scholarly journals Periodic Solutions of a Periodic FitzHugh–Nagumo System

2015 ◽  
Vol 25 (13) ◽  
pp. 1550180 ◽  
Author(s):  
Jaume Llibre ◽  
Claudio Vidal
Keyword(s):  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
Existence Of Periodic Solutions

Recently some interest has appeared for the periodic FitzHugh–Nagumo differential systems. Here, we provide sufficient conditions for the existence of periodic solutions in such differential systems.

scholarly journals Periodic Solutions of Some Polynomial Differential Systems in Dimension 3 via Averaging Theory

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Amar Makhlouf ◽  
Lilia Bousbiat
Keyword(s):  
Small Parameter ◽  
Real Number ◽  
Periodic Solutions ◽  
Differential System ◽  
Sufficient Conditions ◽  
Periodic Functions ◽  
Third Order ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Existence Of Periodic Solutions

We provide sufficient conditions for the existence of periodic solutions of the polynomial third order differential systemx.=-y+εP(x,y,z)+h1(t),  y.=x+εQ(x,y,z)+h2(t),  and  z.=az+εR(x,y,z)+h3(t), whereP,Q, andRare polynomials in the variablesx,y, andzof degreen,  hi(t)=hi(t+2π)withi=1,2,3being periodic functions,ais a real number, andεis a small parameter.

scholarly journals Periodic solutions for differential systems in ℝ3 and ℝ4

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Amina Feddaoui ◽  
Jaume Llibre ◽  
Chemseddine Berhail ◽  
Amar Makhlouf
Keyword(s):  
Small Parameter ◽  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Periodic Functions ◽  
Differential Systems ◽  
Existence Of Periodic Solutions

AbstractWe provide sufficient conditions for the existence of periodic solutions for the differential systems \matrix{{x' = y,\;\;\;y' = z,\;\;\;z' = - y - \varepsilon F(t,x,y,z),\;\;\;{\rm{and}}} \cr {x' = y,\quad y' = - x - \varepsilon G(t,x,y,z,u),\quad z' = u,\quad u' = - z - \varepsilon H(t,x,y,z,u),} \hfill \cr } where F, G and H are 2π–periodic functions in the variable t and ɛ is a small parameter. These differential systems appear frequently in many problems coming from the sciences and the engineering.

scholarly journals Existence of periodic solutions of impulsive differential systems

1991 ◽  
Vol 4 (2) ◽  
pp. 137-146 ◽  
Author(s):  
L. H. Erbe ◽  
Xinzhi Liu
Keyword(s):  
Periodic Solutions ◽  
Comparison Principle ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
Impulsive Differential Systems ◽  
Existence Of Periodic Solutions ◽  
Piecewise Continuous

In this paper, the existence of periodic solutions of impulsive differential systems is considered. Since the solutions of such a system are peicewise continuous, it is necessary to introduce piecewise continuous Lyapunov functions. By means of such functions, together with the comparison principle, some sufficient conditions for the existence of periodic solutions of impulsive differential systems are established.

scholarly journals On the equivalence of three-dimensional differential systems

2020 ◽  
Vol 18 (1) ◽  
pp. 1164-1172
Author(s):  
Jian Zhou ◽  
Shiyin Zhao
Keyword(s):  
Periodic Solutions ◽  
Differential System ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Systems ◽  
Structural Form ◽  
Differential Systems

Abstract In this paper, firstly, we study the structural form of reflective integral for a given system. Then the sufficient conditions are obtained to ensure there exists the reflective integral with these structured form for such system. Secondly, we discuss the necessary conditions for the equivalence of such systems and a general three-dimensional differential system. And then, we apply the obtained results to the study of the behavior of their periodic solutions when such systems are periodic systems in t.

scholarly journals Existence for Singular Periodic Problems: A Survey of Recent Results

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Jifeng Chu ◽  
Juntao Sun ◽  
Patricia J. Y. Wong
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Impulsive Differential Equations ◽  
Differential Systems ◽  
Singular Differential Equations ◽  
Periodic Problems ◽  
Existence Of Periodic Solutions

We present a survey on the existence of periodic solutions of singular differential equations. In particular, we pay our attention to singular scalar differential equations, singular damped differential equations, singular impulsive differential equations, and singular differential systems.

scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

scholarly journals ON THE EXISTENCE OF PERIODIC SOLUTIONS FOR A CLASS OF NONLINEAR DIFFERENTIAL SYSTEMS

1993 ◽  
Vol 24 (2) ◽  
pp. 173-188
Author(s):  
LIHONG HUANG ◽  
JIANSHE YU
Keyword(s):  
Periodic Solutions ◽  
Special Form ◽  
Differential Systems ◽  
Nontrivial Periodic Solutions ◽  
Existence Of Periodic Solutions ◽  
Nonlinear Differential Systems

In this paper three theorems on the existence of nontrivial periodic solutions of the system \[ dx/dt =e(y)\]\[dy/dt =-e(y)f(x)- g(x)\] are proved, which not only generalize some known results on the existence of periodic solutions of Lienard's system (i.e. the special form for $e(y) = y$), but also relax or eliminate some traditional assumptions.

scholarly journals Periodic solutions of a nonlinear oscillatory system with two degrees of freedom

2005 ◽  
Vol 47 (2) ◽  
pp. 249-263
Author(s):  
Zhengqiu Zhang ◽  
Yusen Zhu ◽  
Biwen Li
Keyword(s):  
Periodic Solutions ◽  
Degrees Of Freedom ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Oscillatory System ◽  
Continuation Theorem ◽  
Two Degrees Of Freedom ◽  
Nonlinear Oscillatory System ◽  
Existence Of Periodic Solutions

AbstractWe study a nonlinear oscillatory system with two degrees of freedom. By using the continuation theorem of coincidence degree theory, some sufficient conditions are obtained to establish the existence of periodic solutions of the system.

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