From Small To One Big Chaos Bo Deng Department of Mathematics University of Nebraska – Lincoln Outline: Small Chaos – Logistic Map Poincaré Return Map – Spike Renormalization All Dynamical Systems Considered Big Chaos Reprint/Preprint Download at: http://www.math.unl.edu/~bdeng1

Orbit with initial point Logistic Map Logistic Map: Orbit with initial point Fixed Point: Periodic Point of Period n : A periodic orbit is globally stable if for all non-periodic initial points x0

Cobweb Diagram x2 x1 x0 x1 x2 … Period Doubling Bifurcation Robert May 1976

Period Doubling Bifurcation

rn 4.6692016… Feigenbaum’s Universal Number (1978) Period-Doubling Cascade, and Universality n cycle of period 2n rn 1 2 3 4 3.449490 8 3.544090 16 3.564407 5 32 3.568750 6 64 3.56969 7 128 3.56989 256 3.569934 9 512 3.569943 10 1024 3.5699451 11 2048 3.569945557 ∞ Onset of Chaos r*=3.569945672 Feigenbaum’s Universal Number (1978) i.e. at a geometric rate 4.6692016…

Feigenbaum’s Renormalization, --- Zoom in to the center square of the graph of --- Rotate it 180o if n = odd --- Translate and scale the square to [0,1]x[0,1] --- where U is the set of unimodal maps

Feigenbaum’s Renormalization at

U E u E s The Feigenbaum Number α = 4.669… Geometric View of Renormalization E u E s U The Feigenbaum Number α = 4.669… is the only expanding eigenvalue of the linearization of R at the fixed point g*

At r = 4, f is chaotic in A = [0,1]. 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 ε-neighborhood 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 (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 V c c0 1 f INa Ipump 0 c0 1

Poincaré Return Map V c C -1 c0 1 R( f ) INa Ipump 0 1 C -1/C0

Renormalized Poincaré maps are Poincaré maps, and Poincaré Map Renormalization f R f 2 Renormalized 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 …

e-k/m fm f0 m →0 ym y0=id m 1-m Spike Return Maps 1 0 c0 1 0 c0 1 0 1 0 1 ym 0 1 y0=id m 1-m 0 c0 1 fm e-k/m

μ m = Is /C μ∞=0 ← μn … μ10 μ9 μ8 μ7 μ6 μ5 Scaling Laws : μn ~ 1/n and Bifurcation of Spikes -- Natural Number Progression m = Is /C Silent Phase Discontinuity for Spike Reset 6th 5th 4th 3rd 2nd 1st Spike μ∞=0 ← μn … μ10 μ9 μ8 μ7 μ6 μ5 μ Scaling Laws : μn ~ 1/n and (μn - μn-1)/(μn+1 - μn) → 1

At the limiting bifurcation point μ = 0, an equilibrium Poincaré Return Map At the limiting bifurcation point μ = 0, an equilibrium point of the differential equations invades a family of limit cycles. V c Homoclinic Orbit at μ = 0 c0 1 f0 0 c0 1 INa Ipump

Bifurcation of Spikes 1 / IS ~ n ↔ IS ~ 1 / n

R[y0]=y0 R[ym]=ym / (1-m) R[y1/(n+1) ]= y1/n R m / (1-m) Dynamics of Spike Map Renormalization -- Universal Number 1 Y μn f μn ] R[y0]=y0 R[ym]=ym / (1-m) R[y1/(n+1) ]= y1/n μn f μn ] μ1 μ2 f μn 0 1 1 W = { } , the set of elements of Y , each has at least one fixed point in [0,1]. 0 1 ym R 1 m 1-m ym /(1-m) m / (1-m) universal constant 1

1 is an eigenvalue of DR[y0] R m / (1-m) R[y0]=y0 R[ym]=ym/(1-m) Universal Number 1 μn f μn ] R[y0]=y0 R[ym]=ym/(1-m) R[y1/(n+1) ]= y1/n 1 is an eigenvalue of DR[y0] μn f μn ] μ1 μ2 f μn 0 1 ym R 1 m 1-m m / (1-m) ym /(1-m) m

R m / (1-m) R[y0]=y0 R[ym]=ym/(1-m) R[y1/(n+1) ]= y1/n Universal Number 1 μn f μn ] R[y0]=y0 R[ym]=ym/(1-m) R[y1/(n+1) ]= y1/n 1 is an eigenvalue of DR[y0] Theorem of One (BD, 2011): The first natural number 1 is a new universal number . μn f μn ] μ1 μ2 f μn 0 1 ym R 1 m 1-m m / (1-m) ym /(1-m) m

most fixed point is 0. } Eigenvalue: l = 1 U={ym} X1 X0 Invariant Renormalization Summary 0 1 X0 = { : the right most fixed point is 0. } 1 Eigenvalue: l = 1 0 1 X1 = { } = W \ X0 1 U={ym} Invariant y0 = id Fixed Point chaos X1 X0 W = X0 U X1 Invariant

Cartesian Coordinate (1637), Lorenz Equations (1964) All Dynamical Systems Considered Cartesian Coordinate (1637), Lorenz Equations (1964) and Smale’s Horseshoe Map (1965) Time-1 Map Orbit MatLab Simulation 2 … because of Cartwright-Littlewood-Levinson (1940s)

All Systems ym W id Theorem of Big: Every dynamical system of All Dynamical Systems Considered Theorem of Big: Every dynamical system of any finite dimension can be embedded into the spike renormalization R : X0 → X0 infinitely many times. That is, for any n and every map f : R n → R n there are infinitely many injective maps θ : R n → X0 so that the diagram commutes. ym W id 0 1 X0 = { } 1 X1 All Systems X0

Chaos ym W id Theorem of Chaos: The spike renormalization R : X0 → X0 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. l = 1 ym W chaos id 0 1 X0 = { } 1 X1 Chaos X0

Chaos ym W id Theorem of Chaos: The spike renormalization R : X0 → X0 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. l > 1 … Rn( f ) Rn(g) f g l = 1 ym W chaos id X1 Chaos X0

Chaos ym W id Theorem of Chaos: The spike renormalization R : X0 → X0 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. l = 1 ym W chaos id X1 Chaos X0 Use concatenation on a countable dense set (as L1 is separable) to construct a dense orbit

Pan-Chaos fm gμ ym g0 gμ W id g0 Universal Number Theorem of Almost Universality: Every number is an eigenvalue of the spike renormalization. Slope = λ > 1 gμ l = 1 l > 1 fm ym g0 gμ μ W Pan-Chaos chaos id l < 1 X1 l = 0 0 1 1 c0 y0=id f0 g0 X0

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:

Big Chaos

… X1 For each orbit { x0 , x1= f (x0), x2= f (x1), …} in [0,1], All Dynamical Systems Considered slope = l … Let W = X0 U X1 with y0 q (x0) 0 1 X1 = { } 1 0 1 X0 = { }, 1 y1 l > 1 y2 l = 1 ym Every n-dimensional dynamical system can be conjugate embedded into X0 in infinitely many ways. chaos id X1 For each orbit { x0 , x1= f (x0), x2= f (x1), …} in [0,1], let y0 = S(x0), y1 = R-1S(x1), y2 = R-2S(x2), … X0 W

Bifurcation of Spikes c0 V c I INa Ipump c0 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 c0 V c Isospike of 3 spikes I INa Ipump c0 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].

R m / (1-m) R[y0]=y0 R[ym]=ym/(1-m) R[y1/(n+1) ]= y1/n Universal Number 1 μn f μn ] R[y0]=y0 R[ym]=ym/(1-m) R[y1/(n+1) ]= y1/n 1 is an eigenvalue of DR[y0] Theorem of One (BD, 2011): The first natural number 1 is a new universal number . μn f μn ] μ1 μ2 f μn 0 1 ym R 1 m 1-m m / (1-m) ym /(1-m) m