22 1. Differential Equations and Their Solutions 1.3.3. Remark. The ODEs modeling many physical systems have periodic orbits, and each such orbit defines a physical “clock” whose natural unit is the prime period of the orbit. We simply choose a con- figuration of the system that lies on this periodic orbit and tick off the successive recurrences of that configuration to “tell time”. The resolu- tion to which before and after can be distinguished with such a clock is limited to approximately the prime period of the orbit. There seems to be no limit to the benefits of ever more precise chronometry— each time a clock has been constructed with a significantly shorter period, it has opened up new technological possibilities. Humankind has always had a 24-hour period clock provided by the rotation of the earth on its axis, but it was only about four hundred years ago that reasonably accurate clocks were developed with a period in the 1- second range. In recent decades the resolution of clocks has increased dramatically. For example, the fundamental clock period for the com- puter on which we are writing this text is about 0.4 × 10−9 seconds. The highest resolution (and most accurate) of current clocks is the cesium vapor atomic clocks used by international standards agencies. These have a period of about 10−11 seconds (with a drift error of about 1 second in 300,000 years!). This means that if two events oc- cur only one hundred billionth of a second apart, one of these clocks can in principle tell which came first. 1.4. Continuity with Respect to Initial Conditions We consider next how the maximal solutions σp of a first-order ODE dx dt = V (x) depends on the initial condition p. Eventually we will see that this dependence is as smooth as the vector field V , but as a first step we will content ourselves with proving just continuity. The argument rests on a simple but important general principle called Gronwall’s Inequality. 1.4.1. Gronwall’s Inequality. Let u : [0,T ) → [0, ∞) be a continuous, nonnegative, real-valued function and assume that u(t) ≤ U(t) := C+K t 0 u(s) ds for certain constants C ≥ 0 and K 0. Then u(t) ≤ CeKt.

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