# Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations

Authors: Mervan Pašić; Satoshi Tanaka
Journal: International Journal of Differential Equations
 (2013)

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## Abstract

We derive some simple sufficient conditions on the amplitudea(x), the phase φ(x), and the instantaneous frequency ω(x) such that the so-called chirp function y(x)=a(x)S(φ(x)) is fractal oscillatory near a point x=x0, where φ′(x)=ω(x) and S=S(t) is a periodic function on ℝ. It means that y(x) oscillates near x=x0, and its graph Γ(y) is a fractal curve in ℝ2 such that its box-counting dimension equals a prescribed real number s∈[1,2) and the s-dimensional upper and lower Minkowski contents of Γ(y) are strictly positive and finite. It numerically determines the order of concentration of oscillations of y(x) near x=x0. Next, we give some applications of the main results to the fractal oscillations of solutions of linear differential equations which are generated by the chirp functions taken as the fundamental system of all solutions.


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## Body

## 1. Introduction

The brilliant heuristic approach of Tricot [1] to the fractal curves such as the graph of functions y(x)=xαsinx-β and y(x)=xαcosx-β gave the main motivation for studying the fractal properties near x=0 of graph of oscillatory solutions of various types of differential equations: linear Euler-type equation y′′+λx-σy=0 (see [2]), general second-order linear equation y′′+f(x)y=0 (see [3]) where f(x) satisfies the Hartman-Wintner asymptotic condition near x=0, half-linear equation (|y′|p-2y′)′+f(x)|y|p-2y=0 (see [4]), linear self-adjoint equation (p(x)y′)′+q(x)y=0 (see [5]), and p-Laplace differential equations in an annular domain (see [6]).A functiony(x) is said to be a chirp function if it possesses the form y(x)=a(x)S(φ(x)), where a(x) and φ(x) denote, respectively, the amplitude and phase of y(x), and S(t) is a periodic function on ℝ. In all previously mentioned papers [2–5], authors are dealing with the fractal oscillations of second-order differential equations and are deriving some sufficient conditions on the coefficients of considered equations such that all their solutions y(x) together with the first derivative y′(x) admit asymptotic behaviour near x=0. It is formally written in the form of a chirp function, that is, y(x)=a(x)sin(φ(x)) and y′(x)=b(x)cos(φ(x)) near x=0. According to it, one can say that the asymptotic formula for solutions of considered equations satisfies the chirp-like behaviour near x=0 (on the asymptotic formula for solutions near x=∞, see [7, 8]). Then, in the dependence of a prescribed real number s∈[1,2), authors give some asymptotic conditions on a(x), b(x), and φ(x) such that all solutions y(x) are fractal oscillatory near x=0 with the fractal dimension s.In this paper, independently of the asymptotic theory of differential equations, we firstly study the fractal oscillations of a chirp function; see Theorems8 and 11. Second, taking two linearly independent chirp functions y1(x) and y2(x), we generate some new classes of fractal oscillatory linear differential equations which are not considered in [2–5] and have the general solution in the form of y(x)=c1y1(x)+c2y2(x); see Theorems 16 and 17 (on some detailed description of the solution space of the second-order linear differential equations and on their constructions, we refer the reader to [9, 10]). Finally, we suggest that the reader considers the fractal oscillations near an arbitrary real point x=x0 instead of x=0 and studies the fractal oscillations near x=x0 from the left side and from both sides; see Theorem 24. Many examples are considered to show the originality of obtained results.The chirp functions are also appearing in the time-frequency analysis; see for instance, [11–15] as well as in several applications of the time-frequency analysis; see for instance, [16–20].


## 2. Statement of the Main Results

We study some local asymptotic behaviours of fractal types for the so-called chirp function as follows:(1)y(x)=a(x)S(φ(x)),x∈(0,t0],
where t0>0, the amplitudea=a(x) is a nontrivial function on (0,t0], the phaseφ=φ(x) is singular at x=0 and S=S(t) is a periodic function on ℝ such as cos(t) and sin(t).The chirp function (1) is sometimes written in the following form in which the so-called instantaneous  frequencyω(x) appears instead of phase φ(x):
(2)y(x)=a(x)S(∫xt0ω(ξ)dξ),x∈(0,t0].Let for somet0>0, the functions a∈C((0,t0]), φ∈C2((0,t0]), and S∈C1(ℝ) satisfy the following basic structural conditions:
(3)a(x)>0,a(x)isboundedon(0,t0],(4)limx→+0φ(x)=∞,φ(x)>0,φ′(x)<0,x∈(0,t0],(5)forsomeτ>0,|S(t+τ)|=|S(t)|∀t∈ℝ,forsomeτ0∈ℝ,S(τ0)=0,S(t)≠0∀t∈(τ0,τ0+τ).Definition 1.
A functiony∈C((0,t0]) is oscillatory nearx=0, if there is a decreasing sequence xn∈(0,t0] such that y(xn)=0 for n∈ℕ, and xn↘0 as n→∞; see  Figure 1.
It is easy to prove the next proposition.Figure 1
yis continuous, bounded, and oscillatory near x=0.Proposition 2.
Leta∈C((0,t0]) satisfy a(x)>0 on (0,t0], let φ∈C((0,t0]) be strictly decreasing on (0,t0], and let S∈C(ℝ) satisfy (5). Then, the following two conditions are equivalent:(i)
the chirp functiony(x)=a(x)S(φ(x))is bounded and oscillatory nearx=0;(ii)
the amplitudea(x)is bounded andlimx→+0φ(x)=∞.On the qualitative and oscillatory behaviours of solutions of differential equations of several types, we refer the reader to [7, 8].Example 3.
The following three main types of chirp functions satisfy the conditions (3), (4), and (5): (i)
the so-called(α,β)-chirps
(6)y1(x)=xαcos(x-β),y2(x)=xαsin(x-β),
where α≥0 and β>0; the first studies on the (α,β)-chirps appeared in [1, 13, 14] from different point of views;(ii)
the so-called logarithmic chirps(7)y1(x)=xγcos(ρlogx),y2(x)=xγsin(ρlogx),
where γ>0 and ρ∈ℝ; this type of chirps appears in definition of the Lamperti transform (see [11]) and in the fundamental system of solutions of the famous Euler equation y′′+λx-2y=0 for γ=1/2 and ρ=λ-1/4 (see [21]);(iii)
the chirp function of exponential type(8)y1(x)=xe-1/(2x)cos(e1/x),y2(x)=xe-1/(2x)sin(e1/x).Example 4.
Lety1(x)=a(x)cos(φ(x)) and y2(x)=a(x)sin(φ(x)), x∈(0,t0], where the amplitude a and phase φ satisfy (3) and (4), respectively. Then, y(x)=c1y1(x)+c2y2(x), c12+c22>0, is also a chirp function which is bounded and oscillatory near x=0. Indeed, if c1=0 or c2=0, then y(x)=c2y2(x) or y(x)=c1y1(x); since S(t)=cost and S(t)=sint satisfy (5), both cases y(x) are chirp functions; otherwise, we have y(x)=Aa(x)sin(φ(x)+B), where A=(c12+c22)1/2 and B=arctan(c1/c2); obviously, the amplitude Aa(x) and the phase φ(x)+B satisfy the required conditions (3) and (4), respectively.Next, let(9)Γ(y)={(t1,t2)∈ℝ2:t1∈(0,t0],t2=y(t1)}
denotes the graph of a function y∈C((0,t0]). In this paper, we show that the graph Γ(y) of chirp function (1) is a fractal curve in ℝ2 in respect to the Minkowski-Bouligand dimension dimMΓ(y)=s (box-counting dimension) and the s-dimensional upper and lower Minkowski contents M*s(Γ(y)) and M*s(Γ(y)) defined by
(10)dimMΓ(y)=limɛ→+0(2-log|Γɛ(y)|logɛ),M*s(Γ(y))=limsupɛ→+0(2ɛ)s-2|Γɛ(y)|,M*s(Γ(y))=liminfɛ→+0(2ɛ)s-2|Γɛ(y)|.
Here, Γɛ(y) denotes the ɛ-neighbourhood of graph Γ(y) defined by
(11)Γɛ(y)={(t1,t2)∈ℝ2:d((t1,t2),Γ(y))≤ɛ},ɛ>0,
and d((t1,t2),Γ(y)) denotes the distance from (t1,t2) to Γ(y), and |Γɛ(y)| denotes the Lebesgue measure of Γɛ(y). On the box-counting dimension and the s-dimensional Minkowski content, we refer the reader to [22–27].The main fractal properties considered in the paper are given in the next definitions.Definition 5.
For a given real numbers∈[1,2) and a function y∈C((0,t0]), which is bounded on (0,t0] and oscillatory near x=0, it is said that y(x) is fractal oscillatory nearx=0with the fractal dimensions, if
(12)dimMΓ(y)=s,0<M*s(Γ(y))≤M*s(Γ(y))<∞.
On the contrary, if there is no any s∈[1,2) such that Γ(y) satisfies (12), then y(x)is not fractal oscillatory nearx=0.Fractal oscillations can be understood also as a refinement of rectifiable and nonrectifiable oscillations. They are recently studied in [3, 4, 21, 28–30].Example 6.
The chirp functions(13)y(x)=xcos(1x),y(x)=xsin(1x)
are not fractal oscillatory near x=0. In fact, M*1(Γ(y))=∞ and dimMΓ(y)=1 (see [2]). It also implies that M*s(Γ(y))=0 for all s∈(1,2) (see [1, 23]), and thus the statement (12) is not satisfied for any s∈[1,2). Hence, y1 and y2 are not fractal oscillatory near x=0.Example 7.
Letρ>0. It is clear that the chirp functions
(14)y(x)=xcos(ρlogx),y(x)=xsin(ρlogx)
are oscillatory near x=0. Moreover, the length of Γ(y) is finite (see [21]), and therefore we observe that 0<M*1(Γ(y))≤M*1(Γ(y))<∞ and dimMΓ(y)=1 (see [22]). Thus, such chirp functions are fractal oscillatory near x=0 with the fractal dimension 1.In order to show that the chirp function (1) is fractal oscillatory near x=0, we need to impose on amplitude a(x) the following additional structural condition:
(15)a∈C1((0,t0]),a′(x)≥0forx∈(0,t0].Now we are able to state the first main result of the paper.Theorem 8.
Let the functionsa(x), φ(x), and S(t) satisfy the structural conditions (3), (4), (5), (15), and
(16)φ′′(x)>0on(0,t0],limsupx→0(1-φ′(x))′<∞.
Let the amplitude a(x) and the phase φ(x) satisfy the following asymptotic conditions near x=0:
(17)liminfx→0([-φ′(x)]2-s∫0xa(ξ)dξ)>0,(18)limsupx→0[xa(x)](s-1)/(2-s)∫xt0a(ξ)[-φ′(ξ)]dξ<∞,
where s∈(1,2) is a given real number. Then, the chirp function (1) is fractal oscillatory near x=0 with the fractal dimension s.Remark 9.
Lets∈(1,2). Assume that (15) and (17) hold. Then,
(19)limsupx→0(xa(x)[-φ′(x)]2-s)<∞
implies
(20)limsupx→0[1-φ′(x)]s-1∫xt0a(ξ)[-φ′(ξ)]dξ<∞
and (18). Therefore, condition (18) is better than (19) in Theorem 8.The proof of Remark9 is presented in the Appendix.Theorem8 can be rewritten in the term of ω(x) instead of φ(x) as follows.Corollary 10.
Let the functionsa(x), ω(x), and S(t) satisfy the structural conditions (3), (5), (15), and
(21)ω∈C1((0,t0]),ω(x)>0,ω′(x)<0on(0,t0],ω∉L1((0,t0]),limsupx→0(1ω(x))′<∞.
Let the amplitude a(x) and the instantaneous frequency ω(x) satisfy the following asymptotic conditions near x=0:
(22)liminfx→0([ω(x)]2-s∫0xa(ξ)dξ)>0,limsupx→0[xa(x)](s-1)/(2-s)∫xt0a(ξ)ω(ξ)dξ<∞,
where s∈(1,2) is a given real number. Then, the chirp function (2) is fractal oscillatory near x=0 with the fractal dimension s.Now, we state analogous result to Theorem8 in the case of s=1, which will be proved in Section 3.Theorem 11.
Assume that the functionS(t) satisfies (5), a∈C([0,t0])∩C1((0,t0]), φ∈C2((0,t0]) and limx→+0φ(x)=∞. If a′∈L1((0,t0]), and φ′a∈L1((0,t0]), then the chirp function (1) is fractal oscillatory near x=0 with the fractal dimension 1.In respect to some existing results on the fractal dimension of graph of chirp functions, previous theorems are the most simple and general. It is because in [31, 32] authors require some extra conditions on the chirp function y(x) which are not easy to be satisfied in the application, for instance, the rapid convex-concave properties of y(x) as in [31] and a condition on the curvature of y(x) as in [32].According to previous theorems, we can show the fractal oscillations of the so-called(α,β)-chirp as well as logarithmic chirp functions.Example 12.
We consider the(α,β)-chirp y(x)=xαS(x-β), where S(t)=cost or S(t)=sint and β>α≥0. It is fractal oscillatory near x=0 with the fractal dimension s=2-(1+α)/(1+β). In fact, it is easy to see that a(x)=xα, S(t), and φ(x)=x-β satisfy (3), (4), (5), (15), and (16). When s=2-(1+α)/(1+β), we see that
(23)[-φ′(x)]2-s∫0xa(ξ)dξ=β(1+α)/(1+β)1+α>0,[xa(x)](s-1)/(2-s)∫xt0a(ξ)[-φ′(ξ)]dξ=ββ-α(1-xβ-αt0α-β)≤ββ-α.
From Theorem 8, it follows that y(x)=xαS(x-β), β>α≥0, is fractal oscillatory near x=0 with the fractal dimension s=2-(1+α)/(1+β).Example 13.
We consider the(α,β)-chirp y(x)=xαS(x-β) again, where S(t)=cost or S(t)=sint. Now, we assume that 0<β<α. Applying Theorem 8, we easily see that it is fractal oscillatory near x=0 with the fractal dimension 1.Example 14.
We consider the logarithmic chirp functions(24)y1(x)=xγcos(ρlogx),y2(x)=xγsin(ρlogx),
where γ>0 and ρ>0. Put S(t)=cos(-t) or S(t)=sin(-t), a(x)=xγ, and φ(x)=-ρlogx. Then, we easily see that S(t) satisfies (5) and that a′(x)=γxγ-1∈L1((0,t0]) and a(x)φ′(x)=-ρxγ-1∈L1((0,t0]). Theorem 11 implies that y1(x) and y2(x) are fractal oscillatory near x=0 with the fractal dimension 1.Question 1.
Is it possible to apply Theorem8 on the exponential chirp given in Example 3(iii)?Example 15.
Lety1(x)=a(x)cos(φ(x)) and y2(x)=a(x)sin(φ(x)), x∈(0,t0], where the amplitude a(x) and the phase φ(x) satisfy all assumptions of Theorem 8 for an arbitrary given s∈(1,2). Then, the chirp function y(x)=c1y1(x)+c2y2(x), c12+c22>0, is also fractal oscillatory near x=0 with the fractal dimension s. Indeed, similarly as in Example 4, y(x) can be rewritten in the form y(x)=d1sin(φ(x)+d2), where d1≠0 and d2∈ℝ. It is clear that the function S(t)=d1sin(t+d2) satisfies the required condition (5) and hence Theorem 8 proves that y(x) is fractal oscillatory near x=0 with the fractal dimension s.Next, we pay attention to the fractal oscillations of solutions of linear differential equations generated by the system of functions as follows:(25)y(x)=c1y1(x)+c2y2(x),c12+c22>0,wherey1(x)=a(x)cos(φ(x)),y2(x)=a(x)sin(φ(x)),x∈(0,t0].
It is not difficult to check that (25) is the fundamental system of all solutions of the following linear differential equation:
(26)y′′+(-2Ra-Rφ′)y′+(-Ra′+Ra2+RaRφ′+(φ′)2)y=0,x∈(0,t0],
where Rf=Rf(x)=f′(x)/f(x) for some C1-function f=f(x).Theorem 16.
Let the functionsa, φ∈C2((0,t0]) satisfy structural conditions (3), (4), and (15) as well as the conditions (16), (17), and (18) in respect to a given real number s∈(1,2). Then, every nontrivial solution y∈C2((0,t0]) of (26) is fractal oscillatory near x=0 with the fractal dimension s.With the help of Theorem11, we can state analogous result to Theorem 16 in the case of s=1.Theorem 17.
Assume thata∈C([0,t0])∩C2((0,t0]), a(x)≠0 for x∈(0,t0], φ∈C2((0,t0]), limx→+0φ(x)=∞, a′∈L1((0,t0]), and φ′a∈L1((0,t0]). Then, every nontrivial solution y∈C2((0,t0]) of (26) is fractal oscillatory near x=0 with the fractal dimension 1.The previous two theorems will be proved in Section4. Assumptions on the coefficients of (26) in general are different to those considered in [2–5].Whena′/a=-(1/2)φ′′/φ′, then (26) becomes the undamped equation
(27)y′′+(12Sder(φ′)+(φ′)2)y=0,x∈(0,t0],
where Sder(f) denotes the Schwarzian derivative of f defined by
(28)Sder(f)=f′′f-32(f′f)2.
Hence, from Theorem 16, we obtain the following consequence.Corollary 18.
Let the functionsa∈C2((0,t0]) and φ∈C3((0,t0]) satisfy structural conditions (3), (4), and (15) as well as the conditions (16), (17), and (18) in respect to a given real number s∈(1,2). Let a′/a=-(1/2)φ′′/φ′. Then, every nontrivial solution y∈C2((0,t0]) of (27) is fractal oscillatory near x=0 with the fractal dimension s.As a consequence of Theorem16, and Corollary 18 we derive the following examples for linear differential equations of second order having all the solutions to be fractal oscillatory near x=0.Example 19.
The so-called damped chirp equation(29)y′′+β-2α+1xy′+(β2x2β+2-α(β-α)x2)y=0,x∈(0,t0],
is fractal oscillatory near x=0 with the fractal dimension 2-(1+α)/(1+β), where β>α≥0. When a(x)=xα and φ(x)=x-β, (26) becomes (29). It is easy to see that (3), (4), (15), and (16) are satisfied. In the same as in Example 12, we see that (17) and (18) hold for s=2-(1+α)/(1+β). Hence, Theorem 16 proves that every nontrivial solution of (29) is fractal oscillatory near x=0 with the fractal dimension 2-(1+α)/(1+β).
Now we assume that0<β<α. Then, Theorem 17 implies that (29) is fractal oscillatory near x=0 with the fractal dimension 1.Example 20.
The following equation(30)y′′+1-2γxy′+γ2+ρ2x2y=0,x∈(0,t0],
is fractal oscillatory near x=0 with the fractal dimension 1, where γ>0 and ρ>0. In the case where a(x)=xγ and φ(x)=-ρlogx, (26) becomes (30). We see that a∈C([0,t0])∩C2((0,t0]), φ∈C2((0,t0]), limx→+0φ(x)=∞, a′∈L1((0,t0]), and φ′a∈L1((0,t0]). Therefore Theorem 17 implies that every nontrivial solution of (30) is fractal oscillatory near x=0 with the fractal dimension 1.Question 2.
What can we say about the application of Theorem16 on the case of φ(x) given in Example 3(iii)?At the end of this section, we suggest that the reader studies some invariant properties of fractal oscillations of the chirp function (1) in respect to the translation and reflexion. Analogously to Definitions 1 and 5, one can define the fractal oscillations near an arbitrary real point x=x0 as follows.Definition 21.
Letx0∈ℝ and δ>0. A function y∈C((x0,x0+δ]) is oscillatory nearx=x0, if there is a decreasing sequence bn∈(x0,x0+δ] such that y(bn)=0, for n∈ℕ, and bn↘x0 as n→∞. Moreover, if the graph Γ(y) satisfies the condition (12) for some s∈[1,2), then y(x) is said to be fractal oscillatory nearx=x0with the fractal dimensions.Definition 22.
Letx0∈ℝ and δ>0. It is said that a function y∈C([x0-δ,x0)) is fractal oscillatory nearx=0from the left side with the fractal dimensions∈[1,2), if there is an increasing sequence an∈[x0-δ,x0) such that y(an)=0 for n∈ℕ, an↗x0 as n→∞, and the graph Γ(y) satisfies the condition (12), see Figure 2.Figure 2
y is oscillatory near x=0 from the left side.Definition 23.
Letx0∈ℝ, and let δ>0. It is said that a function y∈C([x0-δ,x0)∪(x0,x0+δ]) is two-sided fractal oscillatory nearx=0with the fractal dimensions∈[1,2), if there is an increasing sequence an∈[x0-δ,x0) and a decreasing sequence bn∈(x0,x0+δ] such that y(an)=y(bn)=0 for n∈ℕ, an↗x0 and bn↘x0 as n→∞, and the graph Γ(y) satisfies the condition (12); see Figure 3.Figure 3
y is two-sided oscillatory near x=0.With the help of Theorem8, we state the following result.Theorem 24.
Letx0∈ℝ, and let δ>0. Let the functions a,ω∈C1((0,δ]) satisfy structural conditions (3), (4), (5), (15), and (16) as well as conditions (17) and (18) in respect to a given real number s∈(1,2). Then, one has(i)
the chirp functiony(x)=a(x-x0)S(φ(x-x0)), x∈(x0,x0+δ] is fractal oscillatory near x=x0 with the fractal dimension s;(ii)
the chirp functiony(x)=a(x0-x)S(φ(x0-x)), x∈[x0-δ,x0) is fractal oscillatory near x=x0 from the left side with the fractal dimension s;(iii)
the chirp functiony(x)=a(|x-x0|)S(φ(|x-x0|)), x∈[x0-δ,x0)∪(x0,x0+δ] is two-sided fractal oscillatory near x=x0 with the fractal dimension s.


## 3. Proof for the Fractal Oscillations of Chirp Functions

In this section, we give the proofs of the main results dealing with the fractal oscillations of chirp functions.By Definition5, it follows that if for a prescribed real number s∈(1,2) there are two positive constants c1 and c2 such that
(31)c1ɛ2-s≤|Γɛ(y)|≤c2ɛ2-s,ɛ∈(0,ɛ0),
for some ɛ0>0, then a function y∈C((0,t0]) is fractal oscillatory near x=0 with the fractal dimension s. The following lemma plays the essential role in the proof of (31).Lemma 25.
Lety∈C((0,t0]) be a bounded function on (0,t0], and let an∈(0,t0] be a decreasing sequence of consecutive zeros of y(x) such that an→0. Let ɛ0>0, and let k=k(ɛ) be an index function satisfying
(32)|an-an+1|≤ɛ∀n≥k(ɛ),ɛ∈(0,ɛ0).
Then
(33)|Γɛ(y)|≥∑n≥k(ɛ)maxx∈[an+1,an]|y(x)|(an-an+1)∀ɛ∈(0,ɛ0).Proof.
It is exactly the same as [3, pp. 2350].Let us remark that for someɛ0>0, we say that a function k=k(ɛ) is an index function on (0,ɛ0) if  k:(0,ɛ0)→ℕ  and limɛ→0k(ɛ)=∞.Lemma 26.
Lety∈C[a,b]. Then,
(34)|Γɛ(y)|≤4πɛlength(y)+4πɛ2,foreveryɛ>0.Proof.
Letɛ>0. Set x1=a, and set
(35)xi+1=max{x∈[xi,b]:d((t,y(t)),(xi,y(xi)))≤ɛfort∈[xi,x]((t,y(t)),(xi,y(xi)))},
for i=1,2,…. Then, there exists n≥2 such that xn=b. Set N=max{n∈N:xn<b}. We see that N≥1,
(36)a=x1<x2<⋯<xi<xi+1<⋯<xN<xN+1=b,
and if N≥2, then
(37)d((xi,y(xi)),(xi+1,y(xi+1)))=ɛ,i=1,2,…,N-1.
We will show that
(38)Γɛ(y)⊂⋃i=1NB2ɛ(xi,y(xi)),
where
(39)B2ɛ(t1,t2)={(τ1,τ2)∈R2:d((t1,t2),(τ1,τ2))≤2ɛ}.
Let (τ1,τ2)∈Γɛ(y). Then there exists ξ∈[a,b] such that d((τ1,τ2),(ξ,y(ξ)))≤ɛ. Because of the definition of xi+1, we find that ξ∈[xi,xi+1] for some i∈{1,2,…,N}, so that
(40)d((ξ,y(ξ)),(xi,y(xi)))≤ɛ.
Hence, it follows that
(41)d((τ1,τ2),(xi,y(xi)))≤d((τ1,τ2),(ξ,y(ξ)))+d((ξ,y(ξ)),(xi,y(xi)))≤2ɛ,
which means that (τ1,τ2)∈B2ɛ(xi,y(xi)). Therefore, we obtain (38). By (38), we conclude that
(42)|Γɛ(y)|≤∑i=1N|B2ɛ(xi,y(xi))|=4Nπɛ2.
When N=1, from (42), it follows that
(43)|Γɛ(y)|≤4πɛ2≤4πɛlength(y)+4πɛ2.
Now, we assume that N≥2. We observe that
(44)length(y)=∑i=1Nlength(y|[xi,xi+1])≥∑i=1Nd((xi,y(xi)),(xi+1,y(xi+1)))≥∑i=1N-1d((xi,y(xi)),(xi+1,y(xi+1)))=(N-1)ɛ,
that is,
(45)Nɛ≤length(y)+ɛ.
Combining (42) with (45), we obtain
(46)|Γɛ(y)|≤4πɛlength(y)+4πɛ2.Lemma 27.
Lety∈C(0,t0] be bounded on (0,t0], and let ɛ>0. Then, there exists C>0 such that, for every ɛ>0 and t∈(0,t0],
(47)|Γɛ(y)|≤C[tsupx∈(0,t]|y(x)|+ɛlength(y|[t,t0])+ɛ+ɛ2].Proof.
Letɛ>0, and let t∈(0,t0]. From the geometry point of view, it is clear that
(48)|Γɛ(y|(0,t])|⊂R,
where R is a rectangle in R2 defined by
(49)R={[-ɛ-supx∈(0,t]|y(x)|,ɛ+supx∈(0,t]|y(x)|](t1,t2):t1∈[-ɛ,t+ɛ],t2∈[-ɛ-supx∈(0,t]|y(x)|,ɛ+supx∈(0,t]|y(x)|]}.
Hence, we have
(50)|Γɛ(y(0,t])|≤(2ɛ+t)(2ɛ+2supx∈(0,t]|y(x)|)=4ɛ2+2(t+2supx∈(0,t]|y(x)|)ɛ+2tsupx∈(0,t]|y(x)|≤4ɛ2+C1ɛ+2tsupx∈(0,t]|y(x)|,
where C1=2(t0+2supx∈(0,t0]|y(x)|). Lemma 26 implies that
(51)|Γɛ(y|[t,t0])|≤4πɛlength(y|[t,t0])+4πɛ2.
Consequently, we see that
(52)|Γɛ(y)|≤|Γɛ(y|(0,t])|+|Γɛ(y|[t,t0])|≤4ɛ2+C1ɛ+2tsupx∈(0,t]|y(x)|+4πɛlength(y|[t,t0])+4πɛ2≤C[tsupx∈(0,t]|y(x)|+ɛlength(y|[t,t0])+ɛ+ɛ2],
where C=max{C1,4(π+1)}.Lemma 28.
Lety∈C1(0,t0] be bounded on (0,t0]. Assume that
(53)∫0t0|y′(x)|dx=∞,(54)limsupx→0[xsupξ∈(0,x]|y(ξ)|](s-1)/(2-s)∫xt0|y′(ξ)|dξ<∞,
for some s∈(1,2). Then, there exists c2>0 such that
(55)|Γɛ(y)|≤c2ɛ2-s,ɛ∈(0,1).Proof.
Letɛ∈(0,1) and let s∈(1,2) be satisfy (54). By (53), there exists x1∈(0,t0), which depends on ɛ and s, such that
(56)∫x1t0|y′(t)|dt=ɛ1-s.
Therefore, we have
(57)length(y|[x1,t0])=∫x1t01+|y′(x)|2dx≤∫x1t0(1+|y′(x)|)dx=t0-x1+ɛ1-s≤t0+ɛ1-s.
By (54), there exists C1>0 such that
(58)[xsupξ∈(0,x]|y(ξ)|](s-1)/(2-s)∫xt0|y′(ξ)|dξ≤C1,x∈(0,t0],
which implies that
(59)[x1supξ∈(0,x1]|y(ξ)|](s-1)/(2-s)∫x1t0|y′(ξ)|dξ≤C1,
that is,
(60)x1supξ∈(0,x1]|y(ξ)|≤C2ɛ2-s,
where C2=C1(2-s)/(s-1). From Lemma 27, (57), and (60), it follows that
(61)|Γɛ(y)|≤C[C2ɛ2-s+ɛ(t0+ɛ1-s)+ɛ+ɛ2]=C[(C2+1)+(t0+1)ɛs-1+ɛs]ɛ2-s≤C(C2+t0+3)ɛ2-s,ɛ∈(0,1).In order to show Theorems8 and 11, we need the following two geometric lemmas.Lemma 29 (see [1]).
IfΓ⊆ℝ2 is a simple curve (i.e., its parameterization is a bijection) and  length(Γ)<∞, then
(62)length(Γ)=limɛ→0|Γɛ|2ɛ,
where Γɛ denotes the ɛ-neighborhood of the graph Γ.Now, we are able to prove Theorem8.Proof of Theorem8.
Lets∈(1,2), and let y(x) be a chirp function given by (1). We note here that it is enough to show that y(x) satisfies (31).
At the first, letxn be a sequence defined by xn=φ-1(τ0+nτ) for all sufficiently large n∈ℕ. From (4), it follows that φ-1(t) is decreasing. Hence, xn is decreasing as well as xn→0 as n→∞ because of limx→+0φ(x)=∞ (see (4)). We note that y(xn)=0 and y(x)≠0 on (xn+1,xn) for all sufficiently large n∈ℕ. Also, -1/φ′(x) is an increasing function because of (16). The mean value theorem shows that
(63)τ-φ′(xn+1)≤xn-xn+1≤τ-φ′(xn).
Now, let ɛ0∈(0,1). Let k(ɛ) be the smallest natural number satisfying
(64)τ-φ′(xk(ɛ))≤ɛ∀ɛ∈(0,ɛ0).
Such k(ɛ) exists for every ɛ∈(0,ɛ0), since xn→0 as n→∞ and limx→+0φ′(x)=-∞ (this equality is true because φ′∉L1(0,t0) since limx→+0φ(x)=∞). Moreover, since xn is decreasing and -1/φ′(x) is increasing, we obtain
(65)-φ′(xn)≥τɛ-1∀n≥k(ɛ).
Combining (63) and (65), it is easy to deduce that such defined k(ɛ) satisfies condition (32).
By (16), there exists L>0 such that (1/(-φ′(x)))′≤L for x∈(0,t0], which means that
(66)--φ′′(x)-φ′(x)≤-Lφ′(x),x∈(0,t0].
Integrating (66) on [φ-1(t+2τ),φ-1(t)], we have
(67)log-φ′(φ-1(t+2τ))-φ′(φ-1(t))≤2Lτ,t∈[φ(t0),∞),
that is,
(68)-φ′(φ-1(t))≥-e-2Lτφ′(φ-1(t+τ)),t∈[φ(t0),∞).
By the definition of k(ɛ) and (64), we see that
(69)τ-φ′(xk(ɛ)-1)>ɛ.
Hence, from (68), it follows that
(70)τɛ-1>-φ′(xk(ɛ)-1)=-φ′(φ-1(τ0+(k(ɛ)-1)τ))≥-e-2Lτφ′(φ-1(τ0+(k(ɛ)-1)τ+2τ))=-e-2Lτφ′(φ-1(τ0+(k(ɛ)+1)τ))=-e-2Lτφ′(xk(ɛ)+1),
which implies that
(71)-φ′(xk(ɛ)+1)≤Mɛ-1,ɛ>0,
where M=τe2Lτ.
Next, it is clear that the assumption (5) ensures a real number M0>0 such that
(72)M0=maxt∈ℝ|S(t)|=maxt∈[0,τ]|S(t)|.
Hence, from (15), we have
(73)maxx∈[xn+1,xn]|y(x)|=maxx∈[xn+1,xn](a(x)|S(φ(x))|)≥a(xn+1)maxx∈[xn+1,xn]|S(φ(x))|=M0a(xn+1).
Now, from (63) and (73), it follows that
(74)maxx∈[xn+1,xn]|y(x)|(xn-xn+1)≥M0a(xn+1)(xn-xn+1)≥τM0a(xn+1)-φ′(xn+1)=τM0a(φ-1(τ0+(n+1)τ))-φ′(φ-1(τ0+(n+1)τ)).
Since φ-1(t) is decreasing as well as -1/φ′(x) is increasing and a(x) is nondecreasing and positive, we conclude that the function
(75)a(φ-1(t))-φ′(φ-1(t))isnonincreasing.
Hence, we observe that
(76)∑n≥k(ɛ)maxx∈[xn+1,xn]|y(x)|(xn-xn+1)≥τM0∑n≥k(ɛ)a(φ-1(τ0+(n+1)τ))-φ′(φ-1(τ0+(n+1)τ))≥τM0∫k(ɛ)∞a(φ-1(τ0+(t+1)τ))-φ′(φ-1(τ0+(t+1)τ))dt=M0∫0φ-1(τ0+(k(ɛ)+1)τ)a(ξ)dξ=M0∫0xk(ɛ)+1a(ξ)dξ.
Next, from assumption (17), we get some M1>0 such that
(77)∫0xa(ξ)dξ≥M1[-φ′(x)]-(2-s),x∈(0,t0].
Now, from (76), (77), and (71), we obtain
(78)∑n≥k(ɛ)maxx∈[xn+1,xn]|y(x)|(xn-xn+1)≥M0M1[-φ′(xk(ɛ)+1)]-(2-s)≥M0M1(Mɛ-1)-(2-s)=M0M1M-(2-s)ɛ2-s.
Thus, using Lemma 25, we have
(79)|Γɛ(y)|≥c1ɛ2-s,ɛ∈(0,ɛ0),
for some c1>0.
Next we prove thaty(x) satisfies (53). Assume to the contrary that
(80)∫0t0|y′(x)|dx<∞.
Then, we have
(81)length(y)=∫0t01+|y′(x)|2dx≤∫0t0(1+|y′(x)|)dx<∞.
From Lemma 29, it follows that
(82)length(y)=limɛ→+0|Γɛ(y)|2ɛ.
On the other hand, by (79), we see that
(83)limɛ→+0|Γɛ(y)|2ɛ=∞.
This is a contaradiction, and hence y(x) satisfies (53).
Finally, we show thaty(x) also satisfies inequality (54). Because of (5), we obtain a constant c>0 such that
(84)maxt∈ℝ|S(t)|=maxt∈[0,τ]|S(t)|≤c,maxt∈ℝ|S′(t)|=maxt∈[0,τ]|S′(t)|≤c.
By (15), we find that
(85)supt∈(0,x]|y(t)|=supt∈(0,x]|a(t)||S(φ(t))|≤ca(x).
Hence, from (1) and (15), it follows that
(86)|y′(x)|≤c(a′(x)+a(x)[-φ′(x)]),
and therefore
(87)∫xt0|y′(ξ)|dξ≤ca(t0)+c∫xt0a(ξ)[-φ′(ξ)]dξ.
Using (85) and (87), we see that
(88)[xsupξ∈(0,x]|y(ξ)|](s-1)/(2-s)∫xt0|y′(ξ)|dξ≤ca(t0)[cxa(x)](s-1)/(2-s)+c[cxa(x)](s-1)/(2-s)∫xt0a(ξ)[-φ′(ξ)]dξ≤ca(t0)[ct0a(t0)](s-1)/(2-s)+c1/(2-s)[xa(x)](s-1)/(2-s)∫xt0a(ξ)[-φ′(ξ)]dξ.
By (18) we conclude that (54) holds.
By Lemma28, there exists c2>0 such that
(89)|Γɛ(y)|≤c2ɛ2-s,ɛ∈(0,ɛ0).
Thus, we have proved that the chirp function y(x) given by (1) satisfies the desired inequality (31). This completes the proof of Theorem 8.Proof of Theorem11.
Lety(x) be the chirp function (1). It is easy to see that y(x) is oscillatory near x=0. By (5), there exists M>0 such that
(90)|S(t)|≤M,|S′(t)|≤M,t∈ℝ.
Hence, we observe that
(91)|y′(x)|≤|a′(x)S(φ(x))|+|a(x)S(φ(x))φ′(x)|≤M|a′(x)|+M|a(x)φ′(x)|,
for x∈(0,t0]. From (81), it follows that length(y)<∞. Lemma 29 shows that
(92)length(y)=limɛ→0|Γɛ(y)|2ɛ,
which implies that ɛlength(Γ)≤|Γɛ(y)|≤4ɛlength(Γ), ɛ∈(0,ɛ0), for some ɛ0>0. Therefore, y(x) is fractal oscillatory near x=0 with the fractal dimension 1.


## 4. Proof for the Fractal Oscillations of (26)

In this section, we give the proofs for the fractal oscillations of the linear second-order differential equation (26) considered as an application of the main results on the fractal oscillation of chirp functions.Before we present the proofs of Theorems16 and 17, we make the following observation. Since
(93)y1(x)=a(x)cos(φ(x)),y2(x)=a(x)sin(φ(x))
are solutions of (26), we see that y(x)=c1y1(x)+c2y2(x) is a fundamental system of all solutions of (26). Assume that c12+c22>0, and set S(t)=c1sint+c2cost. Then, S(t) clearly satisfies (5).Proof of Theorem16.
Applying Theorem8 on y(x)=c1y1(x)+c2y2(x), we conclude that y(x) is fractal oscillatory with the fractal dimension s.Proof of Theorem17.
Theorem11 implies that y(x)=c1y1(x)+c2y2(x) is fractal oscillatory near x=0 with the fractal dimension 1.



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*Source: 857410-2013-02-28.xml*