Period Doubling Cascades

1. Period Doubling Cascades Jim Yorke Joint Work with Evelyn Sander George Mason Univ. Extending earlier work by Alligood, SN Chow, Mallet-Paret, & Franks

2. Period-doubling cascades If this picture were infinitely detailed, it would show infinitely many period-doubling cascades, each with an infinite number of period doublings. My goal is to explain this phenomenon And give examples in 1 and n dimensions.

3. some period doubling cascades Period 1 cascade Period 3 & 5 cascades

4. cascade Period-doubling cascades were first reported by Myrberg in 1962, and popularized by May using the logistic map in the 1970’s. For maps depending on a parameter, a cascade is an infinite sequence of period doubling bifurcations in a connected family of periodic orbits. The periods in the cascade are k, 2k, 4k, 8k,… for some k. • Feigenbaum’s rigorous methods suggest that when period-doubling cascades exist, there is a regular behavior in the sequence of period-doubling values.

5. cascade Period-doubling cascades were first reported by Myrberg in 1962, and popularized by May using the logistic map in the 1970’s. For maps depending on a parameter, a cascade is an infinite sequence of period doubling bifurcations in a connected family of periodic orbits. The periods in the cascade are k, 2k, 4k, 8k,… for some k. • Feigenbaum’s rigorous methods suggest that when period-doubling cascades exist, there is a regular behavior in the sequence of period-doubling values.

6. Needed: new examples • Maps like α - x2 have played a prominent role in the history of cascades. What is so special about these maps? If anything?

7. The topological view for problems depending on a parameter Example of a geometric theorem. Theorem. Assume • g is continuous on [α0 , α1] and • g(α0 ) < 0 and g(α1) > 0. • Then g(x) = 0 for some x between α0 & α1. We find an analogous approach for cascades

8. The topological view for problems depending on a parameter Example of a geometric theorem. Theorem. Assume • g is continuous on [α0 , α1] and • g(α0 ) < 0 and g(α1) > 0. • Then g(x) = 0 for some x between α0 & α1. We find an analogous theorems for cascades

9. A snake is a (non-branching) path of periodic orbits

11. The topological view for cascades Let F: [α0 , α1] X Rn→ Rn be differentiable. Theorem (terms explained later) Assume • there are no periodic orbits at α0 ; and • at α1 the dynamics are horse-shoe-like; and • On [α0 , α1] the set of periodic points is bounded in x. • F has generic orbit behavior; Then if (α1, x1) is periodic and has no eigenvalues < -1, it is on a connected family of orbits which includes a cascade. Distinct such orbits yield distinct cascades.

12. The topological view for cascades Let F: [α0 , α1] X Rn→ Rn be differentiable. Theorem (terms explained later) Assume • there are no periodic orbits at α0 ; and • at α1 the dynamics are horse-shoe-like; and • On [α0 , α1] the set of periodic points is bounded in x. • F has generic orbit behavior; Then if (α1, x1) is periodic and has no eigenvalues < -1, it is on a connected family of orbits which includes a cascade. Distinct such orbits yield distinct cascades.

13. A new example Let F(α;x) =α- x2 + g(α ,x) Assume g(α, x) is a real valued function, differentiable and bounded for α,x in R2, and so are its first partial derivatives. For example g = finite sum of fourier series terms in α,x plus terms like tanh(α+x) Let F(α;x) =α- x2 + g(α ,x)

14. A new example Assume g(α ,x) is differentiable and bounded over all α ,x and so are its first partial derivatives. Let F(α;x) =α- x2 + g(α , x) Then • for α0 sufficiently small, there are no periodic orbits at α0 ; and • for α1 sufficiently large, the dynamics are horse-shoe-like,and • for “almost every” g, F has generic orbit behavior • the set of all periodic orbits in [α0 , α1] is bounded, and Theorem. For such generic g, if (α1, x1) is periodic and its derivative is > +1, Then it is on a connected family of orbits which includes a cascade. Corollary: the map has infinitely many disjoint cascades.

15. A new logistic exampleα x(1-x)g(α, x) for some α

16. A new logistic example We require that g(α, x) is differentiable and positive for x in [0,1], and bounded: For some B1 & B2, 0 < B1 < g(α, x) < B2 and the partial derivatives fo g are also bounded. Then αx(1-x)g(α, x) has cascades of period doublings as the parameter α is varied (for typical g). In fact we show the map has infinitely many disjoint cascades as a is varied. a a

17. Periodic orbits of F(α,x) We say (α,x) is p-periodic if Fp(α,x) = x. If (α,x) is p-periodic, its “eigenvalues” are those of its derivative DFp(α,x). If x is one-dimensional, its “eigenvalue” is the derivative (d/dx)Fp(α,x). An orbit with no eigenvalues on the unit circle is called “hyperbolic”; these include attractors.

18. Periodic orbits of F(α,x) We say (α,x) is p-periodic if Fp(α,x) = x. If (α,x) is p-periodic, its “eigenvalues” are those of its derivative DFp(α,x). If x is one-dimensional, its “eigenvalue” is the derivative (d/dx)Fp(α,x). An orbit with no eigenvalues on the unit circle is called “hyperbolic”; these include attractors.

19. Types of hyperbolic orbits Let (α,x) be a hyperbolic periodic point. It is a flip saddle orbit or point if it has an odd number of eigenvalues < -1. If (α,x) is NOT a flip saddle orbit and the number of eigenvalues with λ > 1 = n or n-2 or n-4 etc, then it is a left orbit; otherwise it is a right orbit. For n=1, right orbits are attractors and left orbits are orbits with derivative > +1.

20. A snake is a (non-branching) path of periodic orbits

21. Following segments of orbits Follow a segment of left orbits to the left (decreasing parameter direction) Follow a segment of right orbits to the right. (increasing parameter direction) Never follow segments of flip orbits.

22. Generic Bifurcations of a path For a family of period k orbits x(α) in Rn, bifurcations can occur when DFk(x) has eigenvalue(s) crossing the unit circle. Generically they are simple. • A Saddle node occurs when an e.v. λ= +1 • A Period doubling . . . λ= -1 • Generically complex pairs cross the unit circle at irrational multiples of angle 2π

23. Possible bifurcations affecting paths Bifurcations for 1 dim x or more

24. Possible bifurcations affecting paths Bifurcations for 1 dim x or more Other Bifurcations only in dim x > 1 In addition each period-doubling bifurcation can have both arrows reversed All low-period segments are “right” segments All new low-period segments are “left” segments

25. Possible bifurcations affecting paths Bifurcations for 1 dim x or more Other Bifurcations only in dim x > 1 All S-N & P-D bifurcation points have one segment approaching and one departing(except the upper-right one). In addition each period-doubling bifurcation can have both arrows reversed

26. Coupling n 1-D maps Coupling n 1-D maps. x = (x1, …,xn) Let F(α;x) = (αa1 - x12 + g1 (α, x1,…,xn), . . . αan - xn2 + gn (α, x1,…,xn)) where each gj is bounded and so are its partial derivatives; Assume aj > 0 for each j = 1,…,n.

27. A new n-Dim example Assume gm : RxRn→ R for each m is differentiable and bounded, and so are its first partial derivatives. Then • for α0 sufficiently small, there are no periodic orbits at α0 ; and • for α1 sufficiently large, the dynamics are the horse-shoe-like behavior of the uncoupled system (i.e. g=0), and • for “almost every” g = (gm), F has generic orbit behavior • the set of all periodic orbits in [α0 , α1] is bounded, and Theorem. For such generic g If (α1, x1) is periodic and has an even number of eigenvalues < -1, (possibly none), Then it is on a connected family of orbits which includes a cascade. Corollary: the map has infinitely many disjoint cascades.

28. A new n-Dim example Assume gm : RxRn→ R for each m is differentiable and bounded, and so are its first partial derivatives. Then • for α0 sufficiently small, there are no periodic orbits at α0 ; and • for α1 sufficiently large, the dynamics are the horse-shoe-like behavior of the uncoupled system (i.e. g=0), and • for “almost every” g = (gm), F has generic orbit behavior • the set of all periodic orbits in [α0 , α1] is bounded,and Theorem. For such generic g if (α1, x1) is periodic and has an even number of eigenvalues < -1, (possibly none), Then it is on a connected family of orbits which includes a cascade. Corollary: the map has infinitely many disjoint cascades.

29. A new n-Dim example Assume gm : RxRn→ R for each m is differentiable and bounded, and so are its first partial derivatives. Then • for α0 sufficiently small, there are no periodic orbits at α0 ; and • for α1 sufficiently large, the dynamics are the horse-shoe-like behavior of the uncoupled system (i.e. g=0), and • for “almost every” g = (gm), F has generic orbit behavior • the set of all periodic orbits in [α0 , α1] is bounded, and Theorem. For such generic g If (α1, x1) is periodic and has an even number of eigenvalues < -1, (possibly none), Then it is on a connected family of orbits which includes a cascade. Corollary: the map has infinitely many disjoint cascades.

30. Following families of period p points Let F : R X Rn→ Rn be differentiable. Assume Fp(α0 ,x0) = x0 When does there exist a continuous path (α, x(α)) of period-p points through (α0 ,x0) for α in some neighborhood (α0 -ε,α0 +ε) of α0? This can answered by trying to compute the path x(α) as the sol’n of an ODE..

31. A p-period Orbit (α0 ,x0) can be continued if +1 is not an eigenvalue If Fp(α, x(α)) - x(α) = 0, then (d/dα) {Fp (α, x(α)) - x(α)} = 0 (*) i.e., Fpα, +Fpx dx/dα – Id dx/dα = 0 If Fpx – Id is invertible, then x(α) satisfies dx/dα = [Fpx – Id]-1 Fpα (**) It is easy to check (*) is satisfied by any solution of (**). If (α0 ,x0) is periodic and +1 is not an eigenvalue, then (α,x(α)) can be continued, ending only when +1 is an eigenvalue.

32. A p-period Orbit (α0 ,x0) can be continued if +1 is not an eigenvalue If Fp(α, x(α)) - x(α) = 0, then (d/dα) {Fp (α, x(α)) - x(α)} = 0 (*) i.e., Fpα, +Fpx dx/dα – Id dx/dα = 0 If Fpx – Id is invertible, then x(α) satisfies dx/dα = [Fpx – Id]-1 Fpα (**) It is easy to check (*) is satisfied by any solution of (**). If (α0 ,x0) is periodic and +1 is not an eigenvalue, then (α,x(α)) can be continued, ending only when +1 is an eigenvalue.

33. A p-period Orbit (α0 ,x0) can be continued if +1 is not an eigenvalue If Fp(α, x(α)) - x(α) = 0, then (d/dα) {Fp (α, x(α)) - x(α)} = 0 (*) i.e., Fpα, +Fpx dx/dα – Id dx/dα = 0 If Fpx – Id is invertible, then x(α) satisfies dx/dα = [Fpx – Id]-1 Fpα (**) It is easy to check (*) is satisfied by any solution of (**). If (α0 ,x0) is periodic and +1 is not an eigenvalue, then (α,x(α)) can be continued, ending only when +1 is an eigenvalue.

34. Snakes of periodic orbits • A snake is a connected directed path of periodic orbits. • Following the “path” allows no choices because it does not branch.

35. A snake is a (non-branching) path of periodic orbits

37. Generic Behavior of F(α,x) In a bounded region of (α,x) space, for each period p, • there are finitely many p-periodic (α,x) having +1 as an eigenvalue and all such are generic saddle-node bifurcation orbits. • there are finitely many p-periodic (α,x) having -1 as an eigenvalue and all such are generic period doubling orbits. • If (α,x) has complex eigenvalues on the unit circle, they are irrational multiples of 2π.

38. Generic Behavior of F(α,x) In a bounded region of (α,x) space, for each period p, • there are finitely many p-periodic (α,x) having +1 as an eigenvalue and all such are generic saddle-node bifurcation orbits. • there are finitely many p-periodic (α,x) having -1 as an eigenvalue and all such are generic period doubling orbits. • If (α,x) has complex eigenvalues on the unit circle, they are irrational multiples of 2π.

39. Generic Behavior of F(α,x) In a bounded region of (α,x) space, for each period p, • there are finitely many p-periodic (α,x) having +1 as an eigenvalue and all such are generic saddle-node bifurcation orbits. • there are finitely many p-periodic (α,x) having -1 as an eigenvalue and all such are generic period doubling orbits. • If (α,x) has complex eigenvalues on the unit circle, they are irrational multiples of 2π.

40. Generic maps • Almost every (in the sense of prevalence) map is generic.

41. The reason why cascades occur • Each left segment must terminate (at a SN or PD bifurcation) because there are no orbits at α0. • Each right segment must terminate (at a SN or PD bifurcation) because there are no right orbits at α1. • The family then continues onto a new segment. This leads to an infinite sequence of segments and corresponding periods (pk). • Each period can occur at most finitely many times, so pk→∞. So it includes ∞-many PDs.