Bo Deng Outline: Department of Mathematics Small Chaos – Logistic Map University of Nebraska – Lincoln Poincaré Return Map – Spike Renormalization All Dynamical Systems Considered Big Chaos Logistic Map Logistic Map: xn1 f r ( xn ) rxn (1 xn ) : [0,1] [0,1] Orbit with initial point x0 x {x0 , x1 , x2 , ... xn , ...} 0 Fixed Point: x0 f ( x0 ) Periodic Point of Period n : x0 xn f n ( x0 ) f ( f (... f ( x0 )...)) A periodic orbit p is globally stable if xn p as n for all non-periodic initial points x0 Cobweb Diagram x2 x1 x0 x1 x2 … Robert May 1976 Period Doubling Bifurcation Period Doubling Bifurcation Period-Doubling Cascade, and Universality cycle of period 2n 1 2 rn 2 4 3.449490 3 8 3.544090 4 16 3.564407 5 32 3.568750 6 64 3.56969 7 128 3.56989 8 256 3.569934 9 512 3.569943 10 1024 3.5699451 11 2048 3.569945557 n 3 ∞ Onset of Chaos r*=3.569945672 Feigenbaum’s Universal Number (1978) rn rn 1 4.6692016 4.6692016… : as n rn 1 rn n r r * 1 / i.e. n at a geometric rate Renormalization Feigenbaum’s Renormalization, R n ( f ) --- Zoom in to the center square of the graph of f 2n --- Rotate it 180o if n = odd --- Translate and scale the square to [0,1]x[0,1] --- R : U U where U is the set of unimodal maps Renormalization Feigenbaum’s Renormalization at r r* R n ( f r* ) g * as n Geometric View of Renormalization E u Rn ( fr ) R( f r ) ... E fr f 3.59 s g* f r* U The Feigenbaum Number α = 4.669… is the only expanding eigenvalue of the linearization of R at the fixed point g* Chaos at r* At r = r* = 3.5699… almost all orbits converge to a chaotic set A which is a Cantor set of zero measure. At r = 4, f is chaotic in A = [0,1]. Def.: A map f : A → A is chaotic if the set of periodic points in A is dense in A it is transitive, i.e. having a dense orbit in A it has the property of sensitive dependence on initial points, i.e. there is a δ0 > 0 so that for every εneighborhood of any x there is a y , both in A with |y-x| < ε, and n so that | f n(y) - f n(x) | > δ0 Period three implies chaos, T.Y. Li & J.A. Yorke, 1975 Poincaré Return Map Poincaré Return Map (1887) reduces the trajectory of a differential equation to an orbit of the map. Poincaré Time-1 Map: φτ(x0) = x(τ, x0), for which x(t, x0) is the solution with initial condition x(0, x0) = x0 Poincaré Return Map I pump Poincaré Return Map Ipump Poincaré Return Map Vc c0 1 f 0 c0 1 Ipump INa Poincaré Return Map Vc C -1 c0 1 R( f ) 0 C -1/C0 1 Ipump INa Poincaré Map Renormalization 1 f c 0 c0 f R f2 Renormalized 1 f 2 c 0 c0 Poincaré maps are Poincaré maps, and every Poincaré map is between two successive renormalizations of a Poincaré map. R : Y → Y, where Y is the set of functions from [0,1] to itself each has at most one discontinuity, is both increasing and not below the diagonal to the left of the discontinuity, but below it to the right. MatLab Simulation 1 … Spike Return Maps 1 1 →0 f 0 1 c0 1 e-k/ f0 0 c0 1 0=id →0 0 1 1 0 x, 0 x 1 x 1 x 1 0, 1 1 Bifurcation of Spikes -- Natural Number Progression Silent Phase Is /C Discontinuity for Spike Reset 6th th 5 4th rd 3 2nd 1st Spike μ∞=0 ← μn … μ10 μ9 μ8 μ7 Scaling Laws : μn ~ 1/n and (μn - μn-1)/(μn+1 - μn) → 1 μ6 μ μ5 Poincaré Return Map At the limiting bifurcation point μ = 0, an equilibrium point of the differential equations invades a family of limit cycles. Vc Homoclinic Orbit at μ = 0 c0 1 f0 0 c0 1 Ipump INa Bifurcation of Spikes 1 / IS ~ n ↔ IS ~ 1 / n Dynamics of Spike Map Renormalization -- Universal Number 1 Y f μn ] R[0]=0 R[]= / 1 R[1/n1 ]= 1/n μ1 f μn ] μ2 f μn 1 W={ 1 R universal 0 1 1 constant 1 0 1 / 1 1 the set of elements of Y , each has at least one fixed point in [0,1]. /1 0 }, 1 Universal Number 1 f μn]] R[ ] 1 ( ) || || || R[ R[ 0]=0 /(1 ) || 0 0 4R[ 3 ]=/1 2 || 0 || 2 R[21/n1 ]= 1/n 1- μ1 f μn ] μ2 1 is an eigenvalue of DR[0] f μn R : Y Y , with Y being equipped with the L1 norm 1 1 R / 1 0 1 1 || f ||Y | f ( x) | dx 1 0 /1 0 1 Universal Number 1 f μn ] R[0]=0 R[]=/1 R[1/n1 ]= 1/n 1 is an eigenvalue of DR[0] μ1 f μn ] μ2 Theorem of One (BD, 2011): f μn 1 R 0 1 / 1 1 1 The first natural number 1 is a new universal number . /1 0 1 n 2 n1 lim 1 n n 1 n Renormalization Summary 1 X0 = { : the right 0 1 most fixed point is 0. } Eigenvalue: l 1 1 U={} X1 = { Invariant } = W \ X0 0 0 = id Fixed Point X1 X0 W = X0 U X1 Invariant R :Y Y 1 All Dynamical Systems Considered Cartesian Coordinate (1637), Lorenz Equations (1964) dX (Y X ) dt dY X ( Z ) Y dt dZ XY Z dt and Smale’s Horseshoe Map (1965) MatLab Simulation 2 … Time-1 Map Orbit All Dynamical Systems Considered Theorem of Big: Every dynamical system of f any finite dimension can be embedded into R n R n the spike renormalization R : X0 → X0 infinitely many times. That is, for any X 0 X 0 n and every map f : R n → R n there are R n infinitely many injective maps θ : R → X0 so that the diagram commutes. W id 1 X1 X0 X0 = { } 0 R: Y Y 1 Big Chaos Theorem of Chaos: The spike renormalization R : X0 → X0 is chaotic, i.e. (1) has a dense set of periodic orbits; (2) has the property of sensitive dependence on initial conditions; and (3) has a dense orbit. l1 W id 1 X1 X0 X0 = { } 0 R: Y Y 1 Big Chaos Theorem of Chaos: The spike renormalization R : X0 → X0 is chaotic, i.e. (1) has a dense set of periodic orbits; (2) has the property of sensitive dependence on initial conditions; and (3) has a dense orbit. l1 n R (f) f g l> 1 W … || f gX || 1 id Y X0 R: Y Y 1 0 | f ( x) g ( x) | dx R n R (g) n l> 1 || R n ( f ) R n ( g ) ||Y > 0 1 / 4 Big Chaos Theorem of Chaos: The spike renormalization R : X0 → X0 is chaotic, i.e. (1) has a dense set of periodic orbits; (2) has the property of sensitive dependence on initial conditions; and (3) has a dense orbit. l1 W id X1 X0 R: Y Y Use concatenation on a countable dense set (as L1 is separable) to construct a dense orbit Universal Number Theorem of Almost Universality: Every number is an eigenvalue of the spike renormalization. Slope = λ > 1 l>1 gμ l1 f gμ μ g0 || R ( g ) R ( g 0 ) l ( g g 0 ) ||Y W id l1 || g g 0 ||Y2 X1 1 g0 R: Y Y X0 1 R f0 0 c0 1 0 0=id 1 Summary Zero is the origin of everything. One is a universal constant. Everything has infinitely many parallel copies. All are connected by a transitive orbit. Summary Zero is the origin of everything. One is a universal constant. Everything has infinitely many parallel copies. All are connected by a transitive orbit. Small chaos is hard to prove, big chaos is easy. Hard infinity is small, easy infinity is big. Phenomenon of Bursting Spikes Rinzel & Wang (1997) Excitable Membranes Phenomenon of Bursting Spikes Food Chains Dimensionless Model: y x x(1 x ) : xf ( x, y ) 1 x x z y y (1 1 y ) : yg ( x, y, z ) x y 1 2 y z z ( 2 2 z ) : zh( y, z ) 2 y Big Chaos All Dynamical Systems Considered slope = l y0 (x0) Let W = X0 U X1 with 1 1 X0 = { y1 l>1 y2 … }, X1 = { 0 1 } 0 l1 Every n-dimensional dynamical system f : D R n D, 1 n id can be conjugate embedded into For each orbit { xX 0 1, x1= f (x0), x2= f (x1), …} in [0,1], X0 yin = infinitely let y = S(x ), y = R-1S(x ), R-2S(xmany ), … ways. 0 0 X0 1 1 W R :Y Y 2 2 f : D D, : D Y , s.t f ( x) R ( x) 1 Bifurcation of Spikes Vc c0 c0 Ipump I INa Def: System is isospiking of n spikes if for every c0 < x0 <=1, there are exactly n points x1, x2, … xn in [0, c0) and xn+1 returns to [c0,1]. Bifurcation of Spikes Vc Isospike of 3 spikes c0 c0 Ipump I INa Def: System is isospiking of n spikes if for every c0 < x0 <=1, there are exactly n points x1, x2, … xn in [0, c0) and xn+1 returns to [c0,1]. Universal Number 1 f μn ] μ1 f μn ] μ2 f μn 1 R 0 1 1 Theorem of One (BD, 2011): The first natural number 1 is a new universal number . 1 / 1 R[0]=0 R[]=/1 R[1/n1 ]= 1/n 1 is an eigenvalue of DR[0] /1 0 n nq2pnn1 q p lim lim 1 n n q n q n 1 n 1