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Eliminación de las Grandes Oscilaciones de un Sistema de Ecuaciones Diferenciales

  • Autores: Jorge Rodríguez, Angélica Arroyo, Lesly Salas
  • Localización: MATUA: Revista de matemática de la universidad del Atlántico, ISSN-e 2389-7422, Vol. 2, Nº. 1, 2015 (Ejemplar dedicado a: Revista de Matemática MATUA), págs. 34-50
  • Idioma: español
  • Títulos paralelos:
    • Major elimination Oscillations of a System of Differential Equations
  • Enlaces
  • Resumen
    • español

      Considerando el sistema No lineal.  que tiene un comportamiento oscilatorio, se demuestra en el caso que f(-􀀀x) =- 􀀀f(x), que al reemplazar la función f(x) por f(x+Bsinwt), y para valores de B y w suficientemente grande el sistema no tiene movimiento oscilatorio de gran amplitud. De hecho todas las soluciones tienden a una vecindad del origen tan pequeña como se quiera.Se prueba que para w suficientemente grande, el sistema promediado es una buena aproximación del sistema perturbado. Esto es que toda solución del sistema perturbado está suficientemente cercana a una solución del sistema promediado. Igualmente se prueba que existe Bo tal que B > Bo, la solución trivial del sistema promediado es asintóticamente estable para valores de w suficientemente grandes. Por último se prueba que para B y w suficientemente grande el sistema perturbado no tiene movimiento oscilatorio de gran amplitud, es decir, la perturbación ha aniquilado las oscilaciones de gran amplitud. Palabras claves: Oscilaciones, soluciones periodicas, Sistema Perturbado, Función Promedio, Eliminación de Oscilaciones, soluciones oscilatorias.

    • English

      Considering the nonlinear system x_ = and y_ = z z_ = 􀀀az 􀀀 by 􀀀 f(x) (2) which it has an oscillatory behavior is demonstrated in the case that f(􀀀x) = 􀀀f(x), that by replacing the f(x) function f(x + B sin !t), and will -lores of B and ! large enough the system is oscillatory motion large amplitude. In fact all solutions tend to Origin neighborhood so small as you like.To make this demonstration we proceed as follows: Initially disturbed function in terms of x is expressed and B sin(!t), to proceed to calculate the average function. Then test for h(; x) = f(x + B sin ) 􀀀 f0(x;B), There is a continuous function H(; x; 1! ) such that jH(; x; 1 ! )j !(!) where (!) ! 0 when ! ! 1 and performing substitution z = s + 1 !H(t; x; 1 ! ) shows that the perturbed system is equivalent to the following system x_ = and y_ = z +1! H(t; x; 1!) z_ = 􀀀az 􀀀 by 􀀀 F0(x;B) 􀀀 a 􀀀 1! H(t; x; !) 􀀀 1! @H @x and Thus it is proved that for ! large enough, the averaging system is a good approximation of the system disturbed. This is that any solution of the perturbed system is sufficiently close to a solution of averaging system. There is also evidence that B0 such that B > B0, the solution trivial averaging system is asymptotically stable values of ! sufficiently large. Finally it is proved that for B and ! enough large system has disturbed oscillatory motion large amplitude, that is, the disturbance has destroyed the large amplitude oscillations.

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