By Lamberto Cesari

In the previous few many years the idea of normal differential equations has grown swiftly below the motion of forces that have been operating either from inside and with no: from inside, as a improvement and deepen ing of the thoughts and of the topological and analytical tools led to via LYAPUNOV, POINCARE, BENDIXSON, and some others on the flip of the century; from with no, within the wake of the technological improvement, relatively in communications, servomechanisms, automobile matic controls, and electronics. The early examine of the authors simply pointed out lay in demanding difficulties of astronomy, however the line of proposal therefore produced discovered the main amazing functions within the new fields. The physique of study now accrued is overwhelming, and plenty of books and experiences have seemed on one or one other of the a number of facets of the recent line of study which a few authors name" qualitative idea of differential equations". the aim of the current quantity is to offer the various view issues and questions in a readable brief document for which completeness isn't claimed. The bibliographical notes in each one part are meant to be a advisor to extra unique expositions and to the unique papers. a few conventional themes equivalent to the Sturm comparability idea were passed over. additionally excluded have been all these papers, facing distinct differential equations inspired by way of and meant for the applications.

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**Extra info for Asymptotic Behavior and Stability Problems in Ordinary Differential Equations**

**Sample text**

The Hurwitz criterion is also a particular case of more comprehensive statements concerning the number of zeros of F(z) whose real parts are above, or below a given number, or between two given numbers. Either theory of residues, or Sturm sequences, are used in the proofs of these statements (M. MARDEN [lJ). 3. ii) are also sufficient conditions for the stability in the sense of LYAPUNOV (resp. 5)]. For n large the use of the Routh-Hurwitz criterion is impractical, and other equivalent processes replace it quite well, namely the very same processes by means of which that criterion is usually proved.

Then, the discussion proceeds as in (a) only now for A =l= 0, B=l= 0, the curves have the equations (u/A)Q' (v/B)Q' = 1 and u-+O, V -> 00 as t -+ 00. The trajectories are represented in the illustrations, and the point (0, 0) is said to be a saddle point. , (a+d)2-4(ad+ d =l= 0, and rx. = (a + d)/2. 1) is a linear combination with conjugate coefficients of complex conjugate solutions u of u' = eu, and v = it of v' = (iv. Now in the complex u-plane the first equation has solutions of the form 11 = A eQt = meinelXtei{Jt = melXtei(n+{JtJ.

2) has all solutions bounded in [0, has all solutions bounded (Dini-Hukuhara theorem). (3·3·3) 37 3. Linear systems with variable coefficients This theorem which may be traced back to U. DINI [1J in more particular situations, was proved by H. SPATH [2J, and M. HUKUHARA [2J for li(t) continuous, by L. CESARI [2J for Ii (t)---+O, and under the conditions above by D. CALI GO [3J, R. BELLMAN [1J, H. WEYL [4J, and N. LEVINSON [6]. For systems the theorem above can be given as follows and was essentially proved by the same authors.