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On the significance of integral properties of orbits in some superlinear fixed-period problems. (English) Zbl 1025.34018
The nonlinear two-point boundary value problem \[ u_{xx}+ u^0,\;u(0)=u(L) =0,\;p<1, \;L>0 \] is considered. For every odd \(p\geq 3\) the corresponding positive solution has a constant moment, i.e. it does not depend on \(L\). These integral properties mean the existence of a Hamiltonian for the system of initial value differential equations in which the above system has been transformed. Two numerical methods (a 2-stage Runge-Kutta and a Stormer-Verlet), which preserve the Hamiltonian, are used to integrate the system the positive solutions of which agree fairly well with the author’s previous results.
34B15 Nonlinear boundary value problems for ordinary differential equations
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
35J60 Nonlinear elliptic equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations