Geodesic
Nonexpansive Uniformly Asymptotically Stable Flows are Linear
Canadian mathematical bulletin, Tome 24 (1981) no. 4, pp. 401-407

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We show that if a flow (R, X, π) on a separable metric space (X, d) satisfies (i) the transition mapping π(t, •): X → X is non-expansive for every t ≥ 0; (ii) X contains a globally uniformly asymptotically stable compact invariant subset, then the flow (R, X, π) is linear in the sense that it can be topologically and equivariantly embedded into a flow () on the Hilbert space l2 for which all of the transition mappings are linear operators on l2. 
Janos, Ludvik; McCann, Roger C.; Solomon, J. L. Nonexpansive Uniformly Asymptotically Stable Flows are Linear. Canadian mathematical bulletin, Tome 24 (1981) no. 4, pp. 401-407. doi: 10.4153/CMB-1981-062-4
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     author = {Janos, Ludvik and McCann, Roger C. and Solomon, J. L.},
     title = {Nonexpansive {Uniformly} {Asymptotically} {Stable} {Flows} are {Linear}},
     journal = {Canadian mathematical bulletin},
     pages = {401--407},
     year = {1981},
     volume = {24},
     number = {4},
     doi = {10.4153/CMB-1981-062-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1981-062-4/}
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