Rocky Mountain Mathematics Consortium Summer Conference University of Wyoming 7-18 July 2003

Transcription

1 Rcky Munain Mahmaics Cnsrium Summr Cnfrnc Univrsiy f Wyming 7-8 July 3 n Discr Dynamical Sysms & Applicains Ppulain Dynamics Principl Spakrs J. M. Cushing Shandll M. Hnsn U f Arizna Andrws Univrsiy Firs wk Discr Dynamical Sysms Scnd wk Applicains and Cas Sudis Spcial Spakrs R. F. Csanin Aarn King U f Arizna U f Tnnss Brian Dnnis U f Idah Firs Wk Discr Dynamical Sysms Lcur : Inrducin (JC) Lcur : Linar Maps (Hnsn) Lcur 3: Linar/Nnngaiv Marix Mdls (JC) Lcur 4: Nnlinar, Aunmus Maps (Hnsn) Lcur 5: Lcal bifurcains (Hnsn) Lcur 6: Nnlinar Marix Mdls (JC) Lcur 7: Pridically frcd maps (Hnsn) Lcur 8: Tpics in Chas I (JC) Lcur 9: Tpics in Chas II (JC) Lcur : Tpics in Chas III(?) and/r Muli-spcis Mdls (JC) Scnd Wk Sudis in Ppulain Dynamics & Eclgy Lcur : Mahmaics & Bilgy (JC & Csanin) Lcur : Th LPA Mdl (JC) Lcur 3: Cnncing Mdls Daa I (Dnnis) Lcur 4: Cnncing Mdls Daa II (Dnnis) Lcur 5: Chas I (Csanin) Lcur 6: Chas II (King) Lcur 7: Parns in Chas (King) Lcur 8: Cmping Spcis (JC) Lcur 9: Pridic Habias (Hnsn) Lcur : Pridic Habias (Hnsn)

2 Supplmnary rading Firs Wk Thms Asympic dynamics Sabiliy hry Bifurcains Chas Elmnary lvl Jams T. Sandfur, Discr Dynamical Sysms: Thry & Applicains Undrgradua lvl Sabr N. Elaydi, Discr Chas Richard A. Hlmgrn, A Firs Curs in Discr Dynamical Sysms Undrgradua/Gradua lvl Sabr N. Elaydi, An Inrducin Diffrnc Equains H. Caswll, Marix Ppulain Mdls: Cnsrucin, Analysis and Inrprain, Scnd diin Gradua lvl S. Wiggins, Inr Applid Nnlinar Dynamical Sysms and Chas Gucknhimr & Hlms, Nnlinar Oscillains, Dynamical Sysms, and Bifurcains f Vcr Filds Lcur # x = vcr f sa variabls = im x m x = R = R R whr R = rals xm {,,,,,, } r Z = {,,, } Z = + x ( = xm ( A Prliminary Exampl + ) = a x R, a R, Z Rcursiv frmula Diffrnc quain x () = a x whr x = )

3 () T p+ q Fr ach Z dfin T : R R by x a x NOTE : ( ) T = idniy map (bvius) = T p T q (smi- grup prpry) ( ) = = ( ) = ( ( ) ) p+ q p+ q p q p q T x a x a a x T T x DEFINITION : L X b a s. A mric is a funcin d : X X R ha saisfis h hr cndiins : d( x, y) = x = y d( x, y) = d( y, x) x, y X d( x, y) d( x, z) + d( z, y) x, y, z X X is calld a mric spac. + DEFINITION: A discr dynamical sysm is a n paramr family f cninuus maps T saisfying : X X = mric spac, T p+ q T = T p = idniy map T q fr all Z p, q Z. A diffrnc quain (rcursiv frmula) ( ) ( r ) x ( + ) = f x ( ) x D, Z Z cns f : D D = subs f a mric spac dfins discr dynamical sysm n X = D (r a discr smi-dynamical sysm) + Rplac Z by Z+ and T is a discr smi-dynamical sysm. 3

4 () Dimnsin: A map Tw Basic Classificains f : D D = pn m R dfins an m - dimnsinal dynamical sysm n D. EXAMPLE f ( x) = b x, b >, c + cx f : R R = r R + + dfins an n dimnsinal dynamical sysm n R + x ( + ) = b, Z+ + c b EXAMPLE x + c x + c x f bi cij x b x + c x+ c x =, >, dfins an w dimnsinal dynamical sysm x f : R R R R x ( + ) = b + c x ( + ) = b + c x ( + c x ( + c x ( x ( x( x ( EXAMPLE () Linariy: A map h R m f ( x) = Ax + h, m f : R R A = m m marix f h frm dfins an m-dimnsinal linar sysm. Th sysm is hmgnus if h =. m x ( + ) = L b b b3 τ L = τ 33 τ 43, τ bi ij b4 τ 44 A fur dimnsinal sysm n h 4 "nn-ngaiv cn" R 4

5 Th frmal hry f dynamical sysms is a pwrful and vry gnral hry. As w hav sn aunmus diffrnc quains x ( + ) = f ( ) dfin discr dynamical sysms. I is n always s bvius hw frmula nnaunmus diffrnc quains x ( + ) = f (, ) in h gnral hry in a usful way. Rwri h m-dimnsinal, nn-aunmus prblm + ) = f (, ) ) = x as h (m+)-dimnsinal aunmus prblm + ) = f ( y(, ) y( + ) = y( + ) = x, y() = Difficuly: all rbis f his aunmus prblm ar unbundd. ASYMPTOTIC DYNAMICS T frmula a prblm as a dynamical sysm n ypically aks advanag f spcial faurs f h quains (.g., pridiciy). W will fcus n diffrncs quains pr s. An (pn) ball : Sm Basic Dfiniins Bxr (, ) = y X: dxy (, ) < r A s S X is "pn" if fr ach x S B( x, r) S x X is a "limi pin" f S X if y S such ha lim d( x, y ) = n n n Th "clsur" S f S X is S all limi pins f S S X is "clsd" if S = S S X is "dns in X" if S = X 5

6 Ox ( ) = Tx: Z = h rbi hrugh x ( ) { : } O x = T x Z = h frward rbi hrugh x + + O+ ( x ) A limi pin f a frward rbi is h mga limi pin f h rbi. ω(x ) = {mga limi pins f O + (x )} is h mga limi s f h rbi. Basic Prpris f Omga Limi Ss f bundd rbis n R m () ω( x ) () ω( x ) is bundd (3) ω( x ) is clsd (4) ω( x ) is frward invarian (frward rbis f mga limi pins rmain in ω( x )) () EXAMPLES x ( + ) = ax ( ), ) = x Frward rbis : O ( x ) = a x : Z + + a = O+ () = : Z+ ω = {} and f ( ) N : O() = {} = {} () {} {( ) } () () a = O () = : Z ω =, + + N : f ({,} ) = {,} DEFINITION A cnsan sluin (pin rbi is calld an quilibrium. Equilibria ar fixd pins f f : f(x) = x DEFINITION A sluin saisfying + p) = fr all and a (smalls ingr p > is calld a p -cycl p-cycls ar drmind by h fixd pins f h cmpsi map ( ( ( ))) ( p f ) ( x) f f f x = x 6

7 DEFINITION: A s A X (a mric spac) is an aracr if (a) f( A) = A (b) hr is an pn s U Asuch ha U ω x A ( ) x (c) n subs f A has prpry (a) A is a glbal aracr if U = X x ( + ) = ax ( ), ) = x () a = O+ ( x) = x: Z+ ω x = {} EXAMPLES ( ) A = (h quilibrium) is a glbal aracr {( ) } () {, } () a = O () = : Z ω =, + + Bu A = is n an aracr. ( ) ( ) f () ( x) f f ( x) = x = x all pins (xcp x = ) ar - pridic pins. {( ) } ω ( ) O ( x ) = x : Z x = x, x + + Nn f h -cycls is an aracr x ( + ) = ax ( ), ) = x < a < h quilibrium x = is a glbal aracr N : all sluins ar mnnic. + ( ) = Frward rbis : O x a x X X 4 X 3 X X X 3 X 4 X X X 7

8 < a< h quilibrium x = X x ( + ) = ax ( ), ) = x is a glbal aracr N : all sluins ar scillary. X 3 X 5 X 4 X + ( ) = Frward rbis : O x a x X X 4 X X X 3 X X X Finally a lim = fr all x R < All sluins ar unbundd. If < a, h sluins ar mnnic. If a < -, h sluins ar scillary. In ihr cas, h quilibrium x = is a rpllr. X 4 X X X X X 3 X 3 X 4 X 5 x= rpllr Oscillary unbundd Graphical Summary: Bifurcain Diagram -cycls -.. x= aracr Oscillary cnvrgn Equilibria (-cycls) x= aracr Mnn cnvrgn.. x= rpllr Mnn unbundd a = - and ar calld bifurcain pins a A Linar Applicain x ( = ppulain numbrs x ( + ) = b + d r dnsiy a im + ) = rcruimn (birhs) + survivrs < b = pr capia rcruimn ra pr uni im < d = fracin ha surviv a uni f im + ) = a, a = b + d a < x = is a glbal aracr (xincin) < a x = is a rpllr (unbundd grwh) a = = quilibrium (bundd survival) 8

9 b Dfin : n = d n< x = is a glbal aracr (xincin) b n = d = b + bd + bd + bd 3 + xpcd pr capia numbr f rcruis pr lifim r h n rprduciv numbr quilibria < n x = is a rpllr (unbundd grwh) n = = x quilibrium (bundd survival) xincin unbundd grwh n A vrical bifurcain diagram. Vrical bifurcains (pin spcra) ar ypical f linar quains Ms applicains invlv nnlinar quains Th fllwing marial was n cvrd in Lcur # Mhds f analysis Sluin frmulas (rar) Gmric analysis (dimnsinally limid) Qualiaiv analysis 9

10 A Nnlinar Applicain Rgulad Ppulain Grwh x ( + ) = b + d x ( + ) = bg( ) + d x ( + ) = bg( ) + dh( ) g( x), h( x) g( x) =, + cx Exampl (Bvrn/Hl d = (nn - vrlapping gnrains) ) = x c > + ) = b, + c b > x Wha ar h asympic dynamics? If Apprach # (Sluin Frmula) b <, hn + ) b b x. Assum b >. Chang variabl : y ( =. Bvrn/Hl quain ransfrms in a linar quain fr y(. y( + ) = b y( + b c By inducin (a hmwrk prblm) : c y( = b y + ( b ) b x ( b ) x ( ) = ( b ) b + x c ( b ) lim = ( b ) / c Hmwrk prblm : Shw x = ( b )/ c is an quilibrium.

11 b > x Summary b < x quilibria (xincin) = quilibrium (bundd survival) y = x Apprach # (a gmric apprach) b < b > y = x x b = c aracr +) +) (x,y ) y = bx/( + cx) y = bx/( + cx) aracr rpllr b rpllr If fr all Apprach #3 (analyic) b <, hn + ) b x ( ) b x. Assum b > and x >. < b x < b fr all x > + cx c x > afr n sp < b / c < sluins ar abv and blw nnngaiv and bundd (by ). f ( x) = b x is mnn incrasingin x. + cx < x < x f ( x ) < f ( x ) = x ) < x x fr all (by inducin) < < x < x b x < ( b ) / c < + cx b x < x = ) + cx is mnn incrasing (by inducin)

12 Similar kinds f argumns shw x < x x < ) is mnn dcrasing CONCLUSION: all sluins ar mnnic and ar bundd abv and blw. Thus, all sluins cnvrg a limi. lim lim = x. + ) = lim b + c x = b x + cx Alil algbra shws x = r x = ( b ) / c Fr b > w shwd < x < x < x is incrasing < x < x x is dcrasing hrfr < < x < x x < x < x x