1.4 Phase Line and Bifurcation Diag


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1 Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes ae often only known appoimately. In paticula they ae geneally detemined by measuements which 42 ae not eact. Fo that eason it is impotant to study the behavio of solutions and eamine thei dependence on the paametes. This study leads to the aea efeed to as bifucation 1.4 Phase Line and Bifucation Diag theoy. It can happen that a slight vaiation in a paamete can have significant impact on the solution. Bifucation theoy is a vey deep and complicated aea involving lots of cuent Technical publications may use special diagams to display qu eseach. A complete eaminationinfomation of of the field would aboutbe the impossible. equilibium points of the diffeential e A fied point (o equilibium point) of a diffeential equation y = f(y) is a oot of the (1) y = f(y). equation f(y) = 0. As we have aleady seenfo autonomous poblems fied points can be vey useful in detemining the long This timequation behavio is of solutions. independent of, hence thee ae no etena tems that depend on. Due to the lack of Qualitative infomation about the equilibium points of the diffeential equation y etenal contols, = f(y) tion is said to be selfgovening o autonomous. can be obtained fom special diagams called phase diagams. A phase line diagam fo the autonomous equation y = f(y A phase line diagam fo the autonomous segment with equation labels y sink, = f(y) souce is a line segment o node, with one labels fo each oot of fo socalled sinks, souces o nodes, i.e. one each foequilibium; each oot of f(y) see= Figue 0, i.e. each 11. equilibium. souce sink Figue 11. A phase line diagam y 0 y 1 autonomous equation y = f(y). The names ae boowed fom The the theoy labels of ae fluids boowed and they fom ae the theoy of fluids, and they following special definitions: 6 defined as follows: 1. Sink An equilibium y 0 which attacts neaby solutions at t =, i.e., thee eists Sink y = y 0 The equilibium y = y 0 attacts neaby solu M > 0 so that if y(0) y 0 < M, then y() y 0 t 0 = : fo some H > 0, y(0) y 0 < H y() y 0 deceases to 0 as. 2. Souce An equilibium y 1 which epels neaby solutions at t =, i.e., hee eists Souce y = y 1 The equilibium y = y 1 epels neaby solut M > 0 so that if y(0) y 1 < M, then y() y 1 inceases as t. = : fo some H > 0, y(0) y 1 < H that y() y 1 inceases as. 3. Node An equilibium y 2 which is neithe a sink o a souce. In fluids, sink means fluid Node y = y 2 The equilibium y = y 2 is neithe a sink no a is lost and souce means fluid is ceated. In fluids, sink 1 means fluid is lost and souce means fluid is c memoy device fo these concepts is the kitchen sink, wheein t is the souce and the dain is the sink. The stability test
2 Stability Test: The tem stable means that solutions that stat nea the equilibium will stay neaby as t. The tem unstable means not stable. Theefoe, a sink is stable and a souce is unstable. Pecisely, an equilibium y 0 is stable povided fo given ɛ > 0 thee eists some δ > 0 such that y(0) y 0 < δ implies y(t) eists fo t 0 and y(t)?y 0 < ɛ. Theoem 2.1 (Stability Conditions). Let f and f be continuous. The equation y = f(y) has a sink at y = y 0 povided f(y 0 ) = 0 and f (y0) < 0. An equilibium y = y 1 is a souce povided f(y 1 ) = 0 and f (y 1 ) > 0. Thee is no test when f is zeo at an equilibium. Ou objective in this section (fo fist ode equations) is to biefly eamine the thee simplest types of bifucations: 1) Saddle Node; 2) Tanscitical; 3) Pitchfok. 2.1 Saddle Bode Bifucation We begin with the Saddle Node bifucation (also called the blue sky bifucation) coesponding to the ceation and destuction of fied points. The nomal fom fo this type of bifucation is given by the eample = + 2 (1) The thee cases of < 0, = 0 and > 0 give vey diffeent stuctue fo the solutions. < 0 = 0 > 0 We obseve that thee is a bifucation at = 0. Fo < 0 thee ae two fied points given by = ±. The equilibium = is stable, i.e., solutions beginning nea this equilibium convege to it as time inceases. Futhe, initial conditions nea divege fom it. 2
3 At = 0 thee is a single fied point at = 0 and initial conditions less than zeo give solutions that convege to zeo while positive initial conditions give solutions that incease without bound. Finally if > 0 thee ae no fied points at all. Fo any initial condition solutions incease without bound. Thee ae seveal ways we depict this type of bifucation one of which is the so called bifucation diagam. Note that if instead we conside = 2 the the socalled phase line can be dawn as < 0 = 0 > 0 Eecise: Analyze the bifucation popeties of the following following poblems. 1. = = cosh() 3. = + ln(1 + ) 3
4 2.2 Tanscitical Bifucation Net we conside the tanscitical bifucation coesponding to the echange of stability of fied points. The nomal fom fo this type of bifucation is given by the eample = 2 (2) In this case thee is eithe one ( = 0) o two ( 0) fied points. When = 0 the only fied point is = 0 which is semistable (i.e., stable fom the ight and unstable fom the left). Fo 0 thee ae two fied points given by = 0 and =. So we note in this case = 0 is a fied point fo all. Fo < 0 the nonzeo fied point is unstable but fo > 0 the nonzeo fied point becomes stable. Thus we say that the stability of this fied point has switched fom unstable to stable. < 0 = 0 > 0 Bifucation diagam fo a tanscitical bifucation. Eecise: Analyze the bifucation popeties of the following following poblems. 1. = = ln(1 + ) 4
5 3. = (1 ) 2.3 Pitchfok Bifucation Finally we conside the pitchfok bifucation. The nomal fom fo this type of bifucation is given by the eample = 3 (3) The cases of 0 and > 0, once again, give vey diffeent stuctue fo the solutions. < 0 = 0 > 0 Supe Citical Pitchfok Bifucation Diagam Now conside the eample = + 3. (4) Fo this eample we obtain the socalled subcitical pitchfok bifucation. Notice that solutions blowup in finite time, i.e., satisfy (t) ± as t a <. 5
6 Sub Citical Pitchfok Bifucation Diagam Eecise: Analyze the bifucation popeties of the following following poblems. 1. = + β tanh() 2. = = sin() 4. = = sinh() 6. = = Hysteesis: a moe complicated bifucation In this subsection we conside an even moe complicated eample which contains pitchfokand saddle node bifucations. Conside the eample = (5) 1. Fo small initial conditions the bifucation diagam looks just like the subcitical bifucation diagam. The oigin is locally stable fo < 0 and the two banches ae unstable. The two backwad unstable banches bifucated fom = 0. The tem 5 6
7 has now ceated a new phenomenon: at a value of < 0, denoted by, the unstable banches tun aound aound and become stable. These new banches eist fo all > 2. Note that fo < < 0 thee ae thee stable solutions. The initial condition detemines which of these thee fied points the solution conveges to as time inceases. 3. This eample demonstates an impotant physically obseved phenomenon known as Hysteesis. If we stat the system with an initial condition close to = 0 Bifucation Diagam showing Hysteesis 7
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