
-
Previous Article
Time-domain analysis of forward obstacle scattering for elastic wave
- DCDS-B Home
- This Issue
-
Next Article
Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain
Bifurcation in the almost periodic $ 2 $D Ricker map
1. | Department of Mathematics, California State University Bakersfield, Bakersfield, CA 93311-1022, USA |
2. | Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA |
This paper studies bifurcations in the coupled $ 2 $ dimensional almost periodic Ricker map. We establish criteria for stability of an almost periodic solution in terms of the Lyapunov exponents of a corresponding dynamical system and use them to find a bifurcation function. We find that if the almost periodic coefficients of all the maps are identical, then the bifurcation function is the same as the one obtained in the one dimensional case treated earlier, and that this result holds in $ N $ dimension under modest coupling constraints. In the general two-dimensional case, we compute the Lyapunov exponents numerically and use them to examine the stability and bifurcations of the almost periodic solutions.
References:
[1] |
A. Avila, J. Santamaria, M. Viana and A. Wilkinson,
Cocycles over partially hyperbolic maps, Asterisque, 358 (2013), 1-12.
|
[2] |
E. C. Balreira, S. Elaydi and R. Luis,
Local stability implies global stability for the planar Ricker competition model, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 323-351.
doi: 10.3934/dcdsb.2014.19.323. |
[3] |
H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N. Y., 1947. |
[4] |
J. W. S. Cassel, An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45. Cambridge University Press, New York, 1957.
![]() |
[5] |
K. Chandrasekharan, Introduction to Analytic Number Theory, Number 148 in Die Grundlehren der matematishen Wissenshaft in Einzeldarstellung. Springer Verlag, New York, 1968. |
[6] |
J.-P. Eckmann and D. Ruelle,
Ergodic theory of chaos and strange attractors, Reviews of Modern Physics, 57 (1985), 617-656.
doi: 10.1103/RevModPhys.57.617. |
[7] |
S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005. |
[8] |
S. Elaydi and R. J. Sacker,
Basin of attraction of periodic orbits of maps on the real line, J Difference Eq and Appl, 10 (2004), 881-888.
doi: 10.1080/10236190410001731443. |
[9] |
S. Gershgorin, Uber die Abgrenzung der Eigenwerte einer Matrix, Bull. Acad. Sci. USSR. Classe des sci. math., 6 (1931), 749-754. Google Scholar |
[10] |
M. Keykobad,
Positive solutions of positive linear systems, Lin. Alg. and its Appl., 64 (1985), 133-140.
doi: 10.1016/0024-3795(85)90271-X. |
[11] |
R. Luis, S. Elaydi and an d H. Oliveira,
Stability of a ricker-type competition model and the competitive exclusion principle, J. of Biological Dynamics, 5 (2011), 636-660.
doi: 10.1080/17513758.2011.581764. |
[12] |
V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math Soc., 19 (1968), 197-231. Google Scholar |
[13] |
W. E. Ricker,
Stock and recruitment, J. Fisheries Research Board of Canada, 11 (1954), 559-623.
doi: 10.1139/f54-039. |
[14] |
B. Ryals,
Dynamics of the Degenerate 2D Ricker Equation, Math. Methods Appl. Sci., 42 (2019), 553-566.
doi: 10.1002/mma.5360. |
[15] |
B. Ryals,
A sufficient condition for stability using slopes of isoclines in planar mappings, J. Difference Eq. and Appl., 26 (2020), 370-383.
doi: 10.1080/10236198.2020.1737034. |
[16] |
B. Ryals and R. J. Sacker,
Global stability in the 2-D Ricker equation, J Difference Eq and Appl, 21 (2015), 1068-1081.
doi: 10.1080/10236198.2015.1065825. |
[17] |
B. Ryals and R. J. Sacker,
Global stability in the 2D Ricker equation revisited, Discrete and Continuous Dynam. Syst.-B, 22 (2017), 585-604.
doi: 10.3934/dcdsb.2017028. |
[18] |
R. J. Sacker,
A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92.
doi: 10.1080/10236190601008752. |
[19] |
R. J. Sacker,
Bifurcation in the almost periodic Ricker map, J. Difference Eq. and Appl., 25 (2019), 599-618.
doi: 10.1080/10236198.2019.1604696. |
[20] |
R. J. Sacker and G. R. Sell,
Almost periodicity, Ricker map, Beverton-Holt map and others, a general method, J Difference Eq and Appl, 23 (2017), 1286-1297.
doi: 10.1080/10236198.2017.1320397. |
[21] |
R. J. Sacker and G. R. Sell, Corrigendum, J. Difference Eq. Appl., 24 (2018), 164.
doi: 10.1080/10236198.2017.1379183. |
[22] |
A. Wilkinson,
What are Lyapunov exponents, and why are they interesting?, Bull. Amer. Math. Soc., 54 (2017), 79-105.
doi: 10.1090/bull/1552. |
[23] |
L. -S. Young, Mathematical theory of Lyapunov exponents, J. Phys. A: Mathematical and Theoretical, 46 (2013), 254001, 17pp.
doi: 10.1088/1751-8113/46/25/254001. |
show all references
References:
[1] |
A. Avila, J. Santamaria, M. Viana and A. Wilkinson,
Cocycles over partially hyperbolic maps, Asterisque, 358 (2013), 1-12.
|
[2] |
E. C. Balreira, S. Elaydi and R. Luis,
Local stability implies global stability for the planar Ricker competition model, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 323-351.
doi: 10.3934/dcdsb.2014.19.323. |
[3] |
H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N. Y., 1947. |
[4] |
J. W. S. Cassel, An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45. Cambridge University Press, New York, 1957.
![]() |
[5] |
K. Chandrasekharan, Introduction to Analytic Number Theory, Number 148 in Die Grundlehren der matematishen Wissenshaft in Einzeldarstellung. Springer Verlag, New York, 1968. |
[6] |
J.-P. Eckmann and D. Ruelle,
Ergodic theory of chaos and strange attractors, Reviews of Modern Physics, 57 (1985), 617-656.
doi: 10.1103/RevModPhys.57.617. |
[7] |
S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005. |
[8] |
S. Elaydi and R. J. Sacker,
Basin of attraction of periodic orbits of maps on the real line, J Difference Eq and Appl, 10 (2004), 881-888.
doi: 10.1080/10236190410001731443. |
[9] |
S. Gershgorin, Uber die Abgrenzung der Eigenwerte einer Matrix, Bull. Acad. Sci. USSR. Classe des sci. math., 6 (1931), 749-754. Google Scholar |
[10] |
M. Keykobad,
Positive solutions of positive linear systems, Lin. Alg. and its Appl., 64 (1985), 133-140.
doi: 10.1016/0024-3795(85)90271-X. |
[11] |
R. Luis, S. Elaydi and an d H. Oliveira,
Stability of a ricker-type competition model and the competitive exclusion principle, J. of Biological Dynamics, 5 (2011), 636-660.
doi: 10.1080/17513758.2011.581764. |
[12] |
V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math Soc., 19 (1968), 197-231. Google Scholar |
[13] |
W. E. Ricker,
Stock and recruitment, J. Fisheries Research Board of Canada, 11 (1954), 559-623.
doi: 10.1139/f54-039. |
[14] |
B. Ryals,
Dynamics of the Degenerate 2D Ricker Equation, Math. Methods Appl. Sci., 42 (2019), 553-566.
doi: 10.1002/mma.5360. |
[15] |
B. Ryals,
A sufficient condition for stability using slopes of isoclines in planar mappings, J. Difference Eq. and Appl., 26 (2020), 370-383.
doi: 10.1080/10236198.2020.1737034. |
[16] |
B. Ryals and R. J. Sacker,
Global stability in the 2-D Ricker equation, J Difference Eq and Appl, 21 (2015), 1068-1081.
doi: 10.1080/10236198.2015.1065825. |
[17] |
B. Ryals and R. J. Sacker,
Global stability in the 2D Ricker equation revisited, Discrete and Continuous Dynam. Syst.-B, 22 (2017), 585-604.
doi: 10.3934/dcdsb.2017028. |
[18] |
R. J. Sacker,
A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92.
doi: 10.1080/10236190601008752. |
[19] |
R. J. Sacker,
Bifurcation in the almost periodic Ricker map, J. Difference Eq. and Appl., 25 (2019), 599-618.
doi: 10.1080/10236198.2019.1604696. |
[20] |
R. J. Sacker and G. R. Sell,
Almost periodicity, Ricker map, Beverton-Holt map and others, a general method, J Difference Eq and Appl, 23 (2017), 1286-1297.
doi: 10.1080/10236198.2017.1320397. |
[21] |
R. J. Sacker and G. R. Sell, Corrigendum, J. Difference Eq. Appl., 24 (2018), 164.
doi: 10.1080/10236198.2017.1379183. |
[22] |
A. Wilkinson,
What are Lyapunov exponents, and why are they interesting?, Bull. Amer. Math. Soc., 54 (2017), 79-105.
doi: 10.1090/bull/1552. |
[23] |
L. -S. Young, Mathematical theory of Lyapunov exponents, J. Phys. A: Mathematical and Theoretical, 46 (2013), 254001, 17pp.
doi: 10.1088/1751-8113/46/25/254001. |





[1] |
Mia Jukić, Hermen Jan Hupkes. Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3163-3209. doi: 10.3934/dcds.2020402 |
[2] |
Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183 |
[3] |
Horst R. Thieme. Discrete-time dynamics of structured populations via Feller kernels. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021082 |
[4] |
Anastasiia Panchuk, Frank Westerhoff. Speculative behavior and chaotic asset price dynamics: On the emergence of a bandcount accretion bifurcation structure. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021117 |
[5] |
Xiaozhong Yang, Xinlong Liu. Numerical analysis of two new finite difference methods for time-fractional telegraph equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3921-3942. doi: 10.3934/dcdsb.2020269 |
[6] |
Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, , () : -. doi: 10.3934/era.2021024 |
[7] |
Rong Rong, Yi Peng. KdV-type equation limit for ion dynamics system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021037 |
[8] |
Wenjing Liu, Rong Yang, Xin-Guang Yang. Dynamics of a 3D Brinkman-Forchheimer equation with infinite delay. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021052 |
[9] |
José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2601-2617. doi: 10.3934/dcds.2020376 |
[10] |
Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 |
[11] |
Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 |
[12] |
Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3905-3919. doi: 10.3934/dcdsb.2020268 |
[13] |
Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026 |
[14] |
Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2891-2905. doi: 10.3934/dcds.2020390 |
[15] |
Nouressadat Touafek, Durhasan Turgut Tollu, Youssouf Akrour. On a general homogeneous three-dimensional system of difference equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2021017 |
[16] |
Ruchika Sehgal, Aparna Mehra. Worst-case analysis of Gini mean difference safety measure. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1613-1637. doi: 10.3934/jimo.2020037 |
[17] |
Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024 |
[18] |
Philippe Jouan, Ronald Manríquez. Solvable approximations of 3-dimensional almost-Riemannian structures. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021023 |
[19] |
Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 |
[20] |
Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225 |
2019 Impact Factor: 1.27
Tools
Article outline
Figures and Tables
[Back to Top]