18.034 at ESG Spring, 2003Comments on Hirsch & Smale Chapter 8First, go to the end of the chapter, where the note discusses the view of thederivative as a linear transformation. This view is that of Apostol, presented inChapter 8 of Volume II, especially Section 8.11. The details involved in presentingthis form of a derivative will not be presented here (possibly later, in a separateset of notes), but one point will be mentioned: At the bottom of the page (H&SPage 178), it’s said that “If this map is continuous, then g is said to be C1.Ifthis map is C1itself, then g is said to be C2.” For now, an example of a mapx → Dg(x), where x ∈ R2,andwhichisnotC1, is linked from the 18.03(4)-ESGweb page.Second, go back to the beginning of the chapter, and pay heed to the sugges-tion to “browse through the chapter, omitting the proofs until the purpose of thetheorems begins to fit into place.” For a further suggestion, consult another text,either before the first reading of H&S or between your first and second readings.Two suggested possibilities are Edwards & Penney or Borrelli & Coleman. If youchoose the E&P text, yet a further suggestion is to take a look at the online notesfor the Existence and Uniqueness Theorem, linked from the 18.03(4)-ESGpage under 18.03IS at ESG.Back to the H&S text. This texts uses the method of Picard iterates,asdoallof the sources mentioned. This method will allow us to say that a solution exists,under certain circumstances, and that the solution will be unique, but will not tellus what the solution is. The honest truth is, this is the best we can do. If we couldprove the Existence and Uniqueness Theorem (EUT) by exhibiting a solution for anarbitrary case meeting the continuity conditions, then we’ve solved most differentialequations, and all of the useful ones, and we wouldn’t need to learn this stuff.So, what we will end up doing is use a variation of Cauchy convergence,firstintroduced way back on Page 76. In effect, and sometimes explicitly, we will becomparing bounds on iterations to an exponential function, the convergence ofwhich we know very well. Keeping this in mind, whenever an example in thischapter refers to etA,whereA is a linear operator, it’s recommended that this beinterpreted as A being a constant scalar (multiplication by a scalar being a specialcase of a linear operator).1§1: The use of “Dynamical Systems” to begin this section anticipates §7; notethat no derivatives are used until Equation (1) on Page 160, and the notationis φt(x)=xtis used instead of x(t). In the paragraph after the first displayedequation, look closely at the sentence “At time zero, x is at x or x0.” Again,this anticipates what’s coming; x without a subscript implies the “starting point”of a trajectory, and emphasizes that we want to associate a trajectory with eachpoint x ∈S. Indeed, this is the motivation for Equation (1),(1) f(x)=ddtφt(x)t=0.The authors own up to what they’re doing in the italicized sentence that ends thatparagraph:Thus every dynamical system gives rise to a differential equation.Many of us, of course, are used to doing things in a different order, which theauthors call a “converse process”, but what’s done here makes fine mathematicalsense. In addition, doing so lets the authors relegate the differential equation (2)to a later chapter (but many texts prove the EUT for Equations (1) and (2) in asingle proof).Full Disclosure: Although H&S do indeed emphasize the autonomous case inthe text, we will reserve the right to investigate equations of the form (2); in fact,we have already done so, when we considered equations of the formdxdt+ P (t) x = Q(t)(first-order linear, x ascalar),orx+ p(t) x+ q(t) x = f(t)(variation of parameters). This latter can be converted into the form of Equa-tion (2), with x ∈ R2; this will be done in good time.§2: From here on, more or less, a distinction is made between the elements xor y of the open set W ⊂ E and the function u(t). Specifically, the equation (1),x= f (x) is a differential equation, and u(t) is a solution. Note, however, thatin the statement of Theorem 1, the implication is that x is a map from an opensubset (−a, a)intoW . In short, we have to watch out for what is meant by thesymbol x; it seems to change.2§3:In defining the Lipschitz condition, keep in mind that in general y and xwill be taken to be elements of Rn, not necessarily scalars.Stating that “... then Df(x) is represented by the n × n matrix of partialderivatives ...” depends on our agreement on the notion of what a derivative is:See the beginning of these notes.The crux of the proof is the paragraph that begins at the bottom of Page 164and continues on Page 165 (including the Lemma from analysis, addressed inthe notes Examples of Non-Uniform Con vergence.§4: As mentioned in the text in §3, the Lemma on Page 169 is an equivalentproof of uniqueness, and indeed many other texts use this as the primary
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