Advanced Partial Differential Equations with Applications

Transcription

1 MIT OpenCourseWare Avance Partial Differential Equations with Applications Fall 2009 For information about citing these materials or our Terms of Use, visit:

2 MIT Hopf Bifurcations. Roolfo R. Rosales Department of Mathematics Massachusetts Institute of Technology Cambrige, Massachusetts MA September 25, 1999 Abstract In two imensions a Hopf bifurcation occurs as a Spiral Point switches from stable to unstable (or vice versa) an a perioic solution appears. There are, however, more etails to the story than this: The fact that a critical point switches from stable to unstable spiral (or vice versa) alone oes not guarantee that a perioic solution will arise, 1 though one almost always oes. Here we will explore these questions in some etail, using the metho of multiple scales to n precise conitions for a limit cycle to occur an to calculate its size. We will use a secon orer scalar equation to illustrate the situation, but the results an methos are quite general an easy to generalize to any number of imensions an general ynamical systems. 1Extra conitions have to be satise. For example, in the ampe penulum equation: ẍ+μ _x+sin x =0, there are no perioic solutions for μ 6= 0! 1

3 MIT (Rosales) Hopf Bifurcations. 2 Contents 1 Hopf bifurcation for secon orer scalar equations Reuction of general phase plane case to secon orer scalar Equilibrium solution an linearization Assumptions on the linear eigenvalues neee for a Hopf bifurcation Weakly Nonlinear things an expansion of the equation near equilibrium Explanation of the iea behin the calculation Calculation of the limit cycle size The Two Timing expansion up to O(ffl 3 ) Calculation of the proper scaling for the slow time Resonances occur rst at thir orer. Non-egeneracy Asymptotic equations at thir orer Supercritical an subcritical Hopf bifurcations Remark on the situation at the critical bifurcation value Remark on higher orers an two timing valiity limits Remark on the problem when the nonlinearity is egenerate

4 MIT (Rosales) Hopf Bifurcations. 3 1 Hopf bifurcation for secon orer scalar equations. 1.1 Reuction of general phase plane case to secon orer scalar. We will consier here equations of the form ẍ + h(_x; x; μ) =0; (1.1) where h is a smooth an μ is a parameter. Note 1 There is not much loss of generality in stuying an equation like (1.1), as oppose to a phase plane general system. For let: _x = f(x; y; μ) an _y = g(x; y; μ) : (1.2) Then we have ẍ = f x _x + f y _y = f x f + f y g = F (x; y; μ) : (1.3) Now, from _x = f(x; y; μ) we can, at least in principle, 2 write y = G(_x; x; μ) : (1.4) Substituting then (1.4) into (1.3) we get an equation of the form (1.1) Equilibrium solution an linearization. Consier now an equilibrium solution 4 for (1.1), that is: x = X(μ) such that h(0;x;μ)= 0; (1.5) 2 We can o this in a neighborhoo of any point (x Λ;y Λ ) (say,a critical point) such that f y (x Λ ;y Λ ;μ)6=0, as follows from the Implicit Function theorem. If f y = 0, but g x 6= 0, then the same ieas yiel an equation of the form ÿ + e h(_y; y; μ) = 0 for some e h. The approach will fail only if both f y = g x = 0. But, for a critical point this last situation implies that the eigenvalues are f x an g y, that is: both real! Since we are intereste in stuying the behavior of phase plane systems near a non egenerate critical point switching from stable to unstable spiral behavior, this cannot happen. 3Vice versa, if we have an equation of the form (1.1), then ening y by y = G(_x; x; μ), for any G such that the equation can be solve to yiel _x = f(x; y; μ) (for example: G = _x), then _y = G _x ẍ + G x _x = g(x; y) upon replacing _x = f an ẍ = h. 4i.e.: a critical point.

5 MIT (Rosales) Hopf Bifurcations. 4 so that x X is a solution for any xe μ. There is no loss of generality in assuming X(μ) 0 for all values of μ; (1.6) since we can always change variables as follows: x ol = X(μ)+xnew. The linearize equation near the equilibrium solution x 0 (that is, the equation for x innitesimal) is now: ẍ 2ff _x + x =0; (1.7) where ff = ff(μ) = 2h 1 _x(0; 0;μ) an = (μ) =h x (0; 0;μ): The critical point is a spiral point if > ff 2. The eigenvalues an linearize solution are then = ff ± ie! (1.8) (where e! = p ff 2 ) an x = ae fft cos (e!(t t 0 )) ; (1.9) where a an t 0 are constants. 1.3 Assumptions on the linear eigenvalues neee for a Hopf bifurcation. Assume now: At μ = 0 the critical pointchanges from a stable to an unstable spiral point (if the change occurs for some other μ = μ c, one can always reene μ ol = μ c +μnew). Thus In fact, assume: ff<0for μ<0 an ff>0 for μ>0, with > 0 for μ small. ffl I. h is smooth. ffl II. ff(0) = 0; (0) > 0 an μ ff(0) > 0 :5 9 >= >; (1.10) We point out that, in aition, there are some restrictions on the behavior of the nonlinear terms near the critical point that are neee for a Hopf bifurcation to occur. See equation (1.22). 5This last is known as the Transversality conition. It guarantees that the eigenvalues cross the imaginary axis as μ varies.

6 MIT (Rosales) Hopf Bifurcations Weakly Nonlinear things an expansion of the equation near equilibrium. Our objective is to stuy what happens near the critical point, for μ small. Since for μ = 0 the critical point is a linear center, the nonlinear terms will be important in this stuy. Since we will be consiering the region near the critical point, the nonlinearity will be weak. Thus we will use the methos introuce in the Weakly Nonlinear Things notes. For x; _x, an μ small we can expan h in (1.1). This yiels ẍ +! 0 2 x + n 1 2 A _x2 + B _xx Cx2o + nd _x 3 +3E_x 2 x+3f_xx 2 +Gx 3o (1.11) 2p 2 _xμ +Ωxμ + O(ffl 4 ;ffl 2 μ; fflμ 2 ) = 0 ; where we have use that h(0; 0;μ) 0 an ff(0) = 0. In this equation we have: A.! 2 h(0; 0; 0) = (0) > 0, with! 0 > 0, B. A 2 h(0; 0; 0); B _x 2 C. p 2 = 1 2 @ h(0; 0; 0) _x@x μ h(0; 0; 0) = μ (0), h(0; 0; 0) ; :::, ff(0) > 0, with p>0, E. ffl is a measure of the size of (x; _x). Further: both ffl an μ are small. 1.5 Explanation of the iea behin the calculation. We now want to stuy the solutions of (1.11). The iea is, again: for ffl an μ small the solutions are going to be ominate by the center in the linearize equation ẍ +! 2 0 x = 0, with a slow rift in the amplitue an small changes to the perio 6 cause by the higher orer terms. Thus we will use an approximation for the solution like the ones in section 2.1 of the Weakly Nonlinear Things notes. 6We will not moel these perio changes here. See section 2.3 of the Weakly Nonlinear Things notes for how tooso.

7 MIT (Rosales) Hopf Bifurcations Calculation of the limit cycle size. An important point to be answere is: What is epsilon? (1.12) This is a parameter that oes not appear in (1.1) or, equivalently, (1.11). In fact, the only parameter in the equation is μ (assume small as we are close to the bifurcation point μ = 0). Thus: ffl must be relate to μ. (1.13) In fact, ffl will be a measure of the size of the limit cycle, which isaproperty of the equation (an thus a function of μ an not arbitrary all). However: We o not know ffl apriori! How o we go about etermining it? The iea is: If we choose ffl too small" in our scaling of (x; _x), then we will be looking too close" to the critical point an thus will n only spiral-like behavior, with no limit cycle at all. Thus, we must choose ffl just large enough so that the terms involving μ in (1.11) (specically 2p 2 μ _x, which is the leaing orer term in proucing the stable/unstable spiral behavior) are balance" by the nonlinearity in such a fashion that a limit cycle is allowe. In the context of Two Timing this means we want μ to kick in" the amping/amplication term 2p 2 μ _x at just the right level" in the sequence of solvability conitions the metho prouces. Thus, going back to (1.11), we see that 7 ffl The linear leaing orer terms ẍ +! 2 x appear at O(ffl). 0 ffl The rst nonlinear terms (quaratic) appear at O(ffl 2 ). However: Quaratic terms prouce no resonances, since sin 2 = 1 (1 cos 2 ) an 2 there are no sine or cosine terms. The same applies to cos 2 an to sin cos. ffl Thus, the rst resonances will occur when the cubic terms in x play a role ) we must have the balance O(x 3 )=O(μ_x); (1.14) ) μ = O(ffl 2 ). 7 This is a crucial argument that must be well unerstoo. Else things look like a bunch of miracles!

8 MIT (Rosales) Hopf Bifurcations The Two Timing expansion up to O(ffl 3 ). We are now reay to start. The expansion to use in (1.11) is x = fflx 1 (; T)+ffl 2 x 2 (; T)+ffl 3 x 3 (; T)+::: ; (1.15) where 0 < ffl 1, 2ß perioicity in T is require, T =! 0 t,! 0 is as in (1.11) 8, is a slow time variable an ffl is relate to μ by μ = νffl 2, where ν = ±1 (which ν we take epens on which sie" of μ =0wewant to investigate). What exactly is? Well, we nee to resolve resonances, which will not occur until the cubic terms kick in into the expansion ) = ffl 2 t: (This is exactly the same argument use to get (1.14)). Then, with (1.11) becomes:! 2 0 x00 +! 2 0 x + n 1 2 A!2 0(x 0 ) 2 + B! 0 xx Cx2 o (x 0 ) 3 +3E! 0(x 2 0 ) 2 x +3F! fd!3 0 x 0 x 2 +Gx 3 g + 2ffl 2! 0 x 0 2ffl 2 νp 2! 0 x 0 + ffl 2 νωx + O(ffl 4 )=0: The rest is now a computational nightmare, but it is fairly straightforwar. getting into any of the messy algebra, this is what will happen: (1.16) Without At O(ffl)! 2 fx 00 + x g =0:Thus x 1 = a 1 ()e it + c:c: (1.17) for some complex value function a 1 (). We use complex notation, as in the Weakly Nonlinear Things notes. At O(ffl 2 )! 2 fx 00 + x g + fquaratic terms in x 1 an x 0 1g =0: (1.18) z } From the rst bracket in (1.16), the quaratic terms here have the form: C 1 a 2 + a ei2t 2 C C Λ 1(a Λ 1) 2 e 2iT ; where C 1 an C 2 are constants that can be compute in terms of! 0, A, B an C. Since the solution an equation are real value, C 2 complex conjugate. is real. Here, as usual, Λ inicates the 8Same as the linear (at μ = 0) frequency. No attempt is mae in this expansion to inclue higher orer nonlinear corrections to the frequency.

9 MIT (Rosales) Hopf Bifurcations. 8 No resonances occur an we have ρ x 2 = a 2 ()e it + 1 3! 2 0 C 1a 2 1 ei2t + c:c: ff! 2 C 0 2 a 2 1 : (1.19) At O(ffl 3 )! 2 (x 00 + x )+2! 0 x 0 1 2νp 2! 0 x 0 + νωx CNLT =0; (1.20) where CNLT stans for Cubic Non Linear Terms, involving proucts of the form x 2 x 1, x 0 x 2 1, x 2 x 0 1, x 0 x0 2 1, (x 0 1) 3, (x 0 1) 2 x 1, x 0 an x 3 1 x These will prouce a term of the form a 2 aλ + eit c:c: plus other terms whose T epenencies are: 1, e ±2iT an e ±3iT, none of 1 1 which is resonant (forces a non perioic response in x 3 ). Here is a constant that can be compute in terms of! 0 ;A;B;C;D;E;F an G: (1.21) This is a big an messy calculation, but it involves only sweat. In general, of course, Im() 6= 0. The case Im() = 0isvery particular, as it requires h in equation (1.1)to be just right, so that the particular combination of its erivatives at x = 0, _x = 0 an μ = 0 that yiels Im() just happens to vanish. Thus Assume a nonegenerate case: Im() 6= 0: (1.22) For equation (1.20) to have solutions x 3 perioic in T, the forcing terms proportional to e ±it must vanish. This leas to the equation: Then write 2! 0 i a 1 2νp 2! 0 ia 1 + νωa 1 + a 2 1 a 1 = ρe i ; with ρ an real ; ρ>0: a 1 =0: (1.23) This yiels an = 1 2 ν! 1 0 Ω+ 1 2! 1 0 Re()ρ 3 (1.24) ρ = νp2 (1 νqρ 2 )ρ; (1.25) where q = 1 2! 1 0 p 2 Im().

10 MIT (Rosales) Hopf Bifurcations. 9 Equation(1.24) provies a correction to the phase of x 1, since x 1 = 2ρ cos(t + ). The rst term on the right of (1.24) correspons to the changes in the linear part of the phase ue to μ 6= 0, away from the phase T =! 0 t at μ = 0. The secon term accounts for the nonlinear effects. The secon equation (1.25) above is more interesting. First of all, it reconrms that for μ<0 (that is, ν = 1) the critical point (ρ= 0) is a stable spiral, an that for μ>0 (that is, ν = 1) it is an unstable spiral. Further If Im() > 0: If Im() < 0: Then a stable limit cycle exists for q μ>0 (i.e. ν =1)with ρ = 2! 0 p (Im()) 2 1 : Supercritical (Soft) Hopf Bifurcation. Then an unstable limit cycle exists for q μ<0 (i.e. ν = 1) with ρ = 2! 0 p (Im()) 2 1 : Subcritical (Har) Hopf Bifurcation. 9 >= >; (1.26) Notice that ρ here is equal to 1 2ffl the raius of the limit cycle Remark on the situation at the critical bifurcation value. Notice that, for μ = 0 (critical value of the bifurcation parameter) 9 we can o a two timing analysis as above toverify what the nonlinear terms o to the center. 10 The calculations are exactly as the ones leaing to equations (1.23) (1.25), except that ν = 0 an ffl is now a small parameter (unrelate to μ, asμ= 0now) simply measuring the strength of the nonlinearity near the critical point. Then we get for ρ = 1 raius of orbit aroun the critical point 2ffl From this the behavior near the critical point follows. ρ = 1 2! 1 0 Im()ρ 3 : (1.27) 9 Then the critical point is a center in the linearize regime. 10This is the way one woul normally go about eciing if a linear center is actually a spiral point an what stability it has.

11 MIT (Rosales) Hopf Bifurcations. 10 Clearly 8 ffl Im() > 0 () Soft bifurcation () Nonlinear terms stabilize. >< For μ =0critical point is a stable spiral. ffl Im() < 0 () Har bifurcation () Nonlinear terms e-stabilize. >: For μ =0critical point is an unstable stable spiral Remark on higher orers an two timing valiity limits. As pointe out in the Weakly Nonlinear Things notes, Two Timing is generally vali for some limite" range in time, here probably jj ffl 1. This is because we have no mechanism for incorporating the higher orer corrections to the perio the nonlinearity prouces. If we are only intereste in calculating the limit cycle in a Hopf bifurcation (not it's stability characteristics), we can always o so using the Poincaré Linsteat Metho. In particular, then we can get the perio to as high an orer as wante Remark on the problem when the nonlinearity is egenerate. What about the egenerate case Im() = 0? In this case there may be a limit cycle, or there may not be one. To ecie the question one must look at the effects of nonlinearities higher than cubic (going beyon O(ffl 3 ) in the expansion) an see if they stabilize or estabilize. If a limit cycle exists, then its size will not be given by q jμj, but something else entirely ifferent (given by the appropriate balance between nonlinearity an the linear amping/amplication prouce by ff 6= 0when μ 6= 0 in equation (1.7)). The etails of the calculation neee in a case like this can be quite hairy. One must use methos like the ones in Section 2.3 of the Weakly Nonlinear Things notes because: even though the nonlinearity may require a high orer before it ecies the issue of stability, moications to the frequency of oscillation will occur at lower orers. 11 We will not get into this sort of stuff here. 11Note that Re() 6= 0in (1.24) prouces such a change, even if Im() =0an there are no nonlinear effects in (1.25).