Dynamical Systems PDF

dynamical systems Artificial Life Luı́s Correia Ciências - ULisboa references ▶ Ott, Edward (2002). Chaos in dynamical systems. 2nd ed. Cambridge university press. ▶ Fractal dimension - https: //www.wahl.org/fe/HTML_version/link/FE4W/c4.htm characterisation a...

dynamical systems Artificial Life Luı́s Correia Ciências - ULisboa references ▶ Ott, Edward (2002). Chaos in dynamical systems. 2nd ed. Cambridge university press. ▶ Fractal dimension - https: //www.wahl.org/fe/HTML_version/link/FE4W/c4.htm characterisation area of mathematics deterministic equations that represent the time evolution of a system ▶ deterministic ▶ qualitatively different behaviours ▶ by varying parameters ▶ situations of long term unpredictable behaviour continuous DS - flows time, t, is a continuous variable dx(t) = F[x(t)] dt meaning  (1) dx = F1 (x (1) , x (2) ,... , x (N) )       dt (2)  dx  = F2 (x (1) , x (2) ,... , x (N) )     dt   ..    .  (N)    dx  = FN (x (1) , x (2) ,... , x (N) )   dt orbit or trajectory flow with N = 3, from time 0 to time t discrete DS - maps time, n, is discrete (instants), n ∈ {0, 1, 2,...} a map is xn+1 = M(xn ) orbit: x0 , x1 , x2 ,... and similarly to flows: (1) (2) (N) xn = (xn , xn ,... , xn ) M = M1 , M2 ,... , MN flows to maps a flow of dimension N can be reduced to a map of dimension N − 1 ▶ via a Poincaré section surface (Cattani, M, et al. (2017). Deterministic Chaos Theory: Basic Concepts. Revista Brasileira de Ensino de Fı́sica, 39(1), e1309) complexity of orbits flows: with N ≥ 3 chaos (chaotic dynamics) may appear maps: with N ≥ 2 chaos may appear however... if the map is noninvertible chaos may appear with N = 1 ex: logistic equation M(x ) = r x (1 − x ) presents chaos for some range of r conservative systems definition all points in an initial closed surface S0 of dimension N − 1, in a system of dimension N evolve to points in a surface St in time t, in which the volumes V (0) and V (t) delimited by S0 and St verify V (0) = V (t) ex: Newton eqs. of frictionless body movement ▶ non conservative systems... do not conserve volume! conservative systems formally divergence theorem dV (t) Z = ∇ · Fd N x dt St R St : integral of the volume delimited by surface St ∇·F≡ PN (1) ,... , x (N) )/∂x (i) i=1 ∂Fi (x dV (t) ▶ if dt = 0 ⇒ conservative ▶ if ∇ · F < 0 in some region of the phase space ⇒ contraction of volume in that region i.e. dissipative system ⇒ attractors in maps: ▶ conservative if J((X) ≡ | det[∂M(X)/∂x ]| = 1 ▶ dissipative if J(X) < 1 in some region some attractors damped harmonic oscillator d 2y dy +v + ω2y = 0 dt 2 dt (brilliant.org/wiki/damped-harmonic-oscillators/) x (1) = y ( in the phase space with x (2) = dy dt attractor is a fixed point dimension= 0 x (1) = x (2) = 0 (inspirehep.net/record/930874/plots) some attractors van der Pol oscillator d 2x dx 2 +(x 2 −η) +ω 2 x = 0 dt dt attractor is a limit cycle dimension= 1 (www.sciencedirect.com/topics/engineering/van-der-pol-oscillator) strange attractors some DS have attractors with a complex geometric structure, having a fractal dimension (non integer) ▶ an attractor with a fractal dimension is called a strange attractor ex: Hénon map (1) (1) (2)  xn+1 = A − (xn )2 + Bxn  x (2) = xn(1)  n+1 successive amplifications in an attractor zone always show local structure of the same kind (typical of fractal images) dependency on initial conditions ( X1 (0) in t = 0 we have X2 (0) = X1 (0) + ∆(0) in t = t ′ the orbits will be in X1 (t ′ ) and X2 (t ′ ) with ∆(t ′ ) = X2 (t ′ ) − X1 (t ′ ) if in the limit |∆(0)| → 0 and for t ≫, the orbits of X1 and X2 are limited in the phase space and |∆(t)| |∆(t)| ↗ exp, i.e. ≈ e ht , with h > 0∗ |∆(0)| then the DS shows a sensitive dependency to the initial conditions and therefore it is chaotic ∗ this is trivial if orbits are not limited when t ≫ the power of chaos in a chaotic system, e.g. Hénon map ▶ if |∆(0)| = 0, i.e. the initial point is the same 2 orbits: one with double precision and the other with simple precision (rounding error ≈ 10−14 ) after 36 iterations: ∆(n) = ∆(36) ≈ value of x (!!) ▶ supposing error only in the 1st iteration (≈ 10−14 ) if from then on ∆ doubles in each (errorless) iteration supposing x ∼ units, then ∆(n) ∼ x ⇔ 2n · 10−14 ≃ 1 ⇒ n ≃ 45 predictions unfeasible for n ≳ 45 (iterations) the exponential growth of the error ▶ to increase the prediction window twice, e.g. from 45 to 90 ⇒ decrease the initial error from 10−14 to 10−28 14 orders of magnitude !!! systematic attractor dynamics fixed point: orbits converge (to a point), exponentially limit cycle: orbits get to a difference in the order of ∆(0) and they maintain it endlessly chaotic: orbits diverge exponentially attractors fractal→ Y N sensitivity (strange) Y (chaotic) E E E = exists N E E chaos: dynamics of the attractor strangeness: geometry of the attractor partial measurement of a DS the components of the system state X(t) may not all be accessible ▶ sol: with one component of X(t) or a scalar function of X(t) g(t) = G(X(t)) one can obtain info about the attractor (geometry & dynamics) enters the delay coordinate vector (1)  y (t) = g(t)   y (2) (t) = g(t − τ )     y (3) (t) = g(t − 2τ )  Y = (y (1) , y (2) ,... , y (M) ) =  ...     y (M) (t)  = g(t − (M − 1)τ )  τ chosen such that it is ∼ characteristic time of g(t) delay coordinate vector in principle we can integrate back in time X(t − m τ ) = Lm (X(t)) thus g(t − m τ ) = G(Lm (X(t))) therefore Y = H(X) delay coordinate phase space (Y ) is a function of the original phase space (X) ▶ it can be shown that if M is large enough the Y attractor structure is qualitatively similar to that of X stability of fixed points 1D maps fixed point: x ∗ = f (x ∗ ) df (x ) < 1 ⇒x ∗ is stable: attracts dx x =x ∗ df (x ) > 1 ⇒x ∗ is unstable: repels dx x =x ∗ df (x ) = 0 ⇒x ∗ is super-stable: little sensitivity to dx x =x ∗ perturbations in x ; high attenuation around x ∗ ; fastest convergence to x ∗ df (x ) = 1 ⇒x ∗ is meta-stable: small perturbation in dx x =x ∗ parameters may produce instability df (x ) dx ∼ Lyapunov exponent (long term average divergence of orbits) logistic map e.g.: simplified model of anual variation of insect population xn+1 = r xn (1 − xn ) 1 r   max = f = 2 4 if 0 ≤ xn ≤ 1 then for 0 ≤ r ≤ 4 we obtain 0 ≤ xn+1 ≤ 1 conserves volume! logistic map attractors to compute the attractors we solve for x in the equilibrium: x = r x (1 − x ) which produces 2 attractors xr∗ = 0 and 1 xr∗ = 1 − r Note: for r < 1 the second attractor falls out of the domain [0, 1] logistic map step to study stability of the attractors considering df (x ) remember: λ= f (x ) = r x (1 − x ) dx we obtain λ = |r (1 − 2x )| and the values of λ for the attractors: xr∗ = 0 ⇒ λ = r 1 xr∗ = 1 − ⇒ λ = |2 − r | r logistic map stability of attractors xr∗ = 0 xr∗ = 1 − 1r (λ = r ) (λ = |2 − r |) r 0 repeller fp df λ≡ |X Lyapunov exponent dX 0 unidimensional flows (no chaos) ▶ attractor: stable fp, attracts close trajectories ▶ repeller: unstable fp, repells close trajectories ▶ saddle point: attracts on one side, repells on the other system counters perturbation x system aggravates perturbation x F (X0 + x ) < 0 ⇒ (X0 + x ) → X0 F (X0 + x ) > 0 ⇒ (X0 + x ) ↗ F (X0 − x ) > 0 ⇒ (X0 − x ) → X0 F (X0 − x ) < 0 ⇒ (X0 − x ) ↘ a rare case in 1D, λ = 0 df (X ) because it requires f (X0 ) = 0 ∧ |X0 = 0 with a single parameter dX usually are structurally unstable - they change type, or disappear with small perturbations 1st case Ẍ has the same signal left and right of X0 a rare case in 1D, λ = 0 df (X ) because it requires f (X0 ) = 0 ∧ |X0 = 0 with a single parameter dX 2nd case Ẍ has contrary signals left and right of X0 vertical movement ↕ makes the attractor disappear in 2D saddle point λ1 λ2 fixed point 0 repeller >0 > we have dxn ∼ e λn dx0 (separation of orbits after n iterations) and the Lyapunov exponent is defined as: 1 dxT λ = lim ln| | T →∞ T dx0 since dxT dxT dx1 = ··· dx0 dxT −1 dx0 = M (xT −1 ) · M ′ (xT −1 ) · · · M ′ (x0 ) ′ we obtain −1 1 TX λ > 0 ⇒ chaos λ = lim ln|M ′ (xn )| T →∞ T λ < 0 ⇒ trajectories n=0 converge Lyapunov exponents in maps value of the unidimensional case... exponential divergence of considering x0 and x0 + dx0 , for n >> we have initial conditions dxn ∼ e λn dx0 (separation of orbits after n iterations) and the Lyapunov exponent is defined as: 1 dxT λ = lim ln| | T →∞ T dx0 since dxT dxT dx1 = ··· dx0 dxT −1 dx0 = M (xT −1 ) · M ′ (xT −1 ) · · · M ′ (x0 ) ′ we obtain −1 1 TX λ > 0 ⇒ chaos λ = lim ln|M ′ (xn )| T →∞ T λ < 0 ⇒ trajectories n=0 converge Kolmogorov-Sinai entropy introduction K-S entropy is another metric of complexity / chaotic behaviour ▶ divide the space in cells of size L ▶ initiate the system with a collection of M initial points, all in the same cell ▶ after n time steps of duration τ each ▶ compute the probability (relative freq.) pr , of the system visiting each cell, and from that X Sn = − pr log pr r ex: if all trajectories evolve closely, in each instant only one cell is occupied (fixed point) limit cycle: only a prn (A) = 1 ∧ prn (i) = 0, ∀n, i ̸= A ⇒ Sn = 0 few cells occupied, Sn ≪ Kolmogorov-Sinai entropy (simple) entropy Sn cases ▶ if number of occupied cells Nn grows with n and all have the same probability 1/Nn then prn (i) = 1/Nn , ∀i ⇒ Sn = log Nn note: Sn grows with logarithm of the number of occupied cells chaotic system, or regular, since Sn is indep. of M for M ≫ ▶ if each trajectory, in all n goes to different cells (in each step there are M occupied cells) prn (i) = 1/M, foralli ⇒ Sn = log M Sn is constant with value M – random system Kolmogorov-Sinai entropy at last measures changes in entropy Sn in each step: 1 Kn = (Sn+1 − Sn ) τ averaging over all attractor for growing number of iterations P 1 P−1 X 1 K = lim (Sn+1 − Sn ) = lim (SP − S0 ) P→∞ Pτ P→∞ Pτ n=0 and in the limit of cells of size 0 and time intervals of duration 0 the K-S entropy (KSE), or K-entropy or metric entropy is 1 > 0: chaotic sys K = lim lim lim (SP − S0 ) τ →0 L→0 P→∞ Pτ 0: non-chaotic sys Kolmogorov-Sinai entropy in a chaotic regime particular cases K -entropy = Lyapunov exponent ▶ if Nn grows exponentially with time and and all occupied cells have identical probability for D-dim system D X 1 K= λi N= N0 e λnτ ps = Nn i λPτ ⇒ S0 = log N0 SP = log N0 e = log N0 + λPτ 1 ⇒ K = lim lim lim  ( logN0 + λ Pτ− logN0) τ →0 L→0 P→∞  Pτ ⇒K =λ ▶ for a random system S0 = log N0 but S1 = S2 = · · · = SP = log M ⇒ K = log M if we consider the initial state, otherwise K = 0 ▶ if all trajectories evolve together (fixed point) or very close in few repeat cells (limit cycle) ⇒ K = 0 other measures of dynamics (chaos) ▶ fractal dimension ▶ correlation dimension ▶ invariance measure ▶... concepts complexity ↔ information ⇒ (needs) a frame of objectivity to compute / estimate note: entropy measures radomness, not complexity(!) coupled fingers oscillation a real world (living) example coupled fingers oscillation a real world (living) example → coupled fingers oscillation a real world (living) example → in the phase space: coupled fingers oscillation a real world (living) example → in the phase space: common phenomenon in other bio-organisms DS in bio-organisms within organisms ▶ molecular cell biology ▶ perception-action systems walking as a limit cycle ▶ perception systems ▶ action systems e.g. fingers oscillation ▶ basic systems e.g. pacemaker between organisms ▶ expression (incl. language) and perception ▶ social coordination 2 individuals clapping in phase opposition... ▶ applause in events fractal dimension calculating fractal dimension methods ▶ self-similarity ▶ geometric method (or scale relation) ▶ box counting ▶ also Minkowski-Bouligand ▶... (many more) including ▶ Hausdorf ▶ Rényi self-similarity useful for generating processes let ▶ D - dimension ▶ e - scale ▶ N - number of identical forms it holds log N eD = N ⇒ D= log e self-similarity examples a fractal... not fractal the Peano curve ▶ initially it is a mere line segment — www.goodmath.org/blog/2007/07/25/fractal-pathology-peanos-space-filling-curve/  −1 1 log 9 N = 9, e= =3 ⇒D= =2 3 log 3 Koch curve N = 4, e=3 ⇒ log 4 D= = 1.26 log 3 by Marcelo Byrro Ribeiro Cantor set N = 2, e=3 ⇒ log 2 math.stackexchange.com/questions/2400345/drawing- D= = 0.63 log 3 a-cantor-based-fractal-set Sierpinski triangle N = 3, e=2 ⇒ log 3 D= = 1.585 www.darrinqualman.com/fractal-collapse/sierpinski- log 2 gasket-or-triangle-fractal-collapse/ geometric method the length of the coast of Great Britain let L - total length s - size of the ruler used N - number of rules needed e.g. s = 4, N = 4 s = 2, N = 10 in general (Mandelbrot) log L(s) = (1 − D) log(s) + b where D is the fractal dimension, and the slope of the log/log plot geometric method the length of some coasts www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html the closer to 1 the more regular is the coast box counting method N = 4 occupied of b = 4 boxes N = 11 occupied of b = 16 boxes successively increasing the scale b (side of boxes h = 1/b) we estimate ∆ log N D = lim b→∞ ∆ log b in this case log 11 − log 4 D≃ = 1.46 log 4 − log 2 L-systems ▶ a class of rewriting systems parallel rewriting operates on strings of symbols or graphical elements proposed by Aristid Lindenmeyer in 1968 to model the developmental process of living organisms L-system components ▶ an alphabet A of symbols ▶ an initial string of symbols - axiom ▶ a set of rewriting or production rules - π = {pi } example of a production rule a → bac note: a symbol without a rule for replacement is maintained across iterations ▶ a stopping condition - usually number of iterations L-system with turtle graphics A = {F , f , +, −} F move forward one step drawing a line f move forward one step without drawing a line + turn left by an angle δ − turn right by an angle δ a L-System fractal with turtles this L-system A = {F , +, −} w =F π = {F → F + F − −F + F } ▶ with δ = 60◦ generates... a L-System fractal with turtles this L-system axiom: A A = {F , +, −} iteration 1 w =F iteration 2 π = {F → F + F − −F + F } iteration 3 ▶ with δ = 60◦ generates... iteration n L-system of the TP exercise ▶ typically all caps letters move forward one step ▶ extends with: ▶ 0 - draw a segment (move fwd one step) ending in a leaf ▶ 1 - draw a segment ▶ [ - push (to control branching in figures) ▶ ] - pop (ditto) L-system bio-realistic example ▶ variables: X F ▶ constants: + − [ ] ▶ start: X ▶ rules: (X → F + [[X ] − X ] − F [−FX ] + X ), (F → FF ) ▶ angle: 25◦ auxiliary references mainly in the bio domain ▶ Winfree, A. T. (2001). The geometry of biological time (Vol. 12). Springer Science & Business Media. ▶ Kelso, J. S. (1995). Dynamic patterns: The self-organization of brain and behavior. MIT press. ▶ Smith, L. B., & Thelen, E. E. (1993). A dynamic systems approach to development: Applications. The MIT Press. ▶ Peitgen, H. O., Jürgens, H., Saupe, D., & Feigenbaum, M. J. (2004). Chaos and fractals: new frontiers of science (Vol. 106). New York: Springer.