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Ordinary Difference Eq of Order R = R-Order Ode

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Ordinary Difference Eq of Order R = R-Order Ode Ordinary Difference Eq of order r = r-order ode: K[yn, yn-1, yn-2,... yn-r ; t]=0 yn = y(t0+nτ) = y(t0+nτ) Solution: y = y(t) , t0+nτ, n takes integer values To solve the r-order difference equation means to find solutions y = y(t) such that the sequence yn, yn-1, yn-2,... yn-r satisfies the given equation The solution of the ode of order r, in general involves r arbitrary constants which must be determined by limit conditions on the yn, yn-1, yn-2,... yn-r, (initial conditions , boundary conditions) ode like ODE describe processes in terms of local rates of change Ordinary Difference Equation of order r, r=1,2,… as an equation on yt and its differences . Βackward Difference Equation = Αναδρομες Εξισωσεις Διαφορων 2 r K[yt, yt-1, yt-2,... yt-r ; t] = 0 ⟺ B[yt ,∇yt , ∇ yt , …∇ yt ; t] =0 ∇r are the Backward Difference Operators of order r=1,2,… ∇ yt = yt − yt −1 = (I−V) yt ∇ = (I−V) , V=S−1 is the Backward Shift Operator on Sequences V(ψ1, ψ2, ψ3,...)= (0, ψ1, ψ2, ...) (Vψ)ν = ψν−1 , ν=1,2,3,… ψ0 = O 2 ∇ yt = ∇(∇ yt ) = ∇ (yt − yt −1) = ∇yt − ∇yt −1 = (yt − yt −1 )−( yt −1 − yt−2 ) 2 2 ∇ yt = (yt − 2yt −1 + yt−2 ) = (1−V) yt 3 ∇ yt = ∇(yt − 2yt −1 + yt−2 ) = (yt − yt −1)−2 (yt −1 − yt−2 )− (yt−2 − yt −3) = 3 = yt − 3yt −1 + yt−2 + yt −3 = (1−V) yt The r-th Backward Difference Operator ∇r ∇r =(I−V)r Forward Difference Equation r r-1 r-2 K[yt+r, yt+r-1, yt+r-2,... yt ; t]=0 ⟺ F[∆ yt , ∆ yt , ∆ yt , …, yt ; t] =0 n n r yt+r = ∑ ( ) ∆ y k=0 k t ∆r are the Forward Difference Operators of order r=1,2,… ∆ yt = yt+1 − yt = (S−I) yt ∆ = (S−I) , S is the Forward Shift Operator on Sequences S(ψ1, ψ2, ψ3,...)=(ψ2, ψ3, ψ4,...) (Sψ)ν = ψν+1 , ν=1,2,3,… ∆2yt = ∆ (∆ yt ) = ∆ (yt+1 − yt) = ∆yt+1 − ∆yt = (yt+2 − yt +1 )−( yt +1− yt) ∆2yt = (yt+2 − 2yt +1 + yt ) = (S+I)2 yt ∆3yt = ∆(yt+2 − 2yt +1 + yt) = (yt+3 − yt +2)−2 (yt +2 − yt+1)− (yt+1 − yt) = 3 = yt+3 − 3yt +2 + yt+1 + yt = (S+I) yt n n n−k n n n−k n k ∆ yt = ∑ (−1) ( ) y = ∑ (−1) ( ) S y k=0 k t+k k=0 k t n ∆r =(S−I)r =∑n (−1)n−k ( ) Sk k=0 k EXAMPLE: 2 2 1 3∇ yt + 2∇yt + 7yt =0 ⟺ yt = yt−1 − yt−2 3 4 Recurrence Equations and DS as de The time independent ode K[yt, yt-1, yt-2,... yt-r]=0 is an Implicit Function Eq for yt For solvable ode, the implicit Function S defines the Recurrence Equation of order r yt =S[yt-1, yt-2,...,yt-r] The Recurrence Equations of order r define the r-order Recursive Sequences = Recurrence Sequences = Aναδρομικες Ακολουθιες r-ταξεως The solution to the Recurrence Equation is obtained by iterating the Map S with initial conditions y0, y1,...,yr−1 yt =S[yt-1, yt-2,...,yt-r] is also called Iteration Formula (Y,S) define a Discrete DS 푑푦 3) The Euler ode of the ODE = 퐹(y(t), 푡) 푑푡 with y0 = y(t0) is the 1st-order RE (Αναδρομικη Ακολουθια) : yν+1 = yν + τ F(yν , t0+ντ) = S(yν), ν = 0,1,2,… with: η0 = y0 = y(t0) η1 = τ F(y0 , t0) η2 = τ F(y1 , t0+τ) ην = τ F(yν−1 , t0+(ν−1)τ) Maps as DS 1-order Recurrence Equation yt+1 =S[yt] , S is the (Forward) Shift Map Linear ode Solutions by Systematic Μethods. The most developed Theory Analogy with Linear ODE EXAMPLES Aριθμητικη Προοδος yt+1 = yt + w , t=0,1,2,3,... , yt Πραγματικοι Αριθμοι yt = y0 + tw Γεωμετρικη Προοδος yt+1 = ayt , t=0,1,2,3,... , yt Πραγματικοι Αριθμοι t yt = a y0 Γραμμικη Αναδρομικη Ακολουθια 1ης ταξεως yt+1 = ayt + w , t=0,1,2,3,... , yt Πραγματικοι Αριθμοι a≠1 t t 1−a yt = a y0 + w 1−a Euler ode Reduction to Linear 2 2 2 2 yt+3 + αyt+2 + βyt+1 + γyt = 0 ⟺ xt+3 +αxt+2 + βxt+1 +γxt = 0 2 with xt = yt Fibonacci Sequence Ft = Ft−1 + Ft−2 F0 = 0 , F1 =1 Solution Binet 1843 흋풕−(−흋)−풕 (ퟏ+√ퟓ)풕−(ퟏ−√ퟓ)풕 Ft = = Aσκ 0,02 √ퟓ ퟐ풕√ퟓ 0.3 ευρεση της λυσης ퟏ+ ퟓ 흋 = √ the Golden Number ퟐ Nοn Linear ode Explicit solutions are known only for Some isolated special classes of ΝL RR and for specific parameters. Solutions for general parameter values are not known. In such ode: we use computers to calculate large numbers of terms we study qualitative questions: -the behaviour of the solutions as n→∞, -Stability (Dynamical Structural), which on the whole is analogous to the stability theory for ordinary differential equations -Statistical Properties Aρμονικη Προοδος 풚풕 풚풕+ퟏ = t=0,1,2,3,... , yt Πραγματικοι Αριθμοι ퟏ−풘풚풕 ퟏ , t=0,1,2,3,... Aριθμητικη Προοδος 풚풕 Generalization to Riccati, Oμογραφικες, The Hofstadter–Conway $10,000 sequence is defined as follows 푎(푛) = 푎(푎(푛 − 1)) + 푎(푛 − 푎(푛 − 1)), 푛 = 3,4, … 푎(1) = 푎(2) = 1 The first few terms of this sequence are 1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 10, 11, 12, ... (A004001 in OEIS) Theorem Conway: 0.5 푎(푛) 1 퐥퐢퐦 = 푛→∞ 푛 2 Conway J. 1988, Some Crazy Sequences, Lecture at AT&T Bell Labs, July 15 Batrachion Functions John Horton Conway offered a prize of $10,000 to find a value of n for which 푎(푛) 1 1 | − | < 푛 2 20 퐧 = ퟏퟒퟖퟗ Mallows Schroeder, M. 1991, "John Horton Conway's 'Death Bet.' " Fractals, Chaos, Power Laws, New York: W. H. Freeman, pp. 57-59 Hofstadter later claimed he had found the sequence and its structure some 10– 15 years before Conway posed his challenge [private communication with Klaus Pinn] Απεικονιση Renyi S: [0,1)→ [0,1): yt+1 = 2yt (mod 1) = 2 yt 1[o, ½)(yt)+(2yt - 1) 1[½, 1)(yt) Γεωμετρικη Προοδος με λογο 2 στο [0,1) Periodic Orbits Error Free Computation Απεικονιση Σκηνης (Tent Map) S: [0,1)→ [0,1): yt+1=2 yt 1[o, ½)(yt)+(2 - 2yt) 1[½, 1)( yt) Συμμετρια S(y)=S(1-y) 1 t Λυση Ulam-Von Neumann: yt = arcsin (cos2 πy0), t=1,2,3,..., ∀ y0 in [0,1) 휋 Aσκ Επαληθευση {0.2} 3 υπολογισμοι {0.8} Computability of Ulam-Von Neumann Analytic Formula {2} Λογιστικη Απεικονιση (Logistic Map) S: [0,1)→ [0,1): yt+1 = α yt (1- yt) has known exact solutions only for α=−2,2 and 4. 2 t Λυση Ulam-Von Neumann yt = sin (2 arcsin √y0), t=1,2,3,... Aσκ {0.2} Ισοδυναμια Λογιστικης Απεικονισης με Απεικονιση Σκηνης Aσκ {0.5} Απεικονιση Chebychev Sm: [-2,2)→ [-2,2): yt+1 = 2cos(m arcos yt) , t=1,2,3,... m = 2,3 .... Απεικονιση Gauss S: [0,1)→ [0,1): 1 1 1 1 1 yt+1 = (mod1) = { } = − [ ] = the Fractional part of ,t=1,2,3,... 푦푡 푦푡 푦푡 푦푡 푦푡 Continued Fractions representations of Real numbers Απεικονιση Gauss 2dim= Απεικονιση Mixmaster 1 1 − [ ] 푥푡 푥푡 x S:[0,1)x[0,1) → [0,1)x[0,1) : (y)↦( 1 ) 1 1 −[ ] 푦푡 푥푡 1 1 x 푥+[ ] S−1 ( )=( 푦푡 ) y 1 1 − [ ] 푦푡 푦푡 Απεικονιση Σφηνας (Cusp) S :[ 1,1] [ 1,1] y S y 12 y Απεικονιση Πλαστη (Baker) 풙 ퟐ풙 B:[0,1)x[0,1) → [0,1)x[0,1) :(풚) ↦ ( 풚 ) mod1 ퟐ Στατιστικη Μη Αναστρεψιμοτητα Ασταθεια στην οριζοντια κατευθυνση Ευσταθεια στην καθετη κατευθυνση Υπερβολικη Δομη Τοπικο Γεωμετρικο Χαρακτηριστικο Χαους Baker Map as Source of Information Information of the Baker Map with respect to the generating partition {Ξ0 = lower, Ξ1 =upper} 1 1 풮0(ξ) = 2[− 푙표푔 ] = 푙표푔 2 = 1bit 2 2 2 2 1 1 풮1(ξ) = 4[− 푙표푔 ] = 푙표푔 4 = 2bit 4 2 4 2 1 1 풮2(ξ) = 8[− 푙표푔 ] = 푙표푔 8 = 3bit 8 2 8 2 풮t(ξ) = (t+1) ℐ0 = (t+1) bits The Linear Hyperbolic Map 2 2 풙 ퟐ풙 S:ℝ → ℝ :(풚) ↦ ( 풚 ) ퟐ B = S mod 1 The Horseshoe map= 2dimensional Tent Map Τhe Cat Map on the Torus x 1 1 x x + y S:[0,1)x[0,1) → [0,1)x[0,1) : ( ) ↦ ( ) ( ) = ( ) mod1 y 1 2 y x + 2y Anosov 1963, Sov. Math. Dokl 4, 1153-6 ퟏ ퟏ J=det( )=1 ퟏ ퟐ 1 1 −1 ퟐ −ퟏ ( ) = ( ) 1 2 −ퟏ ퟏ 1 1 3+√5 3−√5 Eigenvalues of ( ) λ+= >1 λ-= <1 1 2 2 2 λ+λ- =1 1 1 Eigenvectors of ( ) 1 2 1 η+ = ( ) κατευθυνση διαστολης 휑 1 1 η- = (− ) κατευθυνση συστολης 휑 ퟏ+√ퟓ φ= ο Χρυσος Αριθμος ퟐ 2 Iterations of the Cat Map After streching the deformed square is cut into 4 pieces The 4 pieces are assembled and stacked to a square Mixing 2 or more metals by mechanical alloying takes place due to repeated streching and folding described by simple chaotic maps (Baker, Cat) [Shingrou,ea 1995 ] Periodic Points x x T (y) is Periodic Point of S with period T → (y) is fixed point of S x 0 ( ) = ( ) is the only fixed point of S y 0 1 2 3 4 5 5 5 5 Only 4 period 2 points exist (3) , (1) , (4) , (2) 5 5 5 5 x Θ.
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