## 1 Introduction  
In previous work, [PERSON] et al. [1, 2] and [PERSON] et al. [3] derived various techniques for a transformation of uncertain linear systems into cooperative state-space representations. An autonomous set of ordinary differential equations (ODEs)  
\[\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t))\,\ \mathbf{x}\in\mathbb{R}^{n} \tag{1}\]  
is cooperative [4, 5, 6] if the property  
\[x_{i}^{(1)}(t)\geq x_{i}^{(2)}(t)\quad\text{for all}\quad i\in\{1,\ldots,n\} \tag{2}\]  
holds for two vectors \(\mathbf{x}^{(1)}(0)\) and \(\mathbf{x}^{(2)}(0)\) of initial conditions which satisfy the inequalities  
\[x_{i}^{(1)}(0)\geq x_{i}^{(2)}(0)\quad\text{for all}\quad i\in\{1,\ldots,n\}\enspace. \tag{3}\]  
Cooperativity can be checked by the sufficient sign conditions  
\[J_{i,j}(\mathbf{x})\geq 0\,\ i,j\in\{1,\ldots,n\}\,\ i\
eq j \tag{4}\]  
for all off-diagonal elements of the Jacobian  
\[\mathbf{J}(\mathbf{x})=\frac{\partial\mathbf{f}}{\partial\mathbf{x}}(\mathbf{ x}) \tag{5}\]of the right-hand side of the state equations evaluated for all reachable states \(\mathbf{x}=\mathbf{x}(t)\). Matrices satisfying this non-negativity property for the off-diagonal elements are also denoted as _Metzler_ matrices in the literature [1; 5; 7; 8]. For such systems, general-purpose, interval-based or other set-valued solvers [9] (AWA [10; 11], CAPD [12], Cosy-VI [13; 14], DynIBEX [15; 16], ValEnCIA-IVP [17; 18; 19; 20; 21], VNODE-LP [22; 23], VSPODE [24]) for initial value problems (IVPs) can be replaced by point-valued simulations of a finite number of extremal realizations that can be extracted from the interval box  
\[\mathbf{x}(0)\in[\mathbf{x}_{0}]=[\mathbf{x}](0):=[\underline{\mathbf{x}}(0) \;;\;\overline{\mathbf{x}}(0)] \tag{6}\]  
of initial conditions. These point-valued extremal system realizations are denoted as _bounding systems_ throughout this paper. A special case of these cooperative systems are so-called positive systems [25]. For initial conditions starting in the positive orthant  
\[\mathbb{R}^{n}_{+}=\left\{\mathbf{x}\in\mathbb{R}^{n}\;\mid\;x_{i}\geq 0\quad \forall i\in\{1,\ldots,n\}\right\}\;, \tag{7}\]  
their trajectories will remain in the positive orthant for all \(t\geq 0\) due to  
\[\dot{x}_{i}(t)=\mathbf{f}(x_{1},\ldots,x_{i-1},0,x_{i+1},\ldots x_{n})\geq 0 \quad\text{for all}\quad i\in\{1,\ldots,n\}\;\;. \tag{8}\]  
Then, worst-case enclosures in the form \(v_{l}(t)\leq x_{i}(t)\leq w_{i}(t)\) are obtained from the decoupled IVPs [26; 27]  
\[\dot{\mathbf{v}}(t)=\mathbf{f}(\mathbf{v}(t)) \tag{9}\]  
and  
\[\dot{\mathbf{w}}(t)=\mathbf{f}(\mathbf{w}(t)) \tag{10}\]  
with the initial conditions  
\[x_{i}^{(1)}(t)\geq x_{i}^{(2)}(t)\quad\text{for all}\quad i\in\{1,\ldots,n\} \tag{2}\]0  
Transformations of linear state equations which enforce cooperativity can be classified into either point-valued approaches or into techniques that employ an interval-valued change of coordinates [28]. In general, point-valued transformations are applicable for parameter-dependent linear systems with purely real eigenvalues. However, finding such transformations becomes more complex for an increasing degree of uncertainty in the system matrix. A failure of the similarity transformation approach by [PERSON] et al. [1; 2] can be recognized by the alternative formulation proposed by the authors in [29]. There, the transformation was cast into an optimization problem constrained by linear matrix inequalities (LMIs). Those LMIs may become infeasible (as a verification that the solution procedure is not successful) or the step size of the iteration in [29] reduces below a certain threshold leading to an excessively slow progress (a non-strict indicator that a point-valued similarity transformation may not exist). Even if interval-valued Metzler matrices can be found for the transformation of a provable asymptotically stable state equation, the transformed dynamics matrix may consist of an unstable upper bound. This commonly results from the wrapping effect in interval computations [30; 31; 32].