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Dynamical systems modelled by third order differential equations with special respect to the influence of the restoring term on the properties of solutions. (English) Zbl 0706.34032

For the equations: \[ x'''+f(x')x''+g(x)x'+h(x)=0\quad (Li\acute enard), \]
\[ x'''+f(x'')+g(x')+h(x)=0\quad (Rayleigh) \] conditions are given ensuring that: a) all solutions are ultimately bounded, and b) there is some solution which is unbounded.
Reviewer: E.O.Roxin

MSC:

34C11 Growth and boundedness of solutions to ordinary differential equations
37-XX Dynamical systems and ergodic theory
34C25 Periodic solutions to ordinary differential equations
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References:

[1] Barbashin E. A., Tabueva V. A.: Dynamical System with Cylindrical Phase Space. Nauka, Moscow, 1969 · Zbl 0212.43404
[2] Reissig R., Sansone G., Conti R.: Qualitative Theorie nichtlinearerer Differentialgleichungen. Cremonese, Roma, 1963. · Zbl 0114.04302
[3] Barbashin E. A.: Liapunov Functions. Moscow, Nauka, 1970
[4] Voráček J.: Über D’-divergente Lösungen der Differentialgleichung \(x^{(n)}= f(x,x',\ldots, x^{(n-1)}; t)\). Acta UPO 41 (1973), 83-89.
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