From Small To One Big Chaos

1. 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

2. 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

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

4. Period Doubling Bifurcation

5. Period-Doubling Cascade, and Universality Feigenbaum’sUniversal Number (1978) i.e. at a geometric rate 4.6692016…

6. Renormalization 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

7. Renormalization Feigenbaum’s Renormalization at

8. 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*

9. 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

10. Period three implies chaos, T.Y. Li & J.A. Yorke, 1975

11. Poincaré Return Map PoincaréReturn Map (1887) reduces the trajectory of a differential equation to an orbit of the map. PoincaréTime-1 Map: φτ(x0) = x(τ, x0), for which x(t, x0) is the solution with initial condition x(0, x0) = x0

12. Poincaré Return Map Ipump

13. Poincaré Return Map Ipump

14. Poincaré Return Map V c c0 1 f INa Ipump 0 c01

15. Poincaré Return Map c0 V c C -1 1 R( f ) INa Ipump 01 C-1/C0

16. Poincaré Map Renormalization f f2 R • 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 …

17. m Spike Return Maps 1 1 fm f0 m→0 m →0 m e-k/m 0 c01 0 c01 1 ym 1 y0=id m 1-m 01 01

18. 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 …μ10μ9μ8 μ7μ6μ5 μ Scaling Laws : μn ~ 1/n and (μn- μn-1)/(μn+1- μn) → 1

19. Poincaré Return Map c0 • At the limiting bifurcation point μ = 0, an equilibrium • point of the differential equations invades a family • of limit cycles. Homoclinic Orbit at μ= 0 V c 1 f0 0 c01 INa Ipump

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

21. universalconstant 1 Dynamics of Spike Map Renormalization -- Universal Number 1 1 1 R m / (1-m) ym /(1-m) ym 1 m W = { } , the set of elements of Y , each has at least one fixed point in [0,1]. 01 01 1-m 01 Y • R[y0]=y0 • R[ym]=ym / (1-m) • R[y1/(n+1)]= y1/n fμn] fμn] μ1 μn μn μ2 fμn

22. Universal Number 1 1 1 R m / (1-m) ym m 01 01 1-m • R[y0]=y0 • R[ym]=ym/(1-m) • R[y1/(n+1)]= y1/n • 1 is an eigenvalue of DR[y0] fμn] fμn] μ1 μn μn μ2 fμn ym /(1-m) m

23. Universal Number 1 1 1 R m / (1-m) ym m 01 01 1-m • 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 . fμn] fμn] μ1 μn μn μ2 fμn ym /(1-m) m

24. Renormalization Summary 1 1 X0={ : the right most fixed point is 0. } X1={ } = W \ X0 Eigenvalue: l = 1 U={ym} Invariant 01 01 y0 = id Fixed Point X1 X0 W = X0U X1 Invariant

25. 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)

26. 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. 1 X0= { } 01 ym W id X1 All Systems X0

27. 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. 1 X0= { } l = 1 01 ym W id X1 Chaos X0

28. 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. Rn( f ) f Rn(g) g l = 1 ym l > 1 l > 1 … W id X1 Chaos X0

29. 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 id X1 Chaos X0 • Use concatenation on a countable dense set • (as L1 is separable) to construct a dense orbit

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

31. 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

32. 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. Summary

33. Phenomenon of Bursting Spikes Rinzel & Wang (1997) Excitable Membranes

34. Phenomenon of Bursting Spikes Dimensionless Model: Food Chains

35. Big Chaos

36. All Dynamical Systems Considered y0 q (x0) y1 y2 … Every n-dimensional dynamical system can be conjugate embedded into X0in infinitely many ways. 1 1 X1 = { } X0= { }, 01 01 slope = l • Let W = X0U X1with l > 1 l = 1 ym 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

37. Bifurcation of Spikes c0 V c I INa Ipump c0 Def: System is isospikingof 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].

38. Bifurcation of Spikes c0 V c Isospike of 3 spikes I INa Ipump c0 Def: System is isospikingof 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].

39. Universal Number 1 1 1 R m / (1-m) ym m 01 01 1-m • 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 . fμn] fμn] μ1 μn μn μ2 fμn ym /(1-m) m