ON PERIODIC SOLUTIONS OF 2-PERIODIC LYNESS' EQUATIONS

We study the existence of periodic solutions of the nonautonomous periodic Lyness' recurrenceun+2 = (an + un+1)/un, where {an}n is a cycle with positive values a, b and with positive initial conditions. It is known that for a = b = 1 all the sequences generated by this recurrence are 5-periodic. We prove that for each pair (a, b) ≠ (1, 1) there are infinitely many initial conditions giving rise to periodic sequences, and that the family of recurrences have almost all the even periods. If a ≠ b, then any odd period, except 1, appears.

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On the existence of periodic solutions of a certain third-order differential equation

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100034678 ◽

1960 ◽

Vol 56 (4) ◽

pp. 381-389 ◽

Cited By ~ 15

Author(s):

J. O. C. Ezeilo

Keyword(s):

Differential Equation ◽

Periodic Function ◽

Periodic Solutions ◽

Order Differential Equation ◽

Initial Conditions ◽

Continuous Periodic Function ◽

Third Order ◽

Third Order Differential Equation ◽

Existence Of Periodic Solutions ◽

Image Position

In this paper we shall be concerned with the differential equationin which a and b are constants, p(t) is a continuous periodic function of t with a least period ω, and dots indicate differentiation with respect to t. The function h(x) is assumed continuous for all x considered, so that solutions of (1) exist satisfying any assigned initial conditions. In an earlier paper (2) explicit hypotheses on (1) were established, in the two distinct cases:under which every solution x(t) of (1) satisfieswhere t0 depends on the particular x chosen, and D is a constant depending only on a, b, h and p. These hypotheses are, in the case (2),or, in the case (3),In what follows here we shall refer to (2) and (H1) collectively as the (boundedness) hypotheses (BH1), and to (3) and (H2) as the hypotheses (BH2). Our object is to examine whether periodic solutions of (1) exist under the hypotheses (BH1), (BH2).

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Decay of Hamiltonian Breathers Under Dissipation

Communications in Mathematical Physics ◽

10.1007/s00220-020-03848-4 ◽

2020 ◽

Vol 380 (1) ◽

pp. 71-102

Author(s):

Jean-Pierre Eckmann ◽

C. Eugene Wayne

Keyword(s):

Phase Space ◽

Systems Theory ◽

Hamiltonian Systems ◽

Initial Conditions ◽

Asymptotic Estimates ◽

Discrete Nonlinear Schrödinger Equation ◽

Rate Of Evolution ◽

Energy Flows ◽

The Family ◽

Almost All

Abstract We study metastable behavior in a discrete nonlinear Schrödinger equation from the viewpoint of Hamiltonian systems theory. When there are $$n<\infty $$ n < ∞ sites in this equation, we consider initial conditions in which almost all the energy is concentrated in one end of the system. We are interested in understanding how energy flows through the system, so we add a dissipation of size $$\gamma $$ γ at the opposite end of the chain, and we show that the energy decreases extremely slowly. Furthermore, the motion is localized in the phase space near a family of breather solutions for the undamped system. We give rigorous, asymptotic estimates for the rate of evolution along the family of breathers and the width of the neighborhood within which the trajectory is confined.

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Hierarchy of Periodic Solutions for Hamiltonian Systems and their Applications to Celestial Mechanics

International Astronomical Union Colloquium ◽

10.1017/s0252921100066215 ◽

1993 ◽

Vol 132 ◽

pp. 323-337

Author(s):

Yu.V. Barkin

Keyword(s):

Small Parameter ◽

Periodic Solutions ◽

Hamiltonian Systems ◽

Necessary Conditions ◽

Initial Conditions ◽

Standard Form ◽

Systematic Investigation ◽

Canonical Transformations ◽

Existence Of Periodic Solutions ◽

Constructive Conditions

AbstractA systematic investigation has been carried out for periodic solutions for standard-form Hamiltonian systems containing a small parameter/the principal problem of dynamics/. An efficient method of investigation of conditions for periodicity of solutions has been developed. Besides fitting the initial conditions of the action-angle variables, the idea of fitting the values of the parameters of the problem is used. Constructive conditions are obtained for the existence of periodic solutions in both principal and degenerate cases, as well as necessary conditions for their stability; algorithms have been developed for constructing these solutions as series in integer powers of the small parameter. To study particular periodic solutions /by high order resonances/, canonical transformations of the initial equations to a special form are used.

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The Effects of Strong Cubic Nonlinearity on the Existence of Periodic Solutions of the Mathieu–Duffing Equation

Journal of Applied Mechanics ◽

10.1115/1.3112722 ◽

2009 ◽

Vol 76 (5) ◽

Cited By ~ 1

Author(s):

Ivana Kovacic ◽

Livija Cveticanin

Keyword(s):

Periodic Solutions ◽

Initial Conditions ◽

Mathieu Equation ◽

Duffing Equation ◽

Cubic Nonlinearity ◽

Strongly Nonlinear ◽

Numerical Integrations ◽

Strongly Nonlinear System ◽

Existence Of Periodic Solutions ◽

Parameter Values

This paper deals with systems governed by the Mathieu–Duffing equation, with a time-dependent coefficient of the linear term and a constant, not necessarily small coefficient of the cubic term. This coefficient can be positive or negative. The method of strained parameters applied to a linear system governed by the Mathieu equation is extended to a strongly nonlinear system. As a result, the curves corresponding to the parameter values at which periodic solutions exist are obtained. It is shown that they strongly depend on the value of the coefficient of nonlinearity and the initial conditions. The corresponding parameter planes are plotted. Numerical integrations are carried out to confirm the analytical results.

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A Nonlinear Problem in the Bending Vibration of a Rotating Beam

Journal of Applied Mechanics ◽

10.1115/1.4010543 ◽

1952 ◽

Vol 19 (4) ◽

pp. 461-464

Author(s):

Hsu Lo

Keyword(s):

Nonlinear Equation ◽

Periodic Solutions ◽

Nonlinear Problem ◽

Nonlinear Term ◽

Initial Conditions ◽

Percentage Error ◽

Rotating Beam ◽

Bending Vibration ◽

Coriolis Acceleration ◽

Existence Of Periodic Solutions

Abstract The problem of bending vibration of a rotating beam is a nonlinear one when the vibration takes place in a plane not perpendicular to the plane of rotation. The nonlinear term arises from the Coriolis acceleration. From the nonlinear equation established, it is found that the existence of periodic solutions depends on the initial conditions, and the most important parameter which affects the periodic solutions is the nondimensional amplitude Ā. The percentage error in the frequency of vibration due to neglect of the Coriolis acceleration is a function of the parameter Ā only. For most of the present-day applications the error is negligible.

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Analisis Perbandingan Manajemen Keuangan Ojek Online Dan Ojek Reguler Studi Kasus Kecamatan Duren Sawit Jakarta Timur

Liquidity ◽

10.32546/lq.v7i1.173 ◽

2018 ◽

Vol 7 (1) ◽

pp. 41-52

Author(s):

M. Koesmawan ◽

Darwin Erhandy ◽

Dede Dahlan

Keyword(s):

Financial Management ◽

Average Income ◽

Savings And Loans ◽

Financial Difficulties ◽

The Family ◽

Family Finances ◽

The Subject ◽

Almost All

In order to meet the needs of living which consists of primary as well as secondary needs, human can work in either a formal or an informal job. One of the informal jobs that is became the subject of this research was to become an ojek driver. Ojek is a ranting motorcycle. Revenue of ojek drivers, accordingly, should be well managed following the concept of financial management. This research was conducted for the driver of the online motorcycle drivers as well as the regular motorcycle drivers they are called “The Ojek”. Ojek’s location is in Kecamatan (subdistrict) Duren Sawit, East Jakarta with 70 drivers of ojeks. The online ojeks earn an average of Rp 100,000 per day, can save Rp 11,000 to 21,000 per day, while, the regular ojek has an average income per day slightly lower amounted to Rp 78,500, this kind of ojeks generally have other businesses and always record the outflow of theirs money. Both the online and regular ojeks feel a tight competition in getting passengers, but their income can help the family finances and both ojeks want a cooperative especially savings and loans, especially to overcome the urgent financial difficulties. Almost all rivers, do not dare to borrow money. They are afraid of can not refund the money as scheduled.

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A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽

10.2298/fil1711157a ◽

2017 ◽

Vol 31 (11) ◽

pp. 3157-3172

Author(s):

Mujahid Abbas ◽

Bahru Leyew ◽

Safeer Khan

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Fixed Points ◽

Periodic Solutions ◽

Metric Space ◽

Delay Differential Equations ◽

Fixed Point Theorems ◽

Metric Spaces ◽

Existence Of Periodic Solutions ◽

Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

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Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian

Advanced Nonlinear Studies ◽

10.1515/ans-2020-2075 ◽

2020 ◽

Vol 20 (3) ◽

pp. 725-737 ◽

Cited By ~ 2

Author(s):

Zhenping Feng ◽

Zhuoran Du

Keyword(s):

Variational Methods ◽

Periodic Solutions ◽

Smooth Function ◽

Fractional Laplacian ◽

Type Inequality ◽

Energy Functional ◽

Large Integer ◽

Existence Of Periodic Solutions ◽

Double Well Potential

AbstractWe consider periodic solutions of the following problem associated with the fractional Laplacian: {(-\partial_{xx})^{s}u(x)+\partial_{u}F(x,u(x))=0} in {\mathbb{R}}. The smooth function {F(x,u)} is periodic about x and is a double-well potential with respect to u with wells at {+1} and -1 for any {x\in\mathbb{R}}. We prove the existence of periodic solutions whose periods are large integer multiples of the period of F about the variable x by using variational methods. An estimate of the energy functional, Hamiltonian identity and Modica-type inequality for periodic solutions are also established.

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