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We consider the generalized Liénard systemfrac(d x, d t) = frac(1, a (x)) [h (y) - F (x)],frac(d y, d t) = - a (x) g (x), where a, F, g, and h are continuous functions on R and a (x) > 0, for x ∈ R. Under the assumptions that the origin is a unique equilibrium, we study the problem whether all trajectories of this system intersect the vertical isocline h (y) = F (x), which is very important in the global asymptotic stability of the origin, oscillation theory, and existence of periodic solutions. Under quite general assumptions we obtain sufficient and necessary conditions which are very sharp. Our results extend the results of Villari and Zanolin, and Hara and Sugie for this system with h... 

In this paper we consider the nonlinear third-order quasi-linear differential equationx‴+k2x′=εfx,x′,x″and obtain some simple conditions for the existence of a periodic solution for it. In so doing we use the implicit function theorem to prove a theorem about the existence of periodic solutions and consider one example to show the realizability of the conditions. The validity of the conditions for the parameter-free problemx‴+k2x′=fx,x′,x″also is considered. © 2001 Academic Press  

Periodic solutions and their existence are one of the most important subjects in dynamical systems. Fractional order systems like integer ones are no exception to this rule. Tavazoei and Haeri (2009) have shown that a time-invariant fractional order system does not have any periodic solution. In this article, this claim has been investigated and it is shown that although in any finite interval of time the solutions do not show any periodic behavior, when the steady state responses of fractional order systems are considered, periodic orbits can be detected  

In this paper the solutions of a two-endpoint boundary value problem is studied and under suitable assumptions the existence and uniqueness of a solution is proved. As a consequence, a condition to guarantee the existence of at least one periodic solution for a class of Liénard equations is presented  

In this work the second-order generalized forced Li ́enard equationx′′+(f(x)+k(x)x′)x′+g(x)=p(t)is considered and anew condition for guaranteeing the existence of at least one periodic solution for this equation is given  

In this paper we explore the existence of periodic solutions of a nonautonomous semi-ratio-dependent predator-prey dynamical system with functional responses on time scales. To illustrate the utility of this work, we should mention that, in our results this system with a large class of monotone functional responses, always has at least one periodic solution. For instance, this system with some celebrated functional responses such as Holling type-II (or Michaelis-Menten), Holling type-III, Ivlev, mx (Holling type I), sigmoidal [e.g., Real and mx2/((A + x)(B +x))] and some other monotone functions, has always at least one ω-periodic solution. Besides, for some well-known functional responses... 

In this work we study the problem of whether all trajectories of the system ẋ=y-F(x) and ẏ=-g(x) cross the vertical isocline, which is very important for the existence of periodic solutions and oscillation theory. Sufficient conditions are given for all trajectories to cross the vertical isocline. © 2005 Elsevier Ltd. All rights reserved  

In this note, we show that under certain assumptions the matrix Riccati differential equation X′ = A(t)X + XB(t)X + C(t) with periodic coeffi cients admits at least one periodic solution. Also, we give an illustrative example in order to indicate the validity of the assumptions and the novelty of our result. © 2021, Tsing Hua University. All rights reserved  

The existence of periodic solutions for a forced Hill's equation is proved. The proof is then extended to the case of a non-homogeneous matrix valued Hill's equation. Under the stated conditions, using Lyapunov's criteria [Proc. AMS 13 (1962) 601; Hill's Equation, Interscience Publishers, New York, 1966] some results on the stability oh Hill's equation are obtained. © 2004 Elsevier Inc. All rights reserved  

In this paper, it is shown that the fractional-order derivatives of a periodic function with a specific period cannot be a periodic function with the same period. The fractional-order derivative considered here can be obtained based on each of the well-known definitions Grunwald-Letnikov definition, Riemann-Liouville definition and Caputo definition. This concluded point confirms the result of a recently published work proving the non-existence of periodic solutions in a class of fractional-order models. Also, based on this point it can be easily proved the absence of periodic responses in a wider class of fractional-order models. Finally, some examples are presented to show the... 

The aim of this note is to highlight one of the basic differences between fractional order and integer order systems. It is analytically shown that a time invariant fractional order system contrary to its integer order counterpart cannot generate exactly periodic signals. As a result, a limit cycle cannot be expected in the solution of these systems. Our investigation is based on Caputo's definition of the fractional order derivative and includes both the commensurate or incommensurate fractional order systems. © 2009 Elsevier Ltd. All rights reserved  

This paper deals with the question of existence of periodic solutions of nonautonomous predator-prey dynamical systems with Beddington-DeAngelis functional response. We explore the periodicity of this system on time scales. New sufficient conditions are derived for the existence of periodic solutions. These conditions extend previous results presented in [M. Bohner, M. Fan, J. Zhang, Existence of periodic solutions in predator-prey and competition dynamic systems, Nonlinear. Anal.: Real World Appl. 7 (2006) 1193-1204; M. Fan, Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelies functional response, J. Math. Anal. Appl. 295 (2004) 15-39; J. Zhang, J. Wang,... 

Inverted pendulum (IP) has been broadly used to model locomotor systems. In this paper, we demonstrate that an IP-like model could simulate stable periodic handspring maneuvers with passive flight phases. The model is a 2-D symmetric rigid body which is merely controlled during the contact phase. To benefit from an open-loop sensorless strategy, the control policy is implemented only by an unvaried torque input. The system's dynamics is an example of nonlinear impulsive systems studied and analyzed by the Poincaré section method. The numerical results reveal that the stable periodic solutions are sufficiently robust for a broad range of the parameter space. © 2020 Elsevier Ltd  

Here we are concerned with the existence of periodic solution for nonlinear non-autonomous third order system of ordinary differential equations with singular terms. Our method here is based on the topological method in the sense that we conclude some computable results for the focal system from a homotopic nonsingular system. The aim is to obtain sufficient conditions for which the system has periodic solution whenever the value of deformation with respect to the first variation of the nonsingular subsystem is sufficiently small. The method presented here is constructive in the sense that the existence of periodic orbits can be verified numerically as well as computed if any. For this, we... 

Here we are concerned with the problem of the existence of periodic solution for certain second and third-order nonlinear differential equations. Our method here is to consider the problem as an eigenvalue problem and treat it by the topological degree theory. In particular we establish the conditions of the existence of periodic solution first for a simpler system which is homotopic to the original system and then generalize the obtained results for the focal system. The method employed here is applicable also for a system of nonlinear differential equations just with simple modifications. Finally, we present some specific examples numerically to show that the results are valid and...