirsch and H. Smith,
Monotone dynamical systems, Handbook of Differential Equations: Ordinary Differential Equations, Elsevier B. V., Amsterdam, 2 (2005), 239-357.
|
[10] |
M. W. Hirsch and H. Smith,
Monotone maps: A review, J. Difference Equ. Appl., 11 (2005), 379-398.
doi: 10.1080/10236190412331335445. |
[11] |
G. L. Karakostas,
The dynamics of a cooperative difference system with coefficient a Metzler matrix, J. Difference Equ. Appl., 20 (2014), 685-693.
doi: 10.1080/10236198.2013.799153. |
[12] |
M. R. S. Kulenović and O. Merino,
Global bifurcations for competitive systems in the plane, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 133-149.
doi: 10.3934/dcdsb.2009.12.133. |
[13] |
M. R. S. Kulenović and O. Merino,
Invariant manifolds for planar competitive and cooperative maps, J. Difference Equ. Appl., 24 (2018), 898-915.
doi: 10.1080/10236198.2018.1438418. |
[14] |
M. R. S. Kulenović, O. Merino and M. Nurkanović,
Global dynamics of certain competitive system in the plane, J. Difference Equ. Appl., 18 (2012), 1951-1966.
doi: 10.1080/10236198.2011.605357. |
[15] |
M. R. S. Kulenović, S. Moranjkić and Z. Nurkanović,
Global dynamics and bifurcation of a perturbed sigmoid Beverton-Holt difference equation, Math. Methods Appl. Sci., 39 (2016), 2696-2715.
doi: 10.1002/mma.3722. |
[16] |
G. Livadiotis, L. Assas, S. Elaydi, E. Kwessi and D. Ribble,
Competition models with the Allee effect, J. Difference Equ. Appl., 20 (2015), 1127-1151.
doi: 10.1080/10236198.2014.897341. |
[17] |
S. W. McDonald, C. Grebogi, E. Ott and J. A. Yorke,
Fractal basin boundaries, Phys. D, 17 (1985), 125-153.
doi: 10.1016/0167-2789(85)90001-6. |
[18] |
J. W. Milnor, Attractor, Scholarpedia, 1 (2006), 1815. Google Scholar |
[19] |
H. E. Nusse and J. A. Yorke,
Basins of attraction, Science, 271 (1996), 1376-1380.
doi: 10.1126/science.271.5254.1376. |
[20] |
H. E. Nusse and J. A. Yorke,
The structure of basins of attraction and their trapping regions, Ergodic Theory Dynam. Systems, 17 (1997), 463-481.
doi: 10.1017/S0143385797069782. |
[21] |
H. E. Nusse and J. A. Yorke,
Characterizing the basins with the most entangled boundaries, Ergodic Theory Dynam. Systems, 23 (2003), 895-906.
doi: 10.1017/S0143385702001360. |
[22] |
H. E. Nusse and J. A. Yorke,
Bifurcations of basins of attraction from the view point of prime ends, Topology Appl., 154 (2007), 2567-2579.
doi: 10.1016/j.topol.2006.07.019. |
[23] |
H. L. Smith,
Planar competitive and cooperative difference equations, J. Difference Equ. Appl., 3 (1998), 335-357.
doi: 10.1080/10236199708808108. |








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