Universal functions and exactly solvable chaotic systems

Abstract

A universal differential equation is a nontrivial differential equation the solutions of which approximate to arbitrary accuracy any continuous function on any interval of the real line. On the other hand, there has been much interest in exactly solvable chaotic maps. An important problem is to generalize these results to continuous systems. Theoretical analysis would allow us to prove theorems about these systems and predict new phenomena. In the present paper we discuss the concept of universal functions and their relevance to the theory of universal differential equations. We present a connection between universal functions and solutions to chaotic systems. We will show the statistical independence between X(t) and X(t+τ)X(t + \tau) (when τ\tau is not equal to zero) and X(t) is a solution to some chaotic systems. We will construct universal functions that behave as delta-correlated noise. We will construct universal dynamical systems with truly noisy solutions. We will discuss physically realizable dynamical systems with universal-like properties.

Full text

arXiv:0811.1179v1 [nlin.SI] 7 Nov 2008
Universal functions and exactly solvable chaotic systems
M´onica A. Garc´ıa- ˜
Nustes∗and Jorge A. Gonz´alez
Centro de F´ısica, Instituto Venezolano de Investigaciones Cient´ıficas, Apartado 21827, Caracas 1020-A, Venezuela
Emilio Hern´andez-Garc´ıa
Departamento de F´ısica Interdisciplinar,Instituto Mediterrneo de Estudios Avanzados
CSIC-Universidad de las Islas Baleares, E-07122-Palma de Mallorca, Espa˜na
A universal differential equation is a nontrivial differential equation the solutions of which ap-
proximate to arbitrary accuracy any continuous function on any interval of the real line. On the
other hand, there has been much interest in exactly solvable chaotic maps. An important problem
is to generalize these results to continuous systems. Theoretical analysis would allow us to prove
theorems about these systems and predict new phenomena. In the present paper we discuss the
concept of universal functions and their relevance to the theory of universal differential equations.
We present a connection between universal functions and solutions to chaotic systems. We will
show the statistical independence between X(t) and X(t+τ) (when τis not equal to zero) and
X(t) is a solution to some chaotic systems. We will construct universal functions that behave as
delta-correlated noise. We will construct universal dynamical systems with truly noisy solutions.
We will discuss physically realizable dynamical systems with universal-like properties.
I. INTRODUCTION
Recently there has been great interest in exactly solvable chaotic systems [1, 2, 3, 4, 5, 6, 7]. S. Ulam and J.
von Neumann were the first to prove that the general solution to the logistic map can be found [1, 2]. It is very
important to extend these results to continuous systems. Theoretical analysis would allow us to prove theorems about
these systems and predict new phenomena. Another surprising parallel development is that of “universal differential
equations” . A universal differential equation is a nontrivial differential-algebraic equation with the property that
its solutions approximate to arbitrary accuracy any continuous function on any interval of the real line [8, 9, 10, 11,
12, 13, 14, 15, 16]. Rubel found the first known universal differential equation by showing that there are differential
equations of low order (e.g. fourth-order) which have solutions arbitrarily close to any prescribed function [8]. The
existence of universal differential equations illustrate the amazing complexity that solutions of low-order dynamical
systems can have. In the present paper, we will review some developments in the areas of universal differential
equations and exactly solvable chaotic dynamical systems. We will discuss the concept of “universal functions” and
their relevance to the theory of universal differential equations. We will show the statistical independence between
X(t) and X(t+τ) (when τ6= 0) and X(t) is the solution to some chaotic dynamical systems. We will present a
connection between universal functions and solutions to chaotic systems. We will construct universal functions that
behave as δ-correlated noise. We will construct universal differential equations with truly noisy solutions. We will
discuss physically realizable dynamical systems with universal properties and their potential applications in secure
communications and analog computing.
II. UNIVERSAL DIFFERENTIAL EQUATIONS
Rubel’s theorem [8] is:
There exists a nontrivial fourth- order differential equation
P(y′, y′′, y′′′, y′′′′) = 0,(1)
where y′=dy
dt ,Pis polynomial in four variables, with integer coefficients, such that for any continuous function φ(t)
on (−∞,∞) and for any positive continuous function ε(t) on (−∞,∞), there exists a C∞solution y(t) such that
|y(t)−φ(t)|< ε(t) for all ton (−∞,∞). A particular example of Eq.(1) is the following:
3y′4y′′y′′′′2−4y′4y′′′2y′′′′ + 6 y′3y′′2y′′′y′′′′ + 24y′2y′′4y′′′′
−29 y′2y′′2y′′′2−12 y′3y′′y′′′3+ 12 y′′7= 0,(2)
∗Corresponding author. Fax: +58-212-5041566; e-mail: mogarcia@ivic.ve

2
(a) (b)
FIG. 1: Solution y(t) for (a) equation (3), and (b) equation (4) with m= 4.
Duffin [10] has found two additional families of universal differential equations:
my′2y′′′′ + (2 −3m)y′y′′y′′′ + 2(m−1)y′′3= 0 (3)
and
m2y′2y′′′′ + 3m(1 −m)y′y′′y′′′ + (2m2−3m+ 1)y′′3= 0 (4)
where m > 3.
Recently, Briggs [17] has found a new family:
y′2y′′′′ −3y′y′′y′′′ + 2(1 −n−2)y′′3= 0,(5)
for n > 3.
The solutions are Cn.
We would like to make some observations about these equations.
Rubel’s function y(t) in Eq.(2) is C∞but not real-analytic, typically having a countable number of essential
singularities.
The functions used to reconstruct the solutions to Eq.(2) are of the form
y=Af(αt +β) + B, (6)
where f(t) = Rg(t)dt,g(t) = exp h−1
(1−t2)ifor −1< t < 1, with g(t) = 0 for all other t.
The solutions to equations (3) are trigonometric splines. The kernel g(t) is defined as
g(t) = a[cos bt +c]m.(7)
On the other hand, the kernel employed to obtain the solutions to Eq. (4) are polynomial splines:
g(t) = a[1 −(bt +c)2]m.(8)
The solutions are very unstable. This phenomenon can be observed in Fig.(1a) and Fig.(1b).
III. EXACTLY SOLVABLE CHAOTIC MAPS
Ulam and von Neumann [1, 2] proved that the function Xn= sin2(θπ 2n) is the general solution to the logistic map
Xn+1 = 4Xn(1 −Xn).(9)
Recently, many papers have been dedicated to exactly solvable chaotic maps [3, 4, 5, 6, 7, 18, 19, 20, 21]. In
some of these papers, the authors not only find the explicit functions Xnthat solve the maps, but also they discuss

3
the statistical properties of the sequences generated by the chaotic maps. For many of the maps discussed in the
mentioned papers, the exact solution can be written as
Xn=P(θkn),(10)
where P(t) is a periodic function, θis a fixed real parameter and kis an integer.
For instance, Xn= cos(2πθ 3n) is the solution to map Xn+1 =Xn(4Xn2−3). This is a particular case of the
Chebyshev maps [5].
Very interesting chaotic maps of type Xn+1 =f(Xn) can be constructed when function P(t) is a combination of
trigonometric, elliptic and other generalized periodic functions [6].
IV. STATISTICAL INDEPENDENCE
We will consider statistical independence in the sense of M. Kac [22, 23, 24, 25]. In this framework, two functions
f1(t), f2(t) are independent if the proportion of time during which, simultaneously, f1(t)< α1and f2(t)< α2is equal
to the product of the proportions of time during which separately f1(t)< α1, f2(t)< α2.
First, we will present some results about functions of natural argument n.
Consider a generalization to the functions that are exact solutions to the Chebyshev maps:
Xn= cos(2πθZ n),(11)
where Zis a generic real number.
Any set of subsequences Xs, Xs+1,···, Xs+r(for any r) constitutes a set of statistically independent random
variables.
If E(X) is the expected value of quantity X, let us define the r- order correlations [18]:
E(Xn1Xn2···Xnr) =
1
Z
−1
dX0[ρ(X0)Xn1Xn2···Xnr].(12)
Here E(Xn) = 0, −16Xn61, ρ(X) = 1
π√1−X2,X0= cos(2πθ).
In Ref. [27] it is shown that
E(Xn0
sXn1
s+1 ···Xnr
s+r) = E(Xn0
s)E(Xn1
s+1)···E(Xnr
s+r) (13)
for all positive integers n0, n1, n2,· ·· , nr.
The results about the independence of subsequences of function Xn= cos(2πθZ n) can be extended to more general
functions as the following:
Xn=P(θ T Z n),(14)
where P(t) is a periodic function, Tis the period of P(t) and θis a parameter [27].
M. Kac [22, 23, 24, 25] has studied independence of different continuous functions, e.g. f1(t) = cos(t), f2(t) =
cos(√2t). However, these functions are periodic [22, 23, 24, 25, 26].
We are interested in the independence of continuous functions in the sense that the same function can produce
statistically independent random variables if evaluated at different times. That is, the functions f(t) and f(t+τ)
should be statistically independent if τ6= 0.
Periodic functions will never possess this property.
Using the theorems of Refs. [22, 25], we obtain the result that functions f1= cos(λ1t), f2= cos(λ2t),···, fr=
cos(λrt) constitute a set of independent functions when the numbers λ1, λ2,···, λrare linearly independent over the
rationals. This is equivalent to the fact that if α1, α2,···, αrare rational, α1λ1+α2λ2+···+αrλr= 0 only if
α1, α2,···, αrare all zero.
Consider the following continuous function constructed as a continuous analogous to the function (11):
X(t) = cos(θebt ).(15)
It is not difficult to see that the functions X(t) and Y(t) = X(t+τ) are independent (in M. Kac’s sense) for
(Lebesgue) almost all τ > 0. Note that Y(s) = cos λs, X (s) = cos s, where s=θebt,λ=ebτ .
The proof of the fact that Y(s) = cos(λs) and X(s) = cos sare independent is trivial based on the theorems of M.
Kac[22, 23, 24, 25].

4
V. ALGEBRAIC DIFFERENTIAL EQUATIONS AND CHAOTIC SOLUTIONS
There exists a nontrivial fourth- order differential equations of type:
P(y′, y′′, y′′′, y′′′′) = 0,(16)
where Pis a polynomial in four variables, with integer coefficients, such that there exist solutions y(t) with the
property that y(t) and y(t+τ) are statistically independent functions for most τ.
A particular example of Eq.(16) is the following:
2y′y′′ −3y′y′′′ +y′y′′′′ −y′′y′′′ + (y′′)2= 0.(17)
The function y(t) = cos(et) is a solution to this equation.
Here we should make a remark. The differential equations (2), (3), (4), (5), and (17) are similar in the sense that
they are all of the type (1) or (16), and that all the terms are nonlinear.
We have shown that the equation (17) possesses entire solutions as the following y(t) = cos(et) which has the
property that y(t) and y(t+τ) are not correlated. This solution is real-analytic on (−∞,∞). This result shows that
equations of type (16) can generate high complexity.
VI. UNIVERSAL FUNCTIONS
In this section we will follow the approach to universal equations developed by R. C. Buck [9] and M. Boshernitzan
[13]. In this paper, a family of functions will be called universal if these functions are dense in the space C[I] of
all continuous real functions on the interval I⊂R. R. C. Buck has obtained universal partial algebraic differential
equations using a very deep method: Kolmogorov’s solution of Hilbert’s Thirteen Problem. He has found that solutions
to a smooth PDE can be dense in C[I][9].
The advantage of this approach is that we can construct universal functions which are real- analytic on R= (−∞,∞).
Thus, the corresponding universal equations will have solutions that are real- analytic on R= (−∞,∞).
For any interval I⊂R= (−∞,∞), C[I] denotes the Banach space of real- valued continuous bounded functions on
I.
Boshernitzan [13] has studied the following families of functions:
y(t) =
t+a
Z
0
bd
1 + d2−cos(bs)cos(es)ds +c, (18)
where d > 0, a,b, and care real parameters.
Another important family of functions is defined as follows:
y(t) = b n
t+a
Z
0
cos2n2(bs) cos(es)ds +c, (19)
where a,b,care parameters and n≥1 is an integer constant.
We should stress that all the functions in the family given by Eq.(19) are real-analytic on R= (−∞,∞) and entire.
Boshernitzan has proved that each of the families of functions, (18) and (19), is dense in C[I], for any compact
interval I= [a, b]⊂R.
So these functions are universal. Hence it is possible to construct universal systems using these universal families
of functions.
Other results of Boshernitzan are the following. There exists a nontrivial sixth- order algebraic differential equation
of the form
P(y′, y′′,···, y(6)) = 0,(20)
such that any functions in the family (18) is a solution. And there exists a nontrivial seventh-order algebraic differential
equation of the form
P(y′, y′′,···, y(7)) = 0,(21)

5
0 10 30 50
−2
−1
0
1
2
t
y(t)
FIG. 2: Time- series generated by (19).
such that any function in the family (19) is a solution.
From these theorems one can obtain an important results for us: There exists a nontrivial seventh- order differential
equation, the real- analytic entire solutions of which are dense in C[I], for any compact interval I⊂R. Thus this
equation is universal. Boshernitzan has not constructed this equation explicitly.
The possibility of an entire approximation for any dynamics is very promising for practical applications.
VII. UNIVERSAL SYSTEMS OF DIFFERENTIAL EQUATIONS
Using the inverse problem techniques and theorems of the works[8, 9, 10, 11, 12, 13, 14, 15, 16], and the results
contained in the present paper, it is possible to write down a system of differential equations such that any function
of the family (19) is a solution:
P1[x, x′,···, x′′′] = 0,(22)
P2[y, y′,··· , y′′′] = 0,(23)
z′=A x y. (24)
Eq. (22) is constructed in such a way that all the functions x=a[cos(bx+c)]mare solutions. The specific polynomial
P1is
P1=mx′′′x2−(3m−2)x′′x′x+ (2m−2)x′3,(25)
where m= 2n2>2.
Eq.(23) is constructed in such a way that y(t) = cos(et) is a solution. The specific polynomial P2is defined as
P2=yy′′′ −3yy′′ −y′y′′ +y′2+ 2y′y. (26)
The vector solution (x, y, z ) to the system of equations (22-24) is such that for the variable z(t) the functions (19) are
solutions.
Thus, the system of differential equations (22-24) can generate, in variable z(t), universal functions.
Note that this is a seventh- order dynamical system as was predicted by Boshernitzan[13].
The system of equations (22-24) is presented in this paper for the first time.
VIII. NOISY FUNCTIONS
In this section we use several concepts and results from probability theory and mathematical statistics which can
be consulted for instance in the books[28, 29].
In sections (IV) and (V) we discussed the function y(t) = cos(et). Note that the statistical independence between
functions y1(t) = cos(et) and y2(t) = cos(et+τ) implies the following relationship:
E[yk1
1(t)yk2
2(t)] = E[yk1
1(t)]E[yk2
2(t)],(27)

6
for all positive integers k1and k2.
Here E[x(t)] is the expected value of quantity x(t). It can be calculated as follows:
E[x(t)] = lim
T→∞
1
T
T
Z
0
x(t)dt. (28)
As E[y1(t)] = 0, we obtain that
E[y(t)y(t′)] = 0,(29)
for t6=t′.
In fact, a direct calculation of the autocorrelation function confirms this results:
C(τ) = lim
T→∞
1
T
T
Z
0
cos(et) cos(et+τ)dt = 0 (30)
for τ6= 0.
Another important statistical property of the independent functions y1(t) and y2(t) is
η(y1(t), y2(t)) = η(y1(t))η(y2(t)),(31)
where η(y) is the probability density.
In fact,
η(y1) = 1
πp1−y2
1
,(32)
η(y2) = 1
πp1−y2
2
,(33)
η(y1, y2) = 1
π2p(1 −y2
1)(1 −y2
2).(34)
Fig.(3) shows these properties.
Let us introduce a generalized function
y(t) = cos(φ(t)).(35)
We should remark here that the argument of function (35), φ(t), does not need to be exponential all the time for
t→ ∞, in order to generate noisy dynamics.
In fact, it is sufficient for function φ(t) to be a bounded nonperiodic oscillating function which possesses repeating
intervals with truncated exponential behavior[30].
A deep analysis of the Boshernitzan’s proofs of the fact that the families functions (18) and (19) are dense in C[I]
shows that the random behavior of functions of type (35) is crucial[13].
We are going to present here two examples of these functions. The first is defined by the equation
x(t) = cos {Aexp [a1sin(ω1t+φ1) + a2sin(ω2t+φ2) + a3sin(ω3t+φ3)]}.(36)
The second function is given by the equation
y(t) = cos {B1sinh [a1cos(ω1t+φ1) + a2cos(ω2t+φ2)]
+B2cosh [a3cos(ω3t+φ3) + a4cos(ω4t+φ4)]}.(37)
Note that if the frequencies ωiare linearly independent over the rationals, in both cases, the argument function
φ(t) is a nonperiodic function with truncated exponential behavior.
Theoretical and numerical investigations give the result that, when the parameters satisfy certain conditions, they
behave as the solutions to chaotic systems[20]. Figure (4) shows that the time- series generated by (36) is very complex.

7
−1 0 1
0
1000
y1(t)
η(y1(t))
−1 0 1
0
1500
y2(t)
η(y2(t))
(a)
y2(t)
(b)
FIG. 3: (a) Probability density η(y1) and η(y2) , (b) Probability density η(y1, y2).
FIG. 4: Time- series generated by function (37).
Furthermore, they behave in such a way that y1=y(t) and y2=y(t+τ) are statistically independent functions (in
M. Kac’s sense) for τ6= 0. An illustration of these properties is that
η(y1, y2) = η(y1)η(y2).(38)
Other noisy functions can be obtained using another generalization
x(t) = P[φ(t)],(39)

8
where P(y) is a general periodic function, and φ(t) is a nonperiodic exponential-like function as before.
The probability density of these functions depends on the choice of P(y).
An important example is the following
x(t) = ln tan2[φ(t)].(40)
The probability density of the time- series produced by function (40) is a Gaussian-like law[30].
Another remarkable property of function (40) is the following:
E[x(t)x(t′)] = Dδ(t−t′),(41)
where E[x(t)x(t′)] is defined as in Eq.(27), and δ(t−t′) is Dirac’s delta-function.
Thus, function (40) possesses all the properties usually required in the stochastic equations with δ- correlated noisy
perturbations[31, 32].
IX. PROPERTIES OF CHAOTIC SOLUTIONS
A Lyapunov exponent of a dynamical system characterizes the rate of separation of infinitesimally close trajectories.
Suppose δz0is the initial distance between two trajectories, and δz(t) is the distance between the tra jectories at
time t.
The maximal Lyapunov exponent can be defined as follows:
λ= lim
t→∞
1
tln 



δz(t)
δz0



.(42)
Consider the dynamical system
d~x
dt =~
F(~x).(43)
The variational equation is
d~
φ
dt =Dx~
F(~x)~
φ(t).(44)
The Lyapunov exponent λsatisfies the equation
λ= lim
t→∞
1
tln |φ(t, x0)|.(45)
A solution will be considered chaotic if λ > 0.
A related property of chaotic systems is sensitive dependence on initial conditions[33, 34].
Function
y(t) = cos[exp(t+φ)] (46)
has been shown to be the solution to some dynamical system. Here φcan define the initial condition.
The Lyapunov exponent of solution (46) can be calculated exactly λ= ln e= 1 >0.
The functions (36), (37), and (40) are also chaotic solutions in this sense. The initial conditions are defined by the
parameters φ1,φ2and φ3.
Let us discuss sensitive dependence on initial conditions in the context of these functions.
Let Sbe the set of functions defined by one of the families given by equations (36), (37), (40), and (46).
A set Sexhibits sensitive dependence if there is a δsuch that for any ǫ > 0 and each y1(φ, t) in S, there is a y2(φ′, t),
also in S, such that |y1(φ, 0) −y2(φ′,0)|< ǫ, and |y1(φ, t)−y2(φ′, t)|> δ for some t > 0.
The exponential behavior in the arguments of these functions ((36), (37), (40), and (46)) makes them chaotic (see
a discussion in Ref. [35]).
All these functions possess equivalent dynamical and statistical properties.
Following Boshernitzan theory [13] of modulo 1 sequences, it is possible to construct the following families of
universal functions:
y(t) = bn
t+a
Z
0
cos2n2(bs)x(s)ds +c, (47)
where x(s) is one of the functions ((36), (37),(40)).

9
X. DYNAMICAL SYSTEMS WITH NOISY SOLUTIONS
In this section we will construct autonomous dynamical systems, the solutions of which are the noisy functions
discussed in the previous section.
Consider the following dynamical system
x′
1=x1[1 −(x2
1+y2
1)] −ω1y1,(48)
y′
1=y1[1 −(x2
1+y2
1)] + ω1x1,(49)
x′
2=x2[1 −(x2
2+y2
2)] −ω2y2,(50)
y′
2=y2[1 −(x2
2+y2
2)] + ω2x2,(51)
x′
3=x3[1 −(x2
3+y2
3)] −ω3y3,(52)
y′
3=y3[1 −(x2
3+y2
3)] + ω3x3,(53)
z′= [a1x1+a2x2+a3x3]z, (54)
u′=Acos[θz].(55)
Note that the equations (48-53) define three pairs of limit-cycle two-dimensional dynamical systems. The exact
solutions to these limit-cycle systems are well-known.
If we define
Q(t) = a1x1(t) + a2x2(t) + a3x3(t),(56)
the function Q(t) will be a quasiperiodic time- series.
Equation (54) will provide us with the appropriate nonperiodic truncated exponential behavior. Finally, the solution
to equation (55) will have properties equivalent to these of function (36). The solution to Eq.(55) is chaotic in the sense
discussed in section (IX). The maximal Lyapunov exponent of this dynamical system is positive and the solutions
possess sensitive dependence on initial conditions.
If we need more variability in the solutions, we can construct a dynamical system such that the right-hand parts
of the equations depend on function u(t) which is known to be highly nonperiodic.
These ideas lead to the next autonomous dynamical system:
x′
1=x1[1 −(x2
1+y2
1)] −ω1y1+ε1u, (57)
y′
1=y1[1 −(x2
1+y2
1)] + ω1x1,(58)
x′
2=x2[1 −(x2
2+y2
2)] −ω2y2+ε2u, (59)
y′
2=y2[1 −(x2
2+y2
2)] + ω2x2,(60)
x′
3=x3[1 −(x2
3+y2
3)] −ω3y3+ε3u, (61)
y′
3=y3[1 −(x2
3+y2
3)] + ω3x3,(62)
z′= [a1x1+a2x2+a3x3+a4u]z, (63)
u′=A1cos[θ1z] + A2cos[θ2z].(64)
Now all the components of the solutions to system (57-64) are chaotic.
The solution to Eq.(64) is chaotic in the sense that the maximal Lyapunov exponent is positive (see section (IX)).
In the dynamical system (57-64), the limit cycle subsystems (57,58), (59,60) and (61,62) are now coupled to the
chaotic component u(t).
We now address the function (37). We have to construct a dynamical system with solutions that behave like this
function.

10
0 200 400 600
−1
−0.5
0
0.5
1
1.5
u(t)
t
(a)
0 200
−0.1
−0.05
0
0.05
0.1
0.15
u’(t)
t
(b)
−0.5 0 0.5 1 1.5
−0.1
0
0.1
0.15
u’(t)
u(t)
(c)
FIG. 5: (a)Time- series of variable u(t),(b) u′(t) and, (c) Phase portrait generated by the autonomous dynamical
system (57-64).
Consider the autonomous dynamical system:
x′
1=x1[1 −(x2
1+y2
1)] −ω1y1,(65)
y′
1=y1[1 −(x2
1+y2
1)] + ω1x1,(66)
x′
2=x2[1 −(x2
2+y2
2)] −ω2y2,(67)
y′
2=y2[1 −(x2
2+y2
2)] + ω2x2,(68)
x′
3=x3[1 −(x2
3+y2
3)] −ω3y3,(69)
y′
3=y3[1 −(x2
3+y2
3)] + ω3x3,(70)
x′
4=x4[1 −(x2
4+y2
4)] −ω4y4,(71)
y′
4=y4[1 −(x2
4+y2
4)] + ω4x4,(72)
z′
1=−(a1ω1y1+a2ω2y2)z2,(73)
z′
2=−(a1ω1y1+a2ω2y2)z1,(74)
z′
3=−(a3ω3y3+a4ω4y4)z4,(75)
z′
4=−(a3ω3y3+a4ω4y4)z3,(76)
u′=Acos[B1z1+B4z4].(77)
The explanation of this system is similar to that of equations (48-55).

11
XI. UNIVERSAL FUNCTIONS AND DYNAMICAL SYSTEMS
Based on the properties of functions (15), (18), (19), (36), and (37), we propose the following set of equations as a
universal dynamical system:
x′
1=x1[1 −(x2
1+y2
1)] −ω1y1,(78)
y′
1=y1[1 −(x2
1+y2
1)] + ω1x1,(79)
x′
2=x2[1 −(x2
2+y2
2)] −ω2y2,(80)
y′
2=y2[1 −(x2
2+y2
2)] + ω2x2,(81)
x′
3=x3[1 −(x2
3+y2
3)] −ω3y3,(82)
y′
3=y3[1 −(x2
3+y2
3)] + ω3x3,(83)
x′
4=x4[1 −(x2
4+y2
4)] −ω4y4,(84)
y′
4=y4[1 −(x2
4+y2
4)] + ω4x4,(85)
z′
1=−(a1ω1y1+a2ω2y2)z2,(86)
z′
2=−(a1ω1y1+a2ω2y2)z1,(87)
z′
3=−(a3ω3y3+a4ω4y4)z4,(88)
z′
4=−(a3ω3y3+a4ω4y4)z3,(89)
u′=ω1nx2n2
1cos [B1z1+B4z4+a].(90)
The solution to Eq.(90) is a function with all the properties of Boshernitzan’s family of functions (19). Thus this
family of functions is dense in C[I].
The dynamical system (78-90) has been constructed using the same technique developed in the papers [9, 13] about
universal differential equations. That is, a family of functions known to be dense in C[I] is utilized as the starting
point for an inverse-problem procedure that consists in reconstructing differential equations the solutions of which are
the functions that belong to the universal family of functions.
This dynamical system can be realized in practice using nonlinear circuits as discussed in Ref.[21].
XII. CONCLUSIONS
We have discussed the concept of “Universal functions” and their relevance to the theory of universal differential
equations.
We believe that the method of construction of universal differential equations using universal functions is more
powerful that the method based on splines.
We have found a connection between the universal families of functions proposed in a very important paper by
Boshernitzan [13] and recently obtained exact solutions to chaotic systems.
We have shown that some functions x(t) that are exact solutions to chaotic systems possess the property that
y1=x(t) and y2=x(t+τ) are statistically independent functions in the sense of M. Kac.
We have constructed algebraic differential equations that possess solutions with these properties. These equations
have, in fact, solutions that behave like noise.
Some known universal equations can only approximate these functions using “solutions” constructed with polyno-
mial or trigonometric splines. The actual exact solutions to the differential equations are not “noisy”.
We have constructed a system of differential equations, the solutions of which are Boshernitzan’s functions. Bosher-
nitzan’s functions are real analytic on R= (−∞,∞). One of the families of Boshernitzan’s functions are real-analytic
entire functions on R= (−∞,∞).
We have developed universal- like functions that behave as δ- correlated noise.
We have constructed physically realizable dynamical systems that possess solutions that are universal-like functions.
The theory of universal differential equations has been linked from the beginning with applications in analog
computing[8, 10, 11, 12, 13, 15, 16].
We believe that the present results can be of interest in the construction of real analog computers because the
discussed dynamical systems can be realized in practice using nonlinear circuits[21, 30, 35].

12
They can also find applications in chaos- based secure communications technologies [21, 27].
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