Universality of the Route to Chaos -- Exact Analysis

The universality of the route to chaos is analytically proven for countably infinite number of maps by proposing the Super Generalized Boole (SGB) transformations. As one of the route to chaos, intermittency route was studied by Pomeau and Manneville numerically. They conjectured the universality in Type 1 intermittency, that the critical exponent of the Lyapunov exponent is $1/2$ in Type 1 intermittency. In order to prove their conjecture, we showed that for certain parameter ranges, the SGB transformations are exact and preserve the Cauchy distribution. Using the property of exactness, we proved that the critical exponent is $1/2$ for countably infinite number of maps where Type 1 intermittency occurs.

Universality in chaos Route from stable states to chaotic (intermittent) states has caught much attention in broad fields in physics. This issue treats fundamental change of systems from stable state to unstable state and it is an essential theme to analyze the stability of physical systems. Route to chaos is also studied theoretically and experimentally such as Hamiltonian systems [1], map systems [2][3][4][5][6][7], coupled oscillators [8], Belousov-Zhabotinskii reaction [9], Rayleigh-Brnard convection [9], Couette Taylor flow [9], noise induced system [10], thermoacoustic system [11] and optomechanics [12][13][14]. There is the theoretical classification of routes to chaos such as intermittency route, period doubling route, frequency locking route, etc [9]. Frequently these researches have been motivated to discover the universality at the onset of chaos with respect to the critical exponent of the Lyapunov exponent, which is an indicator of chaos. The universality of the critical exponents in each route to chaos has been studied extensively by numerical simulations. For period doubling route, Huberman and Rudnick [5] estimated numerically the critical exponent ν as ν = log 2 log δ where δ represents the Feigenbaum constant. For intermittency route treated in this Letter, Pomeau and Manneville [3,4] classified intermittency into three types and conjectured the universality of ν for each intermittent type. In particular, in Type 1 intermittency, they conjectured the universality that ν = 1 2 , by the numerical simulations. After the work by Pomeau and Manneville, the critical exponent has been researched in various field with relations to intermittency such as Billiard system [15,16], electronic circuit [17], plasma physics [18] and intermittent map [19]. Those studies by numerical simulations suggest that their conjecture ν = 1 2 would be right.
However, in these researches, the critical exponent ν is estimated by numerically or the analytical formulae of the Lyapunov exponent λ was not obtained without any assumption.
On the other hand, the present authors [5] led the analytical formula of the Lyapunov exponent λ as an explicit function in terms of the bifurcation parameter by showing the mixing property for the Generalized Boole (GB) transformation. They proved in the GB transformation that both Type 1 and Type 3 intermittency occur and that the conjecture by Pomeau and Manneville is correct.
In this Letter, it is analytically proved that for countably infinite number of maps, it holds that when Type 1 intermittency occurs where α and α c represent a bifurcation parameter and the critical point, respectively. In order to prove this, we propose more generalized maps, the Super Generalized Boole (SGB) transformations and show that there are parameter ranges in which the SGB transformations are exact (stronger condition than ergodicity). That means one obtains countably infinite number of exact (ergodic) maps. Using this result, one can obtain explicitly the analytical formulae of the Lyapunov exponents and critical exponents.
We define two-parameterized one-dimensional maps, the Super Generalized Boole Transformations (SGB) S K,α : R\B → R\B as follows.
where α > 0, K ∈ N\{1}, the set B is a set on R such that for any point x on B, there is an integer n where S n K,α x reaches a singular point, and the function F K corresponds to K-angle formula of cot function defined in Supplemental material. The GB transformation T α,α in [5] corresponds to the map S 2,α . Figure 1 shows the return maps of S 3, 1 3 , S 4, 1 4 and S 5, 1 5 . Invariant density In this paragraph, it is proven that the SGB transformations preserve the Cauchy distribution for certain condition. According to [1,21], the map S K,α is K to one map as follows. y = Kα cot Kθ = KαF K (x j ), x j = cot θ + j π K , j = 1, 2, · · · , K. If variables {x j } obey the Cauchy distribution f γ (x) = 1 π γ x 2 +γ 2 whose scale parameter is γ, then according to [1], the variable y obeys the density function p(y) = 1 π αKGK (γ) y 2 +α 2 K 2 G 2 K (γ) , where the function G K (x) corresponds to K-angle formula of coth function defined in Supplemental material. Then the scale parameter γ is transformed in one iteration as γ −→ αKG K (γ). Now, for each K, let us obtain the fixed point 0 < γ K,α < ∞ which satisfies the relation , respectively. The explicit forms of these three maps are in Supplemental material.

Definition 1. Condition A is referred to as
where N ∈ N.
Then the following theorem holds.
Theorem A. When the Condition A is satisfied, the SGB transformations {S K,α } preserve the Cauchy distribution and the scale parameter can be chosen uniquely.
The proof of Theorem A is given in Supplemental material. Although it has been proven that the map S K,α preserves the Cauchy distribution and its scale parameter γ K,α can be determined uniquely when the Condition A is satisfied, it is not straightforward to obtain the explicit form of fixed point γ K,α for arbitrary K, since we have to solve the Kth-degree equations. From Theorem A, the condition that there exists only one solution of (3) which satisfied 0 < γ K,α < ∞ is nothing but the Condition A.
Definition 2 (Exactness). A dynamics T on a phase space X with transfer operator P T and unique stationary density f * is called to be exact if and only if for every initial density f ∈ D where D denotes all densities on X . This definition is equivalent to as follows, where B denotes the σ-algebra and µ * denotes the invariant measure corresponding the invariant density f * .
In terms of exactness, we obtain the following theorem.
Theorem B. If the the Condition A is satisfied, the SGB transformations {S K,α } are exact.
The proof is given in Supplemental material. From Theorems A and B, when the Condition A is satisfied, the map S K,α preserves certain Cauchy distribution f * and any initial density function f defined on R\B converges to f * as For example, S 3,α , S 4,α and S 5,α are exact for 1 9 < α < 1, 0 < α < 1 and 1 25 < α < 1, respectively. According to [3], if the SGB transformations are exact, then the corresponding dynamical systems are mixing and ergodic. Therefore the following Corollary holds.
Corollary 3. Suppose that the Condition A is satisfied. Then the dynamical system (R\B, S K,α , µ * ) has the mixing property and it is ergodic where µ * is the invariant measure corresponding to the invariant density f * .
Using the property of exactness, one can obtain the explicit formula of the Lyapunov exponent such that From Pesin's formula, one sees that the Kolmogorov-Sinai entropy is equivalent to the Lyapunov exponent since the SGB transformations are a one-dimensional map. For α > 1, changing variable as z n = 1/x n , one obtains the map S K,α defined as Then one has that d SK,α dz (0) = 1 α < 1, so that for any K, the orbits are attracted into the infinite point for α > 1. Thus, the Lyapunov exponent λ K,α for α > 1 is derived from the inclination at the infinite point as Scaling behavior At the edges of the Condition A, one has that Then the Lyapunov exponent converges to zero at the edges of the Condition A. In order to discuss the critical phenomena, define critical points as α c1 = 1, α c2 = 1 (2N +1) 2 and α c3 = 0 and define critical exponents ν 1 , ν 2 and ν 3 corresponding to α ci , i = 1, 2, 3. In terms of the scaling behavior of the Lyapunov exponents λ ∼ b |α − α ci | νi , b > 0, i = 1, 2, 3 the following theorem holds.
Theorem C. Suppose that the Condition A is satisfied.
In the case of K = 3, 4 and 5 In this paragraph, the examples corresponds to K = 3, 4 and 5 are illustrated. The solutions of (3) which satisfies 0 < γ K,α < ∞ are uniquely determined as follows.
From the above discussion, one knows that in the case of K = 3 and 5, only Type 1 intermittency occurs and in the case of K = 4, both Type 1 and Type 3 intermittency occur. The Lyapunov exponents in the case of K = 3, 4 and 5 are given as follows.
Figures 2a, 2b and 2c show the Lyapunov exponents against α in the case of K = 3, 4 and 5, respectively. One sees that the numerical simulations are exactly consistent with the obtained analytical formulae. Since it holds that at the critical points ∂λ K,α /∂α = ±∞, one sees that the parameter dependence of the Lyapunov exponent diverges at the critical points. This means the computational difficulty in obtaining the true value of the Lyapunov exponent by numerical simulation. Figure 3 shows the scaling behavior of the Lyapunov exponent. One sees that ν 2 = 1 2 and ν 3 = 1 2 in the case of K = 3, 4 and 5. Conclusion This work is the first example in which the conjecture by Pomeau and Manneville expressed in (1) is analytically proven to be true for countably infinite number of maps (the proposed Super Generalized Boole transformations). This work shows the theoretical picture of stable-unstable transition for intermittent maps. In the course of proof, we have shown that the Super Generalized Boole (SGB) transformations preserve the unique Cuachy distribution, together with proving the fact that SGB transformations are exact and that any initial density function converges to the invariant Cauchy distribution when the Condition A is satisfied.
Applying the property of exactness, one can obtain analytical formulae of the Lyapunov exponents for the SGB transformations. In the SGB transformations, the Lyapunov exponents λ K,α are equivalent to the Kolmogorov-Sinai entropy applying to the Pesin's theorem. Using the analytical formulae of the Lyapunov exponents, we have confirmed that for K = 3, 4 and 5, the derivative ∂λ K,α /∂α diverge at the critical points and we obtained ν 1 = ν 2 = ν 3 = 1 2 . Thus, we have proven the universality of the route to chaos for a large class of the chaotic systems.
As future works, clarifying the scaling relation between the critical exponent ν and the other critical exponents, we can obtain a new perspective of chaos in physics.

Supplemental Material
In this Supplemental material it is shown that 1. the Super Generalized Boole transformations preserve the Cauchy distribution for certain parameter ranges and the Cauchy distribution is determined uniquely, 2. the SGB transformations are exact for the parameter ranges, and 3. the critical exponent of Lyapunov exponent is ν = 1 2 for all K ∈ N\{1} when Type 1 intermittency occurs. The definitions of F K , G K , S K,α are written in the following.
Definition 3. The map F K : R\A → R\A is referred to as where K ∈ N\{1} and the set A is a set on R such that for any point x ∈ A, F K (x) reaches a singular point.
Definition 5 (Super Generalized Boole Transformation). Super Generalized Boole Transformation S K,α : R\B → R\B is referred to as where α > 0, K ∈ N\{1} and the set B is a set on R such that for any point x on B, there is an integer n where S n K,α x reaches a singular point.
The derivative of S K,α with respect to x is denoted as

Existence of the invariant density function
In this section, we prove that • when K = 2N, N ∈ N and 0 < α < 1, =⇒ the map S 2N,α preserves the Cauchy distribution and it can be determined uniquely, and • when K = 2N + 1, N ∈ N and 1 (2N +1) 2 < α < 1, =⇒ the map S 2N +1,α preserves the Cauchy distribution and it can be determined uniquely.
If variables {x j } obey the Cauchy distribution then according to [1] it holds that Then the scale parameter γ is transformed in one iteration as Now, for each K let us obtain the fixed point γ K,α which satisfies If we discover a real and positive solution of (S9), it is the evidence that the map S K,α preserves the Cauchy distribution.
The map G K (x) is denoted as . (S11) Since at α = 1 K , the SGB transformation S K,α is equivalent to K-angle formula of cot function, it is obvious that the fixed point of scaling parameter γ K, 1 K is unity by simple calculation. In the following, discuss the case that α = 1 K . Such lemmas hold. Lemma 6. For 1 K < α < 1, fix α and there is only one solution which satisfies (S9) in the range of γ K,α > 1 and for 0 < α < 1 K there is no solution in the range of γ K,α > 1.
In the range where y ≥ 0, since a function coth y decreases monotonically, it holds that coth y > coth(Ky).
In the case of 0 < α < 1 K , it holds that for any y > 0 Then, there is no solution which satisfies γ K,α > 1.

Lemma 7.
In the case of K = 2N for 0 < α < 1 K , fix α. There is only one solution which satisfies (S9) in the range of 0 < γ K,α < 1 and for 1 K < α there is no solution in the range of 0 < γ K,α < 1. Proof. In (S9), change the variable from 0 < γ K,α < 1 into tanh y. One has that tanh y = α(2N ) tanh(2N y) . (S19) The derivative of h 2N (y) is denoted as One has that From (S21) and (S22), one sees that in the case of K = 2N for fixed α which satisfied 0 < α < 1 K , there is only one solution which satisfies (S9) in the range of 0 < γ K,α < 1.
In the case of 1 2N < α, it holds that for all y > 0 Therefore, there is no solution which satisfies 0 < γ K,α < 1.
In the case of 1 K < α, since it holds that for all y > 0, there is no solution in the range of 0 < γ K,α < 1.
Lemma 9. Consider the case of K = 2N . For 1 K ≤ α < 1, there is a unique solution of (S9) and the solution γ K,α is in the range of γ K,α ≥ 1. For 0 < α < 1 K , there is a unique solution of (S9) and the solution γ K,α is in the range of 0 < γ K,α < 1.
Lemma 10. Consider the case of K = 2N + 1. For 1 K ≤ α < 1, there is a unique solution of (S9) and the solution γ K,α is in the range of γ K,α ≥ 1. For 1 K 2 < α < 1 K , there is a unique solution of (S9) and the solution γ K,α is in the range of 0 < γ K,α < 1.
The Condition A is defined as follows Definition 11. Condition A is referred to as Condition A : in the case of K = 2N, α satisfies 0 < α < 1 and in the case of K = 2N + 1, α satisfies 1 K 2 < α < 1. (S33) From Lemmas 9 and 10, such theorem holds.
Theorem A. When the Condition A is satisfied, the SGB transformations {S K,α } preserve the Cauchy distribution and the scale parameter can be chosen uniquely.
Definition 10 (Exactness). A dynamics T on a phase space X with transfer operator P T and unique stationary density f * is called to be exact if and only if for every initial density f ∈ D where D denotes all densities on X . This definition is equivalent to as follows, where B denotes the σ-algebra and µ * denotes the invariant measure corresponding the invariant density f * .
Theorem 11. If the the Condition A is satisfied, the SGB transformations {S K,α } are exact.
Then,S K,α increases monotonously. Since it holds that form ofS K,α has the translational symmetry and it can be constructed by shifting the form on [0, 1 K ). That is, the mapS K,α is also K points to one points map and on any interval I j,1 , the form of the mapS K,α is the same as that on the interval I 0,1 . Then by operatingS −1 K,α , the measure on [0, 1) is divided into K equivalently. We obtain intervals {I j,n } defined below by operatingS −n K,α into [0, 1). The interval I j,n ⊂ [0, 1) is defined to be I j,k def = [η j,n , η j+1,n ) , η j,n < η j+1,n , 0 ≤ j ≤ K n − 1, η 0,n = 0 and η K n ,n = 1, S n K,α (I j,n ) = [0, 1), For any non zero measure subset C ⊂ [0, 1), the set C includes cylinder sets j,n ′ I j,n ′ . Then for an invariant density f * and associated measure µ * [4], it holds that Therefore, the mapS K,α on a phase space [0, 1), is exact. Owing to the topological conjugacy, the map S K,α is also exact.
Proof. Discuss the case of K = 2N + 1 and 1 K 2 < α < 1 K . (S9) is rewritten as In the limit of γ 2N +1,α → 0, it holds that From Lemma 12, 13 and 14, one knows that there are relations between the parameter α and the scaling parameter γ K,α . For all α which satisfied the Condition A, Lyapunov exponent is denoted as Define a function f 1 (θ, γ K,α ) to be f 1 (θ, γ K,α ) = log αK 2 sin 2 θ sin 2 Kθ γ K,α γ 2 K,α sin 2 θ + cos 2 θ (S59) and also define a set of points {a n } K−1 n=1 such that at x = a n ∈ (0, π], the function log αK 2 sin 2 θ sin 2 Kθ is not continuous. By changing variable from x to cot θ, Lyapunov exponent λ K,α is rewritten as Theorem C. Suppose that the Condition A is satisfied.

(S61)
The derivative is continuous on each interval (a n , a n+1 ].
From these results, we can say that only Type 1 intermittency occurs for K = 3. These results are new phenomena since for the Generalized Boole transformation, one can observe two different intermittent type, Type 1 and Type 3 [5].
In the case of K = 4, Lyapunov exponent (S72) is denoted as Let us discuss the scaling behavior of λ 4,α at α = 0. Now the first term of (S77) is rewritten as Then, Therefore, the critical exponent of Lyapunov exponent for K = 4, α = 0 is Next, consider the scaling behavior of λ 4,α at α = 1 − 0. From (S78), it holds that near α = 1, The second and third terms of (S77) are denoted as Therefore, the critical exponent of Lyapunov exponent for K = 4, α = 1 − 0 is In the case of K = 4, fixed points of S 4,α are as follows.
In the case of K = 5, equation (S72) converges to zero in the limit of α → 1 25 + 0 and α → 1 − 0. In addition, it holds that Then the the forth term is not dominant near α = 1 25 and α = 1. Thus in considering the scaling behavior, we do not have to care the forth term.
When the parameter α is close to 1 25 , the Lyapunov exponent grows as follows Therefore, the critical exponent ν 2 is denoted as For α 1, Lyapunov exponent grows as follows.