The Region of (In)Stability of a 2-Delay Equation Is Connected

✍ Scribed by Thomas Elsken

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 233 KB
• Volume: 261
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2001.7536

✦ Synopsis

described the region of stability for fixed delays r , r and variable a, b, c.

They show how the region of stability evolves in the bc-plane changing the w x parameter a. Hale and Huang 2 gave a geometric description of the stable region in the r r -plane for fixed a, b, c. They do this under the 1 2 assumption that this region is connected. This seems reasonable, but to our knowledge has as yet not been proven.

We are interested in proving that both the region of stability and Ž . instability of 1 are connected. There are no general results known to us w x concerning these issues. The only partial result is a counterexample in 3 which shows that for fixed r , r , a and variable b, c the region of stability 1 2 is not connected.

We shall show that both regions mentioned above are connected in the space of all parameters as well as for fixed delays. Also we prove that the region of instability in the r r -plane, i.e., for fixed a, b, c, is connected. 1 2

Unfortunately, as yet we have not been able to prove the assumption of