# The Bohl Spectrum for Linear Nonautonomous Differential Equations

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## Abstract

We develop the Bohl spectrum for nonautonomous linear differential equations on a half line, which is a spectral concept that lies between the Lyapunov and the Sacker–Sell spectra. We prove that the Bohl spectrum is given by the union of finitely many intervals, and we show by means of an explicit example that the Bohl spectrum does not coincide with the Sacker–Sell spectrum in general even for bounded systems. We demonstrate for this example that any higher-order nonlinear perturbation is exponentially stable (which is not evident from the Sacker–Sell spectrum), but we show that in general this is not true. We also analyze in detail situations in which the Bohl spectrum is identical to the Sacker–Sell spectrum.

## Keywords

Bohl exponent Bohl spectrum Lyapunov exponent Nonautonomous linear differential equation Sacker–Sell spectrum## Mathematics Subject Classification

34A30 34D05 37H15## 1 Introduction

The stability theory for linear nonautonomous differential equations has its origin in A.M. Lyapunov’s celebrated PhD Thesis [20], where he introduces characteristic numbers, so-called *Lyapunov exponents*, which are given by accumulation points of exponential growth rates of individual solutions. It is well-known that in case of negative Lyapunov exponents, the stability of nonlinearly perturbed systems is not guaranteed without an additional regularity condition.

In the 1970s, R.S. Sacker and G.R. Sell developed the Sacker–Sell spectrum theory for nonautonomous differential equations. In contrast to the Lyapunov spectrum, the Sacker–Sell spectrum is not a solution-based spectral theory, but rather is based on the concept of an exponential dichotomy, which concerns uniform growth behavior in subspaces and extends the idea of hyperbolicity to explicitly time-dependent systems. If the Sacker–Sell spectrum lies left of zero, then the uniform exponential stability of nonlinearly perturbed systems is guaranteed.

It was shown in [22] that the regularity condition on Lyapunov exponents can be more robustly replaced by a nonuniform exponential dichotomy. Here the nonuniformity refers to time, and in contrast to that, so-called Bohl exponents, introduced by Bohl [11], measure exponential growth along solutions uniformly in time. Bohl exponents have been studied extensively in the literature [14], and current research focuses on applications to differential-algebraic equations and control theory [2, 10, 17, 19, 30], and parabolic partial differential equations [23]. In this paper, we develop the Bohl spectrum as union of all possible Bohl exponents of a nonautonomous linear differential equation on a half line. We show that the Bohl spectrum lies between the Lyapunov and the Sacker–Sell spectrum and that the Bohl spectrum is given by the union of finitely many (not necessarily closed) intervals. Each Bohl spectral interval is associated with a linear subspace, leading to a filtration of subspaces which is finer than the filtration obtained by the Sacker–Sell spectrum.

We show by means of an explicit example that the Bohl spectrum can be a proper subset of the Sacker–Sell spectrum even if the system is bounded. We analyze in detail situations in which the Bohl spectrum is identical to the Sacker–Sell spectrum, and in particular, we obtain this for bounded diagonalizable systems, integrally separated systems, and systems with Sacker–Sell point spectrum. The fact that the Bohl and Sacker–Sell spectra coincide for diagonalizable systems shows that the Bohl spectrum mainly gives information about the asymptotic behaviour of individual solutions whereas the Sacker–Sell also embodies information about the relation between different solutions, in particular, whether or not the angle between solutions is bounded below by a positive number. An interesting problem in this context is to give necessary and sufficient conditions that the Bohl and Sacker–Sell spectra coincide.

The example referred to above shows that the Sacker–Sell spectrum can extend past zero even when the Bohl spectrum is given by a negative number. We demonstrate for this example that any higher-order nonlinear perturbation is exponentially stable, although this is not evident from the Sacker–Sell spectrum. In the last section of this paper, we discuss an example with negative Bohl spectrum such that for a certain nonlinear perturbation, the perturbed system is unstable. This means that it is not possible to prove in general that if the Bohl spectrum lies to the left of zero, then any higher-order nonlinear perturbation is exponentially stable. In a forthcoming paper, we will provide additional conditions on the nonlinearities which give a positive answer to this question, even in situations where the Sacker–Sell spectrum intersects the positive half axis.

This paper is organized as follows. In Sect. 2, we provide basic material on the Lyapunov and Sacker–Sell spectrum, and in Sect. 3, we introduce the Bohl spectrum. Section 4 is devoted to prove the Spectral Theorem, which says that the Bohl spectrum is given by the union of finitely many intervals. We compare the Bohl spectrum and the Sacker–Sell spectrum in Sect. 5, and we discuss nonlinear perturbations to linear systems with negative Bohl spectrum in Sect. 6.

## 2 Lyapunov and Sacker–Sell Spectrum

In this section, we review the definition and basic properties of the two main spectral concepts for nonautonomous differential equations: the Lyapunov spectrum and the Sacker–Sell spectrum.

*fundamental matrix*of (1), i.e. \(X(\cdot )\xi \) solves (1) with the initial value condition \(x(0)=\xi \), where \(\xi \in \mathbb {R}^d\).

The Lyapunov spectrum describes asymptotic growth of individual solutions of (1).

**Definition 1**

*Lyapunov spectrum*) The

*lower*and

*upper characteristic Lyapunov exponents*of a particular non-zero solution \(X(\cdot )\xi \) of (1) are defined by

*Lyapunov spectrum of*(1) is then defined as

**Definition 2**

*Exponential dichotomy*) The linear differential equation (1) admits an

*exponential dichotomy*with growth rate \(\gamma \in \mathbb {R}\) if there exist a projector \(P\in \mathbb {R}^{d\times d}\), and constants \(K\ge 1\) and \(\alpha >0\), such that

*exponential dichotomy*with growth rate \(\infty \) if there exists a \(\gamma \in \mathbb {R}\) such that (1) admits an

*exponential dichotomy*with growth rate \(\gamma \) and projector \(P=\mathbbm {1}\), and (1) is said to admit an

*exponential dichotomy*with growth rate \(-\infty \) if there exists a \(\gamma \in \mathbb {R}\) such that (1) admits an

*exponential dichotomy*with growth rate \(\gamma \) and projector \(P=0\), the zero matrix.

The range of the projector *P* of an exponential dichotomy is called the *pseudo-stable space*, and the null space of the projector *P* is called a *pseudo-unstable space*. Note that in contrast to the pseudo-unstable space, the pseudo-stable space is uniquely determined for exponential dichotomies on \(\mathbb {R}^+_0\) [27].

The Sacker–Sell spectrum is then given by set of all growth rates \(\gamma \) such that the linear system does not admit an exponential dichotomy with growth rate \(\gamma \).

**Definition 3**

*Sacker–Sell spectrum*) The

*Sacker–Sell spectrum*of the linear differential equation (1) is defined by

The Sacker–Sell spectrum was introduced by Sacker and Sell in [28] for skew product flows with compact base. It was generalized to nonautonomous dynamical systems with not necessarily compact base in [3, 29] and for systems defined on a half-line in [27].

The Spectral Theorem (see [18, 27] for the half-line case) describes the structure of the dichotomy spectrum.

**Theorem 4**

Note that the linear space \({\mathcal {W}}_i\) is the pseudo-stable space of the exponential dichotomy with any growth rate taken from the spectral gap interval \((b_i, a_{i+1})\) for \(i\in \{1,\dots ,k-1\}\).

The following result on Sacker–Sell spectra of upper triangular systems follows from [9]. Note that such a statement is only true in the half-line case and does not hold for Sacker–Sell spectra on the entire time axis as demonstrated in [9].

**Proposition 5**

*A*(

*t*) are bounded, then we have the representation

*Remark 6*

*A*(

*t*) are unbounded. As a counter example consider the one-dimensional system

*a*(

*t*) is arbitrarily close to \(-\infty \) and \(\infty \) on intervals of the length one. This shows that the representation (2) does not hold for unbounded coefficient matrices.

## 3 The Bohl Spectrum

We first define the Bohl spectrum for each solution of (1). The Bohl spectrum of (1) is then the union over the Bohl spectra of the solutions.

**Definition 7**

*Bohl spectrum*) Consider the linear nonautonomous differential equation (1) in \(\mathbb {R}^d\). The

*Bohl spectrum of a particular solution*\(X(\cdot )\xi , \xi \not =0\), of (1) is defined as

*Bohl spectrum of*(1) is defined as

*Remark 8*

(i) By Definition 1, we have \(\chi _{-}(\xi ), \chi _{+}(\xi )\in \Sigma _{\xi }\) for any \(\xi \in \mathbb {R}^d\setminus \{0\}\), and we see that in contrast to looking at the asymptotic behavior at infinity of a solution by using the Lyapunov exponent, the Bohl spectrum of this solution provides all possible growth rates of this solution when the length of observation time tends to infinity and the initial time is arbitrary.

*upper*and

*lower Bohl exponent*of a solution \(X(\cdot )\xi \) are defined by

(iii) The definition of Bohl spectrum is independent of the norm in \(\mathbb {R}^d\).

(iv) A different definition of a Bohl spectrum for discrete systems depending on certain invariant splittings was proposed in [25, Definition 3.8.1], and another spectrum between the Lyapunov and Sacker–Sell spectrum based on nonuniform exponential dichotomies was introduced in [12].

**Proposition 9**

- (i)We have the representationi.e. in the definition of Bohl spectrum we can always assume \(s_n\rightarrow \infty \).$$\begin{aligned} \Sigma _{\xi }:=&\Big \{\lambda \in \overline{\mathbb {R}}: \hbox { there exist sequences } \{t_n\}_{n\in \mathbb {N}}\hbox { and }\{s_n\}_{n\in \mathbb {N}}\hbox { with }\\&\quad t_n-s_n\rightarrow \infty \text { and }s_n\rightarrow \infty \hbox { such that } \lim _{n\rightarrow \infty } \tfrac{1}{t_n-s_n}\ln \tfrac{\Vert X(t_n)\xi \Vert }{\Vert X(s_n)\xi \Vert }=\lambda \Big \}\,, \end{aligned}$$
- (ii)
\(\Sigma _{\xi }=\Sigma _{\lambda \xi }\) for all \(\lambda \in \mathbb {R}\setminus \{0\}\).

- (iii)
\(\Sigma _{\xi }=\big [\underline{\beta }(\xi ), \overline{\beta }(\xi )\big ]\).

- (iv)Suppose that there exists a constant \(M>0\) such thatThen \(\Sigma _{\xi } \subset [-M,M]\).$$\begin{aligned} \Vert A(t)\Vert \le M \quad \quad \text {for almost all }\,t\in \mathbb {R}_0^+. \end{aligned}$$(3)

*Proof*

*Case* 1 The sequence \(\{s_n\}_{n\in \mathbb {N}}\) is unbounded. Then there exists a subsequence \(\{s_{k_n}\}_{n\in \mathbb {N}}\) of \(\{s_n\}_{n\in \mathbb {N}}\) such that \(\lim _{n\rightarrow \infty } s_{k_n}=\infty \). Letting \(\widetilde{s}_n:=s_{k_n}\) and \(\widetilde{t}_n:=t_{k_n}\). Then these sequences satisfy (5).

*Case*2 The sequence \(\{s_n\}_{n\in \mathbb {N}}\) is bounded. Let \(\Gamma :=\sup _{n\in \mathbb {N}} s_n\), and let \(n\in \mathbb {N}\) be an arbitrary positive integer. Since \(\lim _{m\rightarrow \infty } t_m-s_m=\infty \) and

(ii) This assertion follows directly from Definition 7.

**Proposition 10**

Consider a linear nonautonomous differential equation \(\dot{x}=A(t)x\) in \(\mathbb {R}^d\), and let *x*(*t*), *y*(*t*) be solutions such that the angle between them is bounded below by a positive number. Then if \(\alpha \beta \ne 0\), the solutions \(t\mapsto \alpha x(t)+\beta y(t)\) all have the same Bohl spectrum.

*Proof*

*Remark 11*

## 4 Spectral Theorem

We prove in this section that the Bohl spectrum of a locally integrable linear nonautonomous differential equation consists of at most finitely many intervals, the number of which is bounded by the dimension of the system, and we associate a filtration of subspaces to these spectral intervals.

**Theorem 12**

*k*(not necessarily closed) disjoint intervals, i.e.

*Proof*

*n*:

- (a)
If \(\pm \infty \in \Sigma _\mathrm{Bohl}\), then \(d_0\ge 1\) and \(d_n\le d-1\) and therefore \(n\le d-2\).

- (b)
If \(-\infty \in \Sigma _\mathrm{Bohl}\) and \(+\infty \not \in \Sigma _\mathrm{Bohl}\), then \(d_0\ge 1\) and therefore \(n\le d-1\).

- (c)
If \(-\infty \not \in \Sigma _\mathrm{Bohl}\) and \(+\infty \in \Sigma _\mathrm{Bohl}\), then \(d_n\le d-1\) and therefore \(n\le d-1\).

- (d)
If \(-\infty \not \in \Sigma _\mathrm{Bohl}\) and \(+\infty \not \in \Sigma _\mathrm{Bohl}\), then \(n\le d\).

*k*denote the number of disjoint intervals \(I_i\) of \(\Sigma _\mathrm{Bohl}\). According to the cases (a–d) above, we have the following dependence of

*k*and

*n*:

- (i)
\(k=n+2\) in case (a) above,

- (ii)
\(k=n+1\) in case (b) and (c) above,

- (iii)
\(k=n\) in case (d) above.

*n*and

*d*established above, we always obtain that \(k\le d\). To conclude the proof, for each \(i\in \{1,\dots ,k\}\), we define the set \({\mathcal {S}}_i\) as in (10) together with \(\{0\}\). Note that the space \(\mathcal {S}_i\) coincides with \(\mathcal {M}_\lambda \) for \(\lambda = \frac{1}{2} (\sup I_i + \inf I_{i+1})\), where \(i\in \{1,\dots ,k-1\}\), and \(\mathcal {S}_k=\mathcal {M}_\lambda =\mathbb {R}^d\) for \(\lambda > \sup I_k\). Then, clearly \({\mathcal {S}}_i\) is a linear subspace and satisfies (9). This finishes the proof. \(\square \)

Next, we concentrate on constructing an example of a nonautonomous differential equation such that its Bohl spectrum is not closed. Our construction is implicit by using a result from [4]:

*uniform upper exponent function*of (13), \(\overline{\beta }_A:\mathbb {R}^{d}\setminus \{0\}\rightarrow \mathbb {R}\), where \(\overline{\beta }_A(\xi )\) is the upper Bohl exponent of the solution \(X(t)\xi \) of (13). A complete description of the set of functions \(\overline{{\mathcal {B}}}_d:=\{\overline{\beta }_A: A\in {\mathcal {M}}_d\}\) is given as follows (see [4, Theorem 1]).

**Theorem 13**

- (i)
\(\beta \) is bounded.

- (ii)
\(\beta (\xi )=\beta (\alpha \xi )\) for any nonzero \(\alpha \in \mathbb {R}\) and \(\xi \in \mathbb {R}^d\setminus \{0\}\).

- (iii)
For any \(q\in \mathbb {R}\), the set \(\{\xi : \beta (\xi )\ge q\}\) is a \(G_{\delta }\) set.

The following example shows that the intervals of the Bohl spectrum do not need to be closed.

*Example 14*

*kinematically similar*to another linear nonautonomous differential equation

*S*and \(S^{-1}\) are bounded, and which satisfies the differential equation

**Proposition 15**

(Invariance of the Bohl spectrum under kinematic similarity transformations) Suppose that (15) and (16) are kinematically similar. Then the Bohl spectra \(\Sigma _\mathrm{Bohl}(A)\) and \(\Sigma _\mathrm{Bohl}(B)\) of (15) and (16) coincide.

*Proof*

*S*(0) is invertible it follows that \(\Sigma _\mathrm{Bohl}(A)=\Sigma _\mathrm{Bohl}(B)\) and the proof is complete. \(\square \)

## 5 Bohl and Sacker–Sell Spectrum

This section is devoted to the comparison of the Bohl spectrum with the Sacker–Sell spectrum. Note that the one-dimensional example discussed in Remark 6 is an unbounded system for which the both spectra do not coincide, since the Bohl spectrum of this differential equation is given by \(\{-\infty \}\). It follows also directly from Example 14 that the Bohl spectrum does not always coincide with the Sacker–Sell spectrum, since the Sacker–Sell spectrum consists of closed intervals. In this section, we give an explicit example of a bounded two-dimensional system for which the Sacker–Sell spectrum is a nontrivial interval and the Bohl spectrum is a single point. We also show that the Bohl spectrum is always a subset of the Sacker–Sell spectrum, and we provide sufficient conditions under which both spectra coincide.

### 5.1 The Bohl Spectrum Can Consist of One Point, When the Sacker–Sell Spectrum is a Non-trivial Interval

**Proposition 16**

Before proving the above proposition, we need the following lemma.

**Lemma 17**

Let \(t\mapsto (x(t),y(t))\) be an arbitrary nonzero solution of (20) with \(y(0)\not =0\). Then there exists \(T>0\) such that *x*(*t*) and *y*(*t*) have the same sign for all \(t\ge T\) .

*Proof*

It follows from (21) that there exists a \(k\in \mathbb {N}\) with \(x(T_{2k+1})\ge 0\), and we also have \(y(T_{2k+1})>0\). Since the first quadrant is positively invariant under both systems, it follows that \(x(t)\ge 0\) and \(y(t)\ge 0\) for \(t\ge T_{2k+1}\). If \(y(0)<0\), we get the required result by considering the solution \(-(x(t),y(t))\). \(\square \)

*Proof of Proposition 16*

*a*(

*t*),

*b*(

*t*)), where \(a(t) = -1\) and \(-1 \le b(t) \le 0\). \(\dot{x}=a(t)x\) has Sacker–Sell spectrum \(\{-1\}\), and the Sacker–Sell spectrum of \(\dot{y}=b(t)y\) is contained in \([-1,0]\). However,

*x*(

*t*) and

*y*(

*t*) have the same sign for \(t\ge T_{2K}\) and

### 5.2 Coincidence of the Bohl and Sacker–Sell Spectrum in Special Cases

We first show that the Bohl spectrum is a subset of the Sacker–Sell spectrum. We then show that the two spectra coincide when the Sacker–Sell spectral intervals are singletons. Finally, we show that the Bohl and Sacker–Sell spectra coincide for bounded diagonalizable, and hence, bounded integrally separated systems.

*integrally separated*if there exists \(K\ge 1\) and \(\alpha >0\) such that

*A*(

*t*) is bounded, then the angle between two such solutions is bounded below by a positive number.

In the next lemma, we show that when the solutions are integrally separated and \(X(t)\xi \) is the bigger solution in the above sense, then the Bohl spectrum of any non-trivial linear combination of \(X(t)\xi \) and \(X(t)\eta \) is always given by \(\Sigma _\xi \).

**Lemma 18**

*Proof*

We first use this lemma to show that the Bohl spectrum is a subset of the Sacker–Sell spectrum. As a consequence, the filtration corresponding to the Bohl spectrum is finer than the filtration corresponding to Sacker–Sell spectrum.

**Theorem 19**

- (i)
The Bohl spectrum is a subset of the Sacker–Sell spectrum.

- (ii)
The filtration associated with the Bohl spectrum is finer than the one of Sacker–Sell spectrum.

*Proof*

*P*, then \(\Sigma _{\xi }\subset [\lambda +\alpha ,\infty )\). Thus, \(\lambda \not \in \Sigma _\mathrm{Bohl}\), which finishes the proof of (i).

**Theorem 20**

(Bohl and Sacker–Sell spectra of diagonalizable systems) Suppose that the bounded linear nonautonomous differential equation (1) is *diagonalizable*, i.e. it is kinematically similar to a (nonautonomous) diagonal system. Then the Bohl and Sacker–Sell spectrum of (1) coincide. In particular, both spectra coincide for bounded one-dimensional systems.

*Proof*

*Remark 21*

Proposition 16 and Theorem 20 also show that the Bohl spectrum of a bounded upper triangular system is, in general, not equal to that for the diagonal part, unlike the situation for the Sacker–Sell spectrum in the bounded half-line case (see also Proposition 5). However the Bohl spectrum of the triangular system is a subset of the Sacker–Sell spectrum (see Theorem 19 above), which equals the Sacker–Sell spectrum of the diagonal part, and the Sacker–Sell spectrum of the diagonal part coincides with its Bohl spectrum (see Theorem 20 above). We conclude that for bounded systems, the Bohl spectrum of an upper triangular system is a subset of the Bohl spectrum of its diagonal part.

The linear nonautonomous differential equation (1) is said to be *integrally separated* if there are *d* independent solutions \(X(t)\xi _1,\dots ,X(t)\xi _d\) such that \(X(t)\xi _i\) and \(X(t)\xi _{i+1}\) are integrally separated for all \(i\in \{1,\dots ,d-1\}\).

We now prove using the previous theorem that the Bohl and Sacker–Sell spectra coincide for bounded integrally separated systems. This means also that the Bohl spectrum depends continuously on parameters for such systems.

**Corollary 22**

Suppose that system (1) is integrally separated, and *A*(*t*) is bounded in \(t\in \mathbb {R}^+_0\). Then the Bohl spectrum coincides with the Sacker–Sell spectrum of (1).

*Proof*

*Remark 23*

*A*(

*t*) in the above corollary is needed, since there exists an unbounded integrally separated system which is not diagonalizable such that its Bohl spectrum and and its Sacker–Sell spectrum are different. Consider the system \(\dot{x}=A(t)x\), where

*A*(

*t*) is defined by

*X*(

*t*) of this system is given by

*X*(

*t*), we see that the system is not reducible and hence \(\Sigma _{\mathrm{SS}}\) is an interval containing the points 0 and 2.

**Corollary 24**

(Coincidence is generic) The Bohl spectrum and the Sacker–Sell spectrum coincide generically for bounded linear nonautonomous differential equations.

We demonstrate by means of a counterexample that the Bohl spectrum in not even upper semi-continuous in general with perturbations to the right-hand side in the \(L^{\infty }\)-norm. Note that the Sacker–Sell spectrum is upper semi-continuous in general, and in [26], sufficient criteria for continuity of the Sacker–Sell spectrum are established.

**Corollary 25**

(Discontinuity of the Bohl spectrum) The mapping \(A\mapsto \Sigma _\mathrm{Bohl}(A)\) is not upper semi-continuous in general.

*Proof*

Suppose the Sacker–Sell spectrum consists of points. Then by Theorem 19, the Bohl spectrum consists of points. We still need to prove each point in the Sacker–Sell spectrum is also in the Bohl spectrum. This follows from the next lemma.

**Lemma 26**

Let [*a*, *b*] be a spectral interval of the Sacker–Sell spectrum of the linear nonautonomous differential equation (1). Then there exists a solution whose Bohl spectrum is contained in [*a*, *b*].

*Proof*

**Corollary 27**

If the Sacker–Sell spectrum consists of points, then it coincides with the Bohl spectrum.

*Remark 28*

## 6 Nonlinear Perturbations

This section is devoted to study whether the trivial solution of a nonlinearly perturbed system with negative Bohl spectrum is asymptotically stable. Note that if the Sacker–Sell spectrum is negative, then nonlinear stability follows directly, but we will show below by means of a counter example that we cannot obtain such a result for the Bohl spectrum. Before doing so, we look at the example from Sect. 5.1 with negative Bohl spectrum, and we prove that the system is exponentially stable for any nonlinear perturbation. Since the Sacker–Sell spectrum of this linear system is not negative, this shows that even in those cases, stability for the nonlinear system can follow. Despite the fact that negative Bohl spectrum does not imply nonlinear stability, in a forthcoming paper, we will discuss additional conditions on the nonlinearity that guarantee nonlinear stability for systems with negative Bohl spectrum, which include cases where the Sacker–Sell spectrum cannot indicate stability.

**Proposition 29**

*Proof*

*Remark 30*

*a*,

*b*] and Bohl spectrum \(\{a\}\) since either spectrum of \(\dot{x}=\gamma A(\gamma t)x\) is \(\gamma \) times the spectrum of \(\dot{x}=A(t)x\), and the spectrum of \(\dot{x}=(A(t)+b)x\) is the translation of the spectrum of \(\dot{x}=A(t)x\) by the number

*b*. Taking \([a,b] = [-1+\varepsilon ,\varepsilon ]\), where \(0<\varepsilon <1\), implies that we get a system with \(\Sigma _\mathrm{Bohl}=\{-1+\varepsilon \}\) and \(\Sigma _\mathrm{SS} = [-1+\varepsilon ,\varepsilon ]\), and we obtain asymptotic stability for nonlinear perturbations similarly to Proposition 29, although the Sacker–Sell spectrum has nontrivial intersection with the position half line.

**Lemma 31**

*x*(

*t*),

*y*(

*t*)) be a solution of (39). Then

*Proof*

**Proposition 32**

The Bohl spectrum of (39) satisfies \(\Sigma _\mathrm{Bohl}\le -\alpha <0\).

*Proof*

Fix an initial condition \((x_0,y_0)\in \mathbb {R}^2\), and let \(\xi (t)=(x(t),y(t))^{\mathrm {T}}\) denote the solution of (39) with \(\xi (0)=(x_0,y_0)^{\mathrm {T}}\). Obviously, \(\Sigma _{\xi }=\Sigma _{-\xi }\) and we thus may assume that \(y_0\ge 0\). Let \(\mathbb {R}^2\) be endowed with the maximum norm for the remainder of this proof. We consider the following two cases.

*Case* 1 \(y_0=0\). Then we have \(\xi (t)=(e^{-\alpha t},0)^{\mathrm {T}}\), which implies \(\Sigma _{\xi }=\{-\alpha \}\).

*Case*2 \(y_0\not =0\). By the variation of constants formula, we have

*Case*2.1 \(x_0\not =-\delta \int _0^{\infty }e^{\alpha s} y(s)\,\mathrm {d}s\). Then

*Case*2.2 \(x_0=-\delta \int _0^{\infty }e^{\alpha s} y(s)\,\mathrm {d}s\). Then

*x*-component of \(\xi \), we now compare \(e^{\alpha t}\Vert \xi (t)\Vert \) and \(e^{\alpha s}\Vert \xi (s)\Vert \) with \(t\ge s\). The following statements hold.

- (i)For all \(t,s\in [2^{2k+1}, 2^{2k+2})\) with \(t\ge s\), we have \(e^{\alpha t}|y(t)|\le e^{\alpha s} |y(s)|\), and with (41), we get$$\begin{aligned} \Vert \xi (t)\Vert \le e^{-\alpha (t-s)}\Vert \xi (s)\Vert \,. \end{aligned}$$
- (ii)For all \(t,s\in [2^{2k}, 2^{2k+1}-1)\) with \(t\ge s\) and \(k\in \mathbb {N}\), note that in the interval \([2^{2k}, 2^{2k+1}]\) the function
*y*(*t*) is increasing, so we havefor all \(t,s\in [2^{2k}, 2^{2k+1}-1)\) with \(t\ge s\). Hence, \(\Vert \xi (t)\Vert =|x(t)|\), and from (41), we obtain$$\begin{aligned} e^{\alpha t}|x(t)|=\delta \int _t^{\infty } e^{\alpha s} y(s)\,\mathrm {d}s \ge \int _t^{2^{2k+1}} e^{\alpha s} y(s)\,\mathrm {d}s \ge e^{\alpha t} y(t) \end{aligned}$$$$\begin{aligned} \Vert \xi (t)\Vert \le e^{-\alpha (t-s)}\Vert \xi (s)\Vert \,. \end{aligned}$$ - (iii)For \(2^{2k+1}-1\le s \le t\le 2^{2k+1}\), we havewhere \(M=\max \{\alpha +\delta ,\gamma \}\) is the operator norm of \(A_2\) with respect to the maximum norm. In particular,$$\begin{aligned} \Vert \xi (t)\Vert \le e^{M(t-s)}\Vert \xi (s)\Vert \,, \end{aligned}$$$$\begin{aligned} \frac{\Vert \xi (2^{2k+1})\Vert }{\Vert \xi (2^{2k})\Vert } = \frac{\Vert \xi (2^{2k+1})\Vert }{\Vert \xi (2^{2k+1}-1)\Vert } \frac{\Vert \xi (2^{2k+1}-1)\Vert }{\Vert \xi (2^{2k})\Vert } \le e^{M+\alpha } e^{-\alpha (2^{2k+1}-2^{2k})}\,. \end{aligned}$$(42)

The following proposition shows that, although the Bohl spectrum is bounded above by \(-\alpha <0\), for certain nonlinear perturbation of (39), the system is unstable.

**Proposition 33**

*Proof*

Let \((x_0,y_0)\) be an initial condition at time \(t=0\) for the solution (*x*(*t*), *y*(*t*)) with \(x_0>0\) and \(y_0>0\). We prove \(\limsup _{t\rightarrow \infty }y(t)=\infty \) with the following two steps.

*Step*1 We show that both

*x*(

*t*) and

*y*(

*t*) are positive for all \(t\ge 0\). If we suppose the contrary, then we can define

*Case*1 If \(x(T^*)=0\), then by variation of constants formula we have

*Case* 2 If \(y(T^*)=0\), then \(\dot{y}(T^*)= x(T^*)^2>0\). Thus, there exists \(\varepsilon >0\) such that \(y(T^*-\varepsilon )<0\). This contradicts the definition of \(T^*\).

*Step*2 We estimate

*y*(

*t*). From the variation of constants formula for the first component, we have

Note that in a forthcoming paper, we will discuss additional conditions on the nonlinearity that guarantee nonlinear stability for systems with negative Bohl spectrum, which include cases where the Sacker–Sell spectrum cannot indicate stability.

## Notes

### Acknowledgments

The authors are grateful to an anonymous referee for useful comments that led to an improvement of this paper. Thai Son Doan was supported by a Marie-Curie IEF Fellowship, and Martin Rasmussen was supported by an EPSRC Career Acceleration Fellowship EP/I004165/1 (2010–2015).

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