ABSTRACT
Let X be a complete metric space with metric d. We call T ( t ) : X → X , t ⩾ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq6730.tif"/> an ω-periodic (autonomous) semiflow on X if there is an ω > 0 (for every ω > 0) such that T (t) x is continuous in ( t , x ) ∈ [ 0 , ∞ ) × X , T ( 0 ) = I https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq6731.tif"/> and T ( t + ω ) = T ( t ) T ( ω ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq6732.tif"/> for all t ⩾ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq6733.tif"/> . A point x 0 corresponds to an ω-periodic orbit (equilibrium point) if T ( t + ω ) x 0 = T ( t ) x 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq6734.tif"/> for all t ⩾ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq6735.tif"/> (and every ω > 0). For an ω-periodic semiflow, these x 0 coincide with the fixed points of its associated Poincaré map T(ω).
