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The existence of periodic solutions
1999Abstract Suppose that the phase diagram for a differential equation contains a single, unstable equilibrium point and a limit cycle surrounding it, as in the case of the van der Pol equation. Then in practice all initial states lead to the periodic oscillation represented by the limit cycle.
D W Jordan, P Smith
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Periodic solutions of periodic differential equations
Journal of the Nigerian Association of Mathematical Physics, 2008In this paper we extend the work of Bello [4] where he considered the periodic solutions of certain dynamical systems inside a cylindrical phase space with differential equations of the form yn-1⓫yn-1+...+αn-1y(1) + f(y1..,yn-1,y) = 0 (\'=ddt (+) with the property that there is a ω>0 and a natural number K such that y (t+w) = y(t) + k ...
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Existence Theorems for Periodic Solutions and Almost Periodic Solutions
1975First of all, we shall state some fixed point theorems without proofs. The following theorem is due to Brouwer. For the proof, see [5].
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2011
This chapter presents existence and stability of almost periodic solutions of the following system \( {\frac{dx(t)}{dt}} = A(t)x(t) + f(t,x(\theta_{\upsilon (t) - p1} ),x(\theta_{\upsilon (t) - p2} ), \ldots ,x(\theta_{\upsilon (t) - pm} )), \) (7.1) where \( x \in \mathbb{R}^{n} ,\;t \in \mathbb{R}, \) υ(t) = 1 if θ i ≤ t < θ i+1, i = …,-2,-1,0,1,2,…,
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This chapter presents existence and stability of almost periodic solutions of the following system \( {\frac{dx(t)}{dt}} = A(t)x(t) + f(t,x(\theta_{\upsilon (t) - p1} ),x(\theta_{\upsilon (t) - p2} ), \ldots ,x(\theta_{\upsilon (t) - pm} )), \) (7.1) where \( x \in \mathbb{R}^{n} ,\;t \in \mathbb{R}, \) υ(t) = 1 if θ i ≤ t < θ i+1, i = …,-2,-1,0,1,2,…,
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2012
In the present chapter, we shall state some basic existence and uniqueness results for almost periodic solutions of impulsive differential equations. Applications to real world problems will also be discussed.
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In the present chapter, we shall state some basic existence and uniqueness results for almost periodic solutions of impulsive differential equations. Applications to real world problems will also be discussed.
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