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New periodic orbits in the planar equal-mass three-body problem
Theory of rotated equations and applications to a population model
1. | Department of Mathematics, Shanghai Normal University, Shanghai 200234, China |
2. | School of Mathematics Sciences, Qufu Normal University, Qufu 273165, China |
3. | Department of Mathematics, Shanghai Normal University Shanghai, Shanghai 200234, China |
4. | College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China |
We consider a family of scalar periodic equations with a parameter and establish theory of rotated equations, studying the behavior of periodic solutions with the change of the parameter. It is shown that a stable (completely unstable) periodic solution of a rotated equation varies monotonically with respect to the parameter and a semi-stable periodic solution splits into two periodic solutions or disappears as the parameter changes in one direction or another. As an application of the obtained results, we give a further study of a piecewise smooth population model verifying the existence of saddle-node bifurcation.
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Averaging methods of arbitrary order, periodic solutions and integrability, J. Differential Equations, 260 (2016), 4130-4156.
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M. Han and D. Zhu, Bifurcation Theory of Differential Equation, Coal Industry Publishing House, Beijing, 1994. |
[8] |
M. Han,
Bifurcation Theory of Limit Cycles, Science Press Beijing, Beijing; Alpha Science International Ltd., Oxford, 2017. |
[9] |
M. Han,
Global behavior of limit cycles in rotated vector fields, Journal of Differential Equations, 151 (1999), 20-35.
doi: 10.1006/jdeq.1998.3508. |
[10] |
P. Liu, J. Shi and Y. Wang,
Periodic solutions of a logistic type population model with harvesting, J. Math. Anal. Appl., 369 (2010), 730-735.
doi: 10.1016/j.jmaa.2010.04.027. |
[11] |
S. Oruganti, J. Shi and R. Shivaji,
Diffusive logistic equaiton with constant yield harvesting, I. Steady States, Trans. Amer. Math. Soc., 354 (2002), 3601-3619.
doi: 10.1090/S0002-9947-02-03005-2. |
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D. Xiao,
Dynamics and bifurcation on a class of population model with seasonal constant-yield harvesting, Discrete Contin. Dyn. Syst., 21 (2016), 699-719.
|
show all references
References:
[1] |
D. Batenkov and G. Binyamini,
Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation, J. Differential Equations, 259 (2015), 5769-5781.
doi: 10.1016/j.jde.2015.07.009. |
[2] |
F. Brauer and D. A. Sanchez,
Periodic environments and periodic harvesting, Natural Resource Modeling, 16 (2003), 233-244.
|
[3] |
F. Brauer and D. A. Sanchez,
Constant rate population harvesting: Equilibrium and stability, Theoret. Pop. biol., 8 (1975), 12-30.
doi: 10.1016/0040-5809(75)90036-2. |
[4] |
D. Campbell and S. R. Kaplan,
A bifurcation problem in differential equations, Math. Mag., 73 (2000), 194-203.
doi: 10.2307/2691522. |
[5] |
G. F. D. Duff,
Limit cycles and rotated vector fields, Ann.of Math., 57 (1953), 15-31.
doi: 10.2307/1969724. |
[6] |
J. Giné, J. Llibre, K. Wu and X. Zhang,
Averaging methods of arbitrary order, periodic solutions and integrability, J. Differential Equations, 260 (2016), 4130-4156.
doi: 10.1016/j.jde.2015.11.005. |
[7] |
M. Han and D. Zhu, Bifurcation Theory of Differential Equation, Coal Industry Publishing House, Beijing, 1994. |
[8] |
M. Han,
Bifurcation Theory of Limit Cycles, Science Press Beijing, Beijing; Alpha Science International Ltd., Oxford, 2017. |
[9] |
M. Han,
Global behavior of limit cycles in rotated vector fields, Journal of Differential Equations, 151 (1999), 20-35.
doi: 10.1006/jdeq.1998.3508. |
[10] |
P. Liu, J. Shi and Y. Wang,
Periodic solutions of a logistic type population model with harvesting, J. Math. Anal. Appl., 369 (2010), 730-735.
doi: 10.1016/j.jmaa.2010.04.027. |
[11] |
S. Oruganti, J. Shi and R. Shivaji,
Diffusive logistic equaiton with constant yield harvesting, I. Steady States, Trans. Amer. Math. Soc., 354 (2002), 3601-3619.
doi: 10.1090/S0002-9947-02-03005-2. |
[12] |
D. Xiao,
Dynamics and bifurcation on a class of population model with seasonal constant-yield harvesting, Discrete Contin. Dyn. Syst., 21 (2016), 699-719.
|





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behavior of periodic solution | stable | completely unstable | upper-stable lower-unstable | upper-unstable lower-stable |
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