The Fixed Point Theory and the Existence of the Periodic Solution on a Nonlinear Differential Equation

Hua, Ni

doi:10.1155/2018/6725989

Cited by 2 publications

(2 citation statements)

References 10 publications

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“…However, for the number of real periodic solutions, we can say from "exchange of stability" reasoning that, If multiplicity μ is even, the origin is stable μ Now we will give some examples that show the feasibility of our results, for more examples see refs. [5][6][7]11]. By using the above mentioned remark, the stability behavior is also discussed here.…”

Section: Examplesmentioning

confidence: 99%

Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients

et al. 2020

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This article concerns with the development of the number of focal values. We analyzed periodic solutions for first-order cubic non-autonomous ordinary differential equations. Bifurcation analysis for periodic solutions from a fine focus {\mathfrak{z}}=0 is also examined. In particular, we are interested to detect the maximum number of periodic solutions for various classes of higher order in which a given solution can bifurcate under perturbation of the coefficients. We calculate the maximum number of periodic solutions for different classes, namely, {C}_{10,5} and {C}_{12,6} with trigonometric coefficients, and they are found with nine and eight multiplicities at most. The classes {C}_{8,3} and {C}_{8,4} with algebraic coefficients have at most eight limit cycles. The new formula {\varkappa }_{10} is developed by which we succeeded to find highest known multiplicity ten for class {C}_{\mathrm{9,3}} with polynomial coefficient. Periodicity is calculated for both trigonometric and algebraic coefficients. Few examples are also considered to explain the applicability and stability of the methods presented.

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Section: Examplesmentioning

confidence: 99%

Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients

et al. 2020

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“…Other examples of self-excited oscillation are the beating of a heart, rhythms in body temperature, hormone secretion, chemical reactions that oscillate spontaneously, and vibrations in bridges and airplane wings. Due to the wide occurrence of limit cycles in science and technology, limit cycle theory has also been extensively studied by physicists, and more recently by chemists, biologists, and economists [9][10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Periodic Solutions of Some Classes of One Dimensional Non-autonomous Equation

et al. 2020

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In this paper, the periodic solutions of a certain one-dimensional differential equation are investigated for the first order cubic non-autonomous equation. The method used here is the bifurcation of periodic solutions from a fine focus z = 0. We aimed to find the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients. For classes C 3, 8 , C 4, 3 , C 7, 5 , C 7, 6 , eight periodic multiplicities have been found. To investigate the multiplicity >9, the formula for the focal value was not available in the literature. We also succeeded in constructing the formula for η 10. By implementing our newly developed formula, we are able to get multiplicity ten for classes C 7, 3 , C 9, 1 , which is the highest known to date. A perturbation method has been properly established for making the maximal number of limit cycles for each class. Some examples are also presented to show the implementation of the newly developed method. By considering all of these facts, it can be concluded that the presented methods are new, authentic, and novel.

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The Fixed Point Theory and the Existence of the Periodic Solution on a Nonlinear Differential Equation

Abstract: This paper deals with a nonlinear differential equation, by using the fixed point theory. The existence of the periodic solution of the nonlinear differential equation is obtained; these results are new.

Cited by 2 publications

References 10 publications

Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients

Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients

Periodic Solutions of Some Classes of One Dimensional Non-autonomous Equation

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