Introduction

In this paper we study the occurrence of any possible stability switch from the increase of the value of the time delay $\tau$ for general delay equation

$$x^{^{\prime }}(t)=F(x(t),\;x(t-\tau ))$$ (1.1)

where $x\in \Re ^{n}$, $\tau \in \Re _{+0}=\left[ 0,+\infty \right)$ is a fixed delay and $F:\mathit{C}$ $(\left[ -\tau ,0\right] ,\Re ^{n})\rightarrow \Re ^{n}$ is of class $\mathit{C}^{1}$ with respect both $x(t)$ and $x(t-\tau )$.

We assume that any equilibrium $x^{*}$ of (1.1) is delay dependent, i.e. $F(x_{t})=0$ gives a constant solution

$$x^{*}=x^{*}(\tau )$$ (1.2)

which is continuous and differentiable in $\tau$.

The variation equation around $x^{*}$ (set $u(t)=x(t)-x^{*}$)

$$\dot{u}(t)=\left( \frac{\partial F}{\partial x(t)}\right) _{... ...rtial F}{\partial x(t-\tau )}\right) _{x^{*}(\tau )}u(t-\tau )$$ (1.3)

gives the characteristic equation

$$\det \left\{ \left( \frac{\partial F}{\partial x(t)}\right) ... ...rtial x(t-\tau )} \right) _{x^{*}(\tau )}-\lambda I\right\} =0$$ (1.4)

which in general has delay dependent coefficients, where $\det$ denotes the determinant of a matrix, $I$ is an identity matrix and $\lambda$ are the corresponding characteristic roots.

We give the following definition:

Definition 1.1 A stability switch occurs at $\tau ^{*}\in \Re _{+0}$ if crossing $\tau ^{*}$ for increasing $\tau$ the stability of $x^{*}(\tau )$ changes from asymptotic stability to instability or vice versa.

The most general structure of characteristic equation (1.4) results to be

$$D(\lambda ,\tau )=0$$ (1.5)

where

$$\left\{ \begin{array}{l} D(\lambda ,\tau )=P(\lambda ,\tau )... ...,\ldots ,m,\\ m,n,m_{k}\in \mathit{N}_{0}, \end{array}\right.$$ (1.6)

and $p_{j}(\tau )$, $q_{j}^{(k)}(\tau )$: $\Re _{+0}\rightarrow \Re$ are continuous and differentiable functions of $\tau \in \Re _{+0}$. Generally in (1.6) is $m\leq n$ but herefollowing we remove this condition.

We assume that $\lambda =0$ cannot be a characteristic root, i.e.

$$D(0,\tau )\neq 0\qquad \forall \tau \in \Re _{+0}.$$ (1.7)

In the study of the occurrence of stability switches the following is an essential result. Assume that we rewrite (1.5) as

$$D(\lambda ,\tau )=\lambda ^{n}+g(\lambda ,\tau ).$$ (1.8)

Then, the following theorem holds (see Freedman and Kuang [5]):

Theorem 1.1   Assume that $g(\lambda ,\tau )$ in (1.8) is an analytic function in $\lambda$ and continuous in $\tau$ such that

$$\alpha =\limsup_{_{\Re e\lambda \geq 0,\left\vert \lambda \r... ...fty }}\left\vert \lambda ^{-n}g(\lambda ,\tau )\right\vert <1.$$ (1.9)

Then, as $\tau$ varies in $\Re _{+0},$the sum of multiplicities of roots of $D(\lambda ,\tau )=0$ in the open right half-plane can change only if a root appears on or crosses the imaginary axis.

It is to be noticed that if in (1.6)

$$m_{k}<n,k=1,\ldots ,n,$$ (1.10)

i.e. the degree of polynomial $Q^{(k)}$ in $\lambda$ is lower than the degree $n$ polynomial $P$ in $\lambda$, then assumption (1.9) holds true and Theorem 1.1 applies to the characteristic equation (1.5) and (1.6).

Rewrite $D(\lambda ,\tau )$ in (1.6) like

$$D(\lambda ,\tau )=p_{n}(\tau )\lambda ^{n}+\left[ \sum_{j=0}... ...{j}^{(k)}(\tau )\lambda ^{j}\right) e^{-k\lambda \tau }\right]$$

without loss of generality we assume $p_{n}(\tau )\equiv 1$ and define

$$g(\lambda ,\tau )=\sum_{j=0}^{n-1}p_{j}(\tau )\lambda ^{j}+\... ...{k}}q_{j}^{(k)}(\tau )\lambda ^{j}\right) e^{-k\lambda \tau }.$$ (1.11)

Assume $\lambda$ such that $Re(\lambda )\geq 0$. Then, for any $\tau \geq 0:$

$$\left\vert \lambda ^{-n}g(\lambda ,\tau )\right\vert \leq \sum_{j=0}^{n-1}\frac{ \left\vert p_{j}(\tau )\right\vert }{\left... ... q_{j}^{(k)}(\tau )\right\vert }{\left\vert \lambda \right\vert ^{n-j}} \right)$$

$$\leq \sum_{j=0}^{n-1}\frac{\left\vert p_{j}(\tau )\right\vert }{\left\... ...{n-j}}\right) \rightarrow 0,\left\vert \lambda \right\vert \rightarrow \infty ,$$ (1.12)

since on the right hand side of (1.12) $n-j>n-m_{k}>0$. Therefore, the assumption (1.9) of the Theorem 1.1 holds true. Hence, characteristic equations (1.5) with structure (1.6) which satisfy (1.7) and (1.10), i.e.

$$(A.1)\qquad \left\{ \begin{array}{c} p_{0}(\tau )+\sum_{j=0}... ...u \geq 0 \\ n>m_{k},\qquad k=1,2,\ldots ,m \end{array}\right.$$

may have a stability switch for some $\tau \geq 0$, say $\tau ^{*}$, only if $\lambda =\pm i\omega (\tau ^{*}),$ $\omega (\tau ^{*})\in \Re _{+}$ are characteristic roots.

In the following of the paper we still study the occurrence of stability switches for the subclass of characteristic equations (1.5) where

$$D(\lambda ,\tau )=P(\lambda ,\tau )+Q^{(1)}(\lambda ,\tau )e^{-\lambda \tau }+Q^{(2)}(\lambda ,\tau )e^{-2\lambda \tau }$$ (1.13)

which is already sufficiently general to include as a particular case the characteristic equation

$$D(\lambda ,\tau )=0,\quad D(\lambda ,\tau )=P(\lambda ,\tau )+Q(\lambda ,\tau )e^{-\lambda \tau }$$ (1.14)

recently studied by Beretta and Kuang [3]. An application is also shown in a paper by Beretta, Carletti and Solimano [2]. Therefore, we extend the geometric stability switch criterion developed by Beretta and Kuang [3] for (1.14) to the more general case (1.13). This will be done in the next section.

However, we don't feel that the method we are presenting in the next section could be applied to general characteristic equations (1.6) with $m>2$.

In Section 3 we present an application of the geometric stability switch criterion.

We conclude the paper with Section 4 showing that many of the characteristic equations known in literature are included in the case with structure (1.13), and that related stability switch results are obtained as particular cases of the geometric stability switch criterion presented in Section 2.