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| THE BEHAVIOR OF SOLUTIONS IN THE VICINITY OF A BOUNDED SOLUTION TO AUTONOMOUS DIFFERENTIAL EQUATIONS |
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Citation： |
Lin Guotian.THE BEHAVIOR OF SOLUTIONS IN THE VICINITY OF A BOUNDED SOLUTION TO AUTONOMOUS DIFFERENTIAL EQUATIONS[J].Chinese Annals of Mathematics B,1986,7(2):205~212 |
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Net amount： 688 |
Authors： |
Lin Guotian; |
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| Abstract： |
By using the exponential dichotomy, this paper investigates the behavior of solutions in the vicinity of a bounded.solution to the autonomous differerntial system:
$$\[\frac{{dx}}{{dt}} = f(x)\]$$
Suppose $\[x = u(t)\]$ is a nontrivial bounded solution of system (1), By discussing the equivalent equations of system(1)
$$\[\frac{{d\theta }}{{dt}} = 1 + {\overline f _1}(\rho ,\theta )\]$$
$$\[\frac{{d\rho }}{{dt}} = A(\theta )\rho + {\overline f _2}(\rho ,\theta )\]$$
with respect to the moving orthonormal transformation
$$\[x = u(\theta ) + s(\theta )\rho \]$$
the author proves that if linear system corresponding to (2) admits exponential dichotomy, then the given bounded solution $\[x = u(t)\]$ should be periodic. The author also discusses the
stadility of the obtained periodic solution. Finally, this paper discusses perturbation of the bounded solution of autonomous system (l). |
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