Major elimination Oscillations of a System of Differential Equations
Abstract
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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Andronov A. A., Vitt A. A. and Chaikin S. E., Theory of Oscillators, Fizmatgiz., Moscow (1959). Translation: Pergamon Press, Oxford - New York, 1966.
Barbashin, E. A., Conditions for Existence of Recurrent Trajectories in Dynamical Systems with a Cilindrical Phase Space. Differential Equations. 3, 10, 843 - 846.
Bogoliubov, N. N. and Mitropolsky Y. A., Asymptotic Methods in the Theory of Non-Linear Oscillations, Hindustan Publishing Corp. (India), Delhi-6. 1961. Gordon and Breach Science Publishers, Inc. New Yok.
Boyer, R. C., Sinusoidal Signal Stabilization, Master’s Thesis, Purdue University, January, 1960.
Oldenburger, R., and R. C. Boyer, Effects of Extra Sinusoidal Inputs to Nonlinear Systems, Paper 61-Wa-66 presented at Winter Annual Meeting of American Society of Mechanical Engineers, New York, Nov. 26-Dec. 1, 1961.
Cartwright, M. L., On the Stability of Solutions of Certain Differential Equations of the Fourth Order. The Quaterly Journal of Mechanic and Applied Mathematics, 9(1956), pp. 185-194.
Coddington, E.A., and Levinson, N., Perturbations of Linear Sistems with Constant Coefficients Possesing Periodic Solutions, Contributions to the Theory of Nonlinear Oscillations, Vol. 2, pp. 19-37, Annals of
Mathematics Studies, Nº 20, Princeton University Press, Princeton, N.J., 1952.
Coddington, E.A., and Levinson N., Theory of Ordinary Differential Equations. McGraw-Hill Book Company, Inc., New York, 1955.
Ezeilo, J. O., On the Existence of Periodic Solutions of a Certain Third Order Differential Equation. Camb. Philos. 56, 1959.
Ezeilo, J. O., On the Boundedness of Solution of a Certain Differential Equation of the Third Order, Proc. London Math. Soc. (3), 9, 1959.
Ezeilo, J. O., On the Stability of Certain Differential Equations of the Third Order. Quart. J. Math. Oxford (2)11, 64-69 (1960).
Ezeilo, J. O., A Note on a Boundedness Theorem for some Third Order Differential Equations. J. London Math. Soc. 36, 439-444 (1961).
Ezeilo, J. O., A Boundedness Theorem for some Nonlinear Differential Equations of the Third Order. J. London Math. Soc. 37, 469-474 (1962).
Ezeilo, J. O., A Property of the Phase Space Trajectories of a Third Order Nonlinear Differential Equations. J. London Math. Soc. 37 (1962), 33-41.
Ezeilo, J. O., A Stability Result for Solutions of a Certain Fourth Order Differential Equation. J. London Math. Soc. 37(1962), 28-32.
Ezeilo, J. O., Periodic Solutions of a Certain Third Order Differential Equation. Ann. Mat. Pura Appl. 34-41 (1973).
Ezeilo, J. O., A Further Result on the Existence of Periodic Solutions of the Equations ... x + (x_ )x+(x)x_ + (t; x; x_ ; x) = p(t) con acotada. Ann. Mat. Pura Appl. 51-57, (1978).
Friedrichs, K., On Nonlinear Vibrations of Third Order. Studies in Nonlinear Vibration Theory, New York University Lecture Notes 1946, 65-103.
Guckenheimer J. and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer Verlag, New-York, 1983.
Haas V., A Stability Result for a Third Order Nonlinear Differential Equation. Journal London Math. Soc., 40(1965), 31-33.
Hale, J. K., Ordinary Differential Equations.Wiley Intersciencie, New York, 1969.
Hale, J. K., On the Stability of Periodic Solutions of Weakly Nonlinear Periodic and Autonomous Differential Systems. Contributions to the Theory of Nonlinear Oscillations, Vol. 5, pp. 91-114, Annals of Mathematics Studies, Princeton University Press, Princeton, N.J., 1960.
Hale, J. K., Oscillations in Nonlinear Systems, McGraw-Hill Book Company, Inc. New York, 1963.
Krylov, N., and N. Bogoliubov, Introduction to Nonlinear Mechanics, Annals of Mathematics Studies, No 11, Princeton University, Princeton, N.J., 1947.
La Salle, J., and Lefschetz, S., Stability by Lyapunov Direct Method. Academic Press, New York, 1961.
Lefschetz, S., Introduction to Topology, Princeton University Press, Princeton, 1949, pp. 117 -119.
Levinson, N., On the Existence of Periodic Solutions of Second Order Differential Equations with a Forcing Term. J. Math. Phys. (1943), 41-48.
McCarthy, J., A Method for the Calculation of Limit Cycles by Succesive Approximation, Contributions to the Theory of Nonlinear Oscillations, Annals of Matehmatics Studies Nº 29, Vol. 2, Princeton University, Princeton 1952, pp. 75-81.
Nemytskii, V. V., Lyapunov method of rotating functions for investigating oscillations, Dokl. Akad. Nauk SSSR 97 (1954) 33-36.
Perelló, C., Periodic Solutions of Odinary Differential Equations with and without time Lag, Ph. D. Thesis. Brown University, 1965.
Perko L., Differential Equations and Dynamical Systems, 2nd. ed.. Springer Verlag, New-York, 1996.
Pliss, V. A., Nonlocal Problems of the Theory of Oscillations. Academic Press, New York and London, 1966.
Qing, L., On the Construction of Globally Asymptotically Stable Lyapunovs Functions of a Type of Nonlinear Third-Order Systems. Ann. of Diff. Eqs. pp. 39-51, 1991.
Rauch, L. L., Oscillation of a Third Order Nonlinear Autonomous System, Contributions to the Theory of Nonlinear Oscillations, Vol. 1, Princeton University Press, Princeton, 1950, pp. 39-88.
Reuter, G.E.H. Note on the Existence of Periodic Solutions of Certain Differential Equations.
Reissig, R., On the Existence of Periodic Solutions of a Certain Non-Autonomous Differential Equation. Ann. Mat. Pura Appl. (IV) 85 (1969), 235-240.
Reissig, R., Periodic Solutions of a Nonlinear n-th Order Vector Differential Equation. Ann. Mat. Pura Appl. July 1970, pp. 111-123.
Reissig, R., Periodic Solutions of a Third Order Nonlinear Differential Equation. Ann. Mat. Pura Appl. September 1971, pp. 194 - 198.
Reissig, R., Perturbation of Certain Critical n-th Order Differential Equation. Bolletino U. M. I. (4)11, Suppl. fasc. 3(1975), 131 - 141.
Wolfgang Wasow, The Construction of Periodic Solutions of Singular Perturbation Problems, Contributions to the Theory of Nonlinear Oscillations, Vol. 1, Princeton University, Princeton, 1950, pp. 313-350.
