Periodic solutions for differential systems in ℝ³ and ℝ⁴

Abstract

We 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.

Limit cycles of perturbed global isochronous center

“…The first-order averaging method theory, that we summarize in the sequel, can be found in a more extended way in [2]. Similar works where the perturbations via polynomials play an important role are for instance [7] and [20].…”

“…We observe that this system is into the normal form of the averaging method (5), with t = θ and x = (ρ, s, ω) that the all assumptions of the Theorem 2 are satisfied for the system (7). We compute the average functions of the first-order associated with the system (7)…”

Zero-Hopf Bifurcation in a Generalized Genesio Differential Equation

“…Here, we present the main results on the second-order averaging theory of dynamical systems that will play a key role in the proof of our main results Theorem 1. For more information on this interesting theory and its application, see for instance [7] or [8] and references therein. For a proof of Theorem 2 that we are going to state, see Theorem 3.5.1 of Sanders and Verhulst [2], or [9] for a formulation in modern terminology.…”

Optimization Simulation System of University Science Education Based on Finite Differential Equations

“…In 1941, the former Soviet mathematician Kolmogorov further analyzed and defined fractional Brownian motion based on the Hilbert space theory. At the same time, they gave the stochastic integral expression based on standard Brownian movement [4]. After that, British hydrologist Hirst found that using the fractional Brownian motion method can more appropriately describe the actual situation when studying the long-term storage capacity of the Nile Reservoir.…”