Periodic Solutions For ẋ = Ax + G(x, t) + ∊p(t)

1971 ◽  
Vol 14 (4) ◽  
pp. 575-577
Author(s):  
Peter J. Ponzo
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Stable Equilibrium ◽  
Asymptotically Stable ◽  
Constant Matrix ◽  
Scalar Parameter ◽  
Locally Lipschitz ◽  
Image Position

We wish to establish the existence of a periodic solution to1where x, g and p are n-vectors, A is an n × n constant matrix, and ∊ is a small scalar parameter. We assume that g and p are locally Lipschitz in x and continuous and T-periodic in t, and that the origin is a point of asymptotically stable equilibrium, when ∊ = 0.

T-periodic solutions for some second order differential equations with singularities

1992 ◽  
Vol 120 (3-4) ◽  
pp. 231-243 ◽  
Author(s):  
Manuel del Pino ◽  
Raúl Manásevich ◽  
Alberto Montero
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Positive Solutions ◽  
Resonance Condition ◽  
Second Order ◽  
Fredholm Alternative ◽  
Positive Slope ◽  
Second Order Differential Equations ◽  
Image Position

SynopsisWe study the existence of T-periodic positive solutions of the equationwhere f(t, .) has a singularity of repulsive type near the origin. Under the assumption that f(t, x) lies between two lines of positive slope for large and positive x, we find a non-resonance condition which predicts the existence of one T-periodic solution.Our main result gives a Fredholm alternative-like result for the existence of T-periodic positive solutions for

scholarly journals Periodic Solutions of a System of Delay Differential Equations for a Small Delay

2002 ◽  
Vol 7 (2) ◽  
pp. 295
Author(s):  
Adu A.M. Wasike ◽  
Wandera Ogana
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Delay Differential Equations ◽  
Original System ◽  
System Of Equations ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Stable Periodic ◽  
Delay Differential

We prove the existence of an asymptotically stable periodic solution of a system of delay differential equations with a small time delay t > 0. To achieve this, we transform the system of equations into a system of perturbed ordinary differential equations and then use perturbation results to show the existence of an asymptotically stable periodic solution. This approach is contingent on the fact that the system of equations with t = 0 has a stable limit cycle. We also provide a comparative study of the solutions of the original system and the perturbed system.  This comparison lays the ground for proving the existence of periodic solutions of the original system by Schauder's fixed point theorem.   

Bifurcation of Nontrivial Periodic Solutions for a Food Chain Beddington–DeAngelis Interference Model with Impulsive Effect

2018 ◽  
Vol 28 (11) ◽  
pp. 1850131 ◽  
Author(s):  
Wang Shuai ◽  
Huang Qingdao
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Release Rate ◽  
Food Chain ◽  
Interference Model ◽  
Impulsive Effect ◽  
Asymptotically Stable ◽  
Nontrivial Periodic Solutions

In this paper, a food chain Beddington–DeAngelis interference model with impulsive effect is studied. The trivial periodic solution is locally asymptotically stable if the release rate or the release period is suitable. Conditions for permanence of the model are obtained. The existence of nontrivial periodic solutions and semi-trivial periodic solutions are established when the trivial periodic solution loses its stability under different conditions.

ANALYSIS OF A CHEMOSTAT MODEL WITH PULSED INPUT

2006 ◽  
Vol 14 (04) ◽  
pp. 583-598 ◽  
Author(s):  
XIANGYUN SHI ◽  
XINYU SONG
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Continuous System ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Chemostat Model ◽  
Globally Asymptotically Stable

In this paper, we consider a chemostat model with pulsed input. We find a critical value of the period of pulses. If the period is more than the critical value, the microorganism-free periodic solution is globally asymptotically stable. If less, the system is permanent. Moreover, the nutrient and the microorganism can co-exist on a periodic solution of period τ. Finally, by comparing the corresponding continuous system, we find that the periodically pulsed input destroys the equilibria of the continuous system and initiates periodic solutions. Our results are valuable for the manufacture of products by genetically altered organisms.

Periodic solutions for a second-order differential equation with indefinite weak singularity

2019 ◽  
Vol 149 (5) ◽  
pp. 1135-1152 ◽  
Author(s):  
José Godoy ◽  
Manuel Zamora
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Second Order ◽  
Constant Sign ◽  
Piecewise Constant ◽  
Weak Singularity ◽  
Second Order Differential Equation ◽  
Image Position

AbstractAs a consequence of the main result of this paper efficient conditions guaranteeing the existence of a T −periodic solution to the second-order differential equation $${u}^{\prime \prime} = \displaystyle{{h(t)} \over {u^\lambda }}$$are established. Here, h ∈ L(ℝ/Tℤ) is a piecewise-constant sign-changing function and the non-linear term presents a weak singularity at 0 (i.e. λ ∈ (0, 1)).

scholarly journals Periodic solutions of a second order nonlinear differential equation

1989 ◽  
Vol 40 (3) ◽  
pp. 357-361 ◽  
Author(s):  
Bahman Mehri
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Fixed Point ◽  
Periodic Solutions ◽  
Nonlinear Differential Equation ◽  
Order Differential Equation ◽  
Second Order ◽  
Fixed Point Technique ◽  
Image Position ◽  
Schauder Fixed Point

We consider the following non-linear nonautonomous second order differential equationwhere h(x) is continuous, f, p are continuous and periodic with respect to t of period w. Using the Leray-Schauder fixed point technique we prove that the above equation possesses at least one non-trivial periodic solution of period w.

Periodic solutions of a nonlinear wave equation

1980 ◽  
Vol 85 (3-4) ◽  
pp. 313-320 ◽  
Author(s):  
Abbas Bahri ◽  
Haïm Brezis
Keyword(s):  
Periodic Solution ◽  
Wave Equation ◽  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Linear Growth ◽  
Necessary Condition ◽  
Image Position

SynopsisWe provide a sufficient and “almost” necessary condition for the existence of a periodic solution of the equationwhere F is nondecreasing in u and has a small linear growth as |u|→∞.

scholarly journals Periodic solutions of quasilinear non-autonomous systems with impulses

1985 ◽  
Vol 31 (2) ◽  
pp. 185-197 ◽  
Author(s):  
S.G. Hristova ◽  
D.D. Bainov
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Small Parameter ◽  
Periodic Solutions ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
System Of Differential Equations ◽  
Image Position ◽  
The Given

The paper considers a system of differential equations with impulse perturbations at fixed moments in time of the formwhere x ∈ Rn, ε is a small parameter,Sufficient conditions are found for the existence of the periodic solution of the given system in the critical and non-critical cases.

scholarly journals Stability and periodicity in a mosquito population suppression model composed of two sub-models

2021 ◽  
Author(s):  
Zhongcai Zhu ◽  
Bo Zheng ◽  
Yantao Shi ◽  
Rong Yan ◽  
Jianshe Yu
Keyword(s):  
Periodic Solution ◽  
Stable Equilibrium ◽  
Mosquito Population ◽  
Sexually Active ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Examples ◽  
Suppression Process ◽  
Population Suppression ◽  
Theoretical Results

AbstractIn this paper, we propose a mosquito population suppression model which is composed of two sub-models switching each other. We assume that the releases of sterile mosquitoes are periodic and impulsive, only sexually active sterile mosquitoes play a role in the mosquito population suppression process, and the survival probability is density-dependent. For the release waiting period T and the release amount c, we find three thresholds denoted by $$T^*$$ T ∗ , $$g^*$$ g ∗ , and $$c^*$$ c ∗ with $$c^*>g^*$$ c ∗ > g ∗ . We show that the origin is a globally or locally asymptotically stable equilibrium when $$c\ge c^*$$ c ≥ c ∗ and $$T\le T^*$$ T ≤ T ∗ , or $$c\in (g^*, c^*)$$ c ∈ ( g ∗ , c ∗ ) and $$T<T^*$$ T < T ∗ . We prove that the model generates a unique globally asymptotically stable T-periodic solution when either $$c\in (g^*, c^*)$$ c ∈ ( g ∗ , c ∗ ) and $$T=T^*$$ T = T ∗ , or $$c>g^*$$ c > g ∗ and $$T>T^*$$ T > T ∗ . Two numerical examples are provided to illustrate our theoretical results.

Reducibility of a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies

2020 ◽  
pp. 1-38
Author(s):  
DONGFENG ZHANG ◽  
JUNXIANG XU
Keyword(s):  
Periodic Solution ◽  
Small Parameter ◽  
Small Perturbation ◽  
Periodic System ◽  
Order Term ◽  
High Order ◽  
Periodic Systems ◽  
Elliptic Type ◽  
Constant Matrix ◽  
Image Position

In this paper we consider the following nonlinear quasi-periodic system: $$\begin{eqnarray}{\dot{x}}=(A+\unicode[STIX]{x1D716}P(t,\unicode[STIX]{x1D716}))x+\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})+h(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $A$ is a $d\times d$ constant matrix of elliptic type,  $\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})$ is a small perturbation with $\unicode[STIX]{x1D716}$ as a small parameter, $h(x,t,\unicode[STIX]{x1D716})=O(x^{2})$ as $x\rightarrow 0$ , and $P,g$ and $h$ are all analytic quasi-periodic in $t$ with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$ , where $\unicode[STIX]{x1D6FC}$ is irrational. It is proved that for most sufficiently small $\unicode[STIX]{x1D716}$ , the system is reducible to the following form: $$\begin{eqnarray}{\dot{x}}=(A+B_{\ast }(t))x+h_{\ast }(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $h_{\ast }(x,t,\unicode[STIX]{x1D716})=O(x^{2})~(x\rightarrow 0)$ is a high-order term. Therefore, the system has a quasi-periodic solution with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$ , such that it goes to zero when $\unicode[STIX]{x1D716}$ does.

Sign in / Sign up

Export Citation Format

Share Document