Trajectories joining critical points

Zvi Artstein, Marshall Slemrod
1982 Journal of Differential Equations  
A semiflow on a metric space X is a continuous mapping 7+,x): [O, al) xx-+x satisfying ~(0, x) =x and n(t + s, x) = n(t, n(s, x)) for x & X and s, t in 10, a). Typically X is the state space for a differential equation and n(t, x0) is the solution x(r) in positive time with the initial condition x(0) =x,. The properties of n reflect the uniqueness and global existence of solutions for positive time, the well-posedness and the autonomacy of the equation. Let Z be an interval in the real line. A
more » ... unction U(t): Z + X is an orbit on Z if U(r + s) = rr(t, U(s)) whenever t > 0 and s and s + t belong to I. If Z = (-co, co) then U is called a full orbit or a full solution of n. A subset B c: X is positively i~uuri~nt with respect to 71 if x" E B implies x(t, x0) E B for all t > 0. It is i~~~ri~n~ if for every x0 E B a full solution U exists such that U(0) =x" and U(t) E B for t E (-co, co). A point x" is a rest point if n(t, x,) =x0 for all t > 0. Let U be an orbit on an interval (r, co). The o-limit set of CJ, denoted w(U) is the set of all limits z = lim U(t,) for sequences t, -+ co. If U is a full orbit then its o-limit set is the set a(U) of all limits z = lim U(tJ for sequences t, -+ --co. It follows from the continuity of n that the a-limit sets and u-limit sets are positively invariant. Let B, and B, be two subsets of X. The full solution U of n is a connecting orbit between B, and B, if both o(U) and o(U) are not empty. dist(U(f), B,) -+ 0 as f + -co and dist(U(t), B,)-+ 0 as t -+ fco. Here dist(a, B) = inf{d(a, b): b E B} and d(', -) is the metric on X. The orbit U on an interval [r, s] is an &-connecting orbit between B, and B, if dist(U(r), B,) < E and dist(U(s), B,) 6 c. LIMITS OF E-CONNECTING ORBITS Let B, and B, be two disjoint, compact and positively invariant sets, with respect to the semiflow 72. Suppose also that far every E > 0 there is an Econnecting orbit between B, and B,. We intend to derive a connecting orbit between B, and B, as an appropriate limit of the s-connecting orbits, as c--f Cl. This scheme would not work out unless some restrictions are imposed on the flow, as indicated by the following examples. EXAMPLE 3.1. Consider the flow on R2 generated by the ordinary differential equations e = 0 and < = q* + r(c -1)" (2 -{). Clearly, for every E > 0 there is an s-connecting orbit between the rest point (0,O) and the rest point (2,0). For instance, the solution with initial condition (0, E) will generate such an orbit. However, a connecting orbit does not exist. The TRAJECTORIES JOINING CRITICAL POINTS
doi:10.1016/0022-0396(82)90024-9 fatcat:zteilm6o4nbdjcldwwqqgf7bl4