#### Abstract

We derive some simple sufficient conditions on the amplitude , the phase and the instantaneous frequency such that the so-called chirp function is fractal oscillatory near a point , where and is a periodic function on . It means that oscillates near , and its graph is a fractal curve in such that its box-counting dimension equals a prescribed real number and the -dimensional upper and lower Minkowski contents of are strictly positive and finite. It numerically determines the order of concentration of oscillations of near . 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.

#### 1. Introduction

The brilliant heuristic approach of Tricot [1] to the fractal curves such as the graph of functions and gave the main motivation for studying the fractal properties near of graph of oscillatory solutions of various types of differential equations: linear Euler-type equation (see [2]), general second-order linear equation (see [3]) where satisfies the Hartman-Wintner asymptotic condition near , half-linear equation (see [4]), linear self-adjoint equation (see [5]), and -Laplace differential equations in an annular domain (see [6]).

A function is said to be a chirp function if it possesses the form , where and denote, respectively, the amplitude and phase of , and is a periodic function on . In all previously mentioned papers [2â€“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 together with the first derivative admit asymptotic behaviour near . It is formally written in the form of a chirp function, that is, and near . According to it, one can say that the asymptotic formula for solutions of considered equations satisfies the chirp-like behaviour near (on the asymptotic formula for solutions near , see [7, 8]). Then, in the dependence of a prescribed real number , authors give some asymptotic conditions on , , and such that all solutions are fractal oscillatory near with the fractal dimension .

In this paper, independently of the asymptotic theory of differential equations, we firstly study the fractal oscillations of a chirp function; see Theorems 8 and 11. Second, taking two linearly independent chirp functions and , we generate some new classes of fractal oscillatory linear differential equations which are not considered in [2â€“5] and have the general solution in the form of ; 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 instead of and studies the fractal oscillations near 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, [11â€“15] as well as in several applications of the time-frequency analysis; see for instance, [16â€“20].

#### 2. Statement of the Main Results

We study some local asymptotic behaviours of fractal types for the so-called chirp function as follows:
where , the * amplitude * is a nontrivial function on , the * phase * is singular at and is a periodic function on such as and .

The chirp function (1) is sometimes written in the following form in which the so-called * instantaneous frequency * appears instead of phase :

Let for some , the functions , , and satisfy the following basic structural conditions:

*Definition 1. * A function is * oscillatory near *, if there is a decreasing sequence such that for , and as ; see Figure 1.

It is easy to prove the next proposition.

Proposition 2. * Let satisfy on , let be strictly decreasing on , and let satisfy (5). Then, the following two conditions are equivalent:*(i)* the chirp function ** is bounded and oscillatory near **;*(ii)* the amplitude ** is bounded and **.*

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 -chirps
where and ; the first studies on the -chirps appeared in [1, 13, 14] from different point of views; (ii)the so-called logarithmic chirps
where and ; 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 for and (see [21]); (iii)the chirp function of exponential type

*Example 4. * Let and , , where the amplitude and phase satisfy (3) and (4), respectively. Then, , , is also a chirp function which is bounded and oscillatory near . Indeed, if or , then or ; since and satisfy (5), both cases are chirp functions; otherwise, we have , where and ; obviously, the amplitude and the phase satisfy the required conditions (3) and (4), respectively.

Next, let denotes the graph of a function . In this paper, we show that the graph of chirp function (1) is a fractal curve in in respect to the Minkowski-Bouligand dimension (box-counting dimension) and the -dimensional upper and lower Minkowski contents and defined by Here, denotes the -neighbourhood of graph defined by and denotes the distance from to , and denotes the Lebesgue measure of . On the box-counting dimension and the -dimensional Minkowski content, we refer the reader to [22â€“27].

The main fractal properties considered in the paper are given in the next definitions.

*Definition 5. * For a given real number and a function , which is bounded on and oscillatory near , it is said that is * fractal oscillatory near ** with the fractal dimension *, if
On the contrary, if there is no any such that satisfies (12), then * is not fractal oscillatory near *.

Fractal oscillations can be understood also as a refinement of rectifiable and nonrectifiable oscillations. They are recently studied in [3, 4, 21, 28â€“30].

*Example 6. * The chirp functions
are not fractal oscillatory near . In fact, and (see [2]). It also implies that for all (see [1, 23]), and thus the statement (12) is not satisfied for any . Hence, and are not fractal oscillatory near .

*Example 7. * Let . It is clear that the chirp functions
are oscillatory near . Moreover, the length of is finite (see [21]), and therefore we observe that and (see [22]). Thus, such chirp functions are fractal oscillatory near with the fractal dimension .

In order to show that the chirp function (1) is fractal oscillatory near , we need to impose on amplitude the following additional structural condition:

Now we are able to state the first main result of the paper.

Theorem 8. *Let the functions , , and satisfy the structural conditions (3), (4), (5), (15), and
**
Let the amplitude and the phase satisfy the following asymptotic conditions near :
**
where is a given real number. Then, the chirp function (1) is fractal oscillatory near with the fractal dimension .*

*Remark 9. * Let . Assume that (15) and (17) hold. Then,
implies
and (18). Therefore, condition (18) is better than (19) in Theorem 8.

The proof of Remark 9 is presented in the Appendix.

Theorem 8 can be rewritten in the term of instead of as follows.

Corollary 10. *Let the functions , , and satisfy the structural conditions (3), (5), (15), and
**
Let the amplitude and the instantaneous frequency satisfy the following asymptotic conditions near :
**
where is a given real number. Then, the chirp function (2) is fractal oscillatory near with the fractal dimension .*

Now, we state analogous result to Theorem 8 in the case of , which will be proved in Section 3.

Theorem 11. *Assume that the function satisfies (5), , and . If , and , then the chirp function (1) is fractal oscillatory near with the fractal dimension .*

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 which are not easy to be satisfied in the application, for instance, the rapid convex-concave properties of as in [31] and a condition on the curvature of as in [32].

According to previous theorems, we can show the fractal oscillations of the so-called -chirp as well as logarithmic chirp functions.

*Example 12. * We consider the -chirp , where or and . It is fractal oscillatory near with the fractal dimension . In fact, it is easy to see that , , and satisfy (3), (4), (5), (15), and (16). When , we see that
From Theorem 8, it follows that , , is fractal oscillatory near with the fractal dimension .

*Example 13. * We consider the -chirp again, where or . Now, we assume that . Applying Theorem 8, we easily see that it is fractal oscillatory near with the fractal dimension .

*Example 14. * We consider the logarithmic chirp functions
where and . Put or , , and . Then, we easily see that satisfies (5) and that and . Theorem 11 implies that and are fractal oscillatory near with the fractal dimension .

*Question 1. *Is it possible to apply Theorem 8 on the exponential chirp given in Example 3(iii)?

*Example 15. * Let and , , where the amplitude and the phase satisfy all assumptions of Theorem 8 for an arbitrary given . Then, the chirp function , , is also fractal oscillatory near with the fractal dimension . Indeed, similarly as in Example 4, can be rewritten in the form , where and . It is clear that the function satisfies the required condition (5) and hence Theorem 8 proves that is fractal oscillatory near with the fractal dimension .

Next, we pay attention to the fractal oscillations of solutions of linear differential equations generated by the system of functions as follows: It is not difficult to check that (25) is the fundamental system of all solutions of the following linear differential equation: where for some -function .

Theorem 16. *Let the functions , satisfy structural conditions (3), (4), and (15) as well as the conditions (16), (17), and (18) in respect to a given real number . Then, every nontrivial solution of (26) is fractal oscillatory near with the fractal dimension .*

With the help of Theorem 11, we can state analogous result to Theorem 16 in the case of .

Theorem 17. *Assume that , for , , , , and . Then, every nontrivial solution of (26) is fractal oscillatory near with the fractal dimension .*

The previous two theorems will be proved in Section 4. Assumptions on the coefficients of (26) in general are different to those considered in [2â€“5].

When , then (26) becomes the undamped equation where denotes the Schwarzian derivative of defined by Hence, from Theorem 16, we obtain the following consequence.

Corollary 18. *Let the functions and satisfy structural conditions (3), (4), and (15) as well as the conditions (16), (17), and (18) in respect to a given real number . Let . Then, every nontrivial solution of (27) is fractal oscillatory near with the fractal dimension .*

As a consequence of Theorem 16, and Corollary 18 we derive the following examples for linear differential equations of second order having all the solutions to be fractal oscillatory near .

*Example 19. * The so-called damped chirp equation
is fractal oscillatory near with the fractal dimension , where . When and , (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 . Hence, Theorem 16 proves that every nontrivial solution of (29) is fractal oscillatory near with the fractal dimension .

Now we assume that . Then, Theorem 17 implies that (29) is fractal oscillatory near with the fractal dimension .

*Example 20. * The following equation
is fractal oscillatory near with the fractal dimension , where and . In the case where and , (26) becomes (30). We see that , , , , and . Therefore Theorem 17 implies that every nontrivial solution of (30) is fractal oscillatory near with the fractal dimension .

*Question 2. *What can we say about the application of Theorem 16 on the case of 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 as follows.

*Definition 21. * Let and . A function is * oscillatory near *, if there is a decreasing sequence such that , for , and as . Moreover, if the graph satisfies the condition (12) for some , then is said to be * fractal oscillatory near ** with the fractal dimension *.

*Definition 22. * Let and . It is said that a function is * fractal oscillatory near ** from the left side with the fractal dimension *, if there is an increasing sequence such that for , as , and the graph satisfies the condition (12), see Figure 2.

*Definition 23. * Let , and let . It is said that a function is * two-sided fractal oscillatory near ** with the fractal dimension *, if there is an increasing sequence and a decreasing sequence such that for , and as , and the graph satisfies the condition (12); see Figure 3.

With the help of Theorem 8, we state the following result.

Theorem 24. *Let , and let . Let the functions satisfy structural conditions (3), (4), (5), (15), and (16) as well as conditions (17) and (18) in respect to a given real number . Then, one has*(i)*the chirp function , is fractal oscillatory near with the fractal dimension ; *(ii)*the chirp function , is fractal oscillatory near from the left side with the fractal dimension ; *(iii)*the chirp function , is two-sided fractal oscillatory near with the fractal dimension .*

#### 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 Definition 5, it follows that if for a prescribed real number there are two positive constants and such that for some , then a function is fractal oscillatory near with the fractal dimension . The following lemma plays the essential role in the proof of (31).

Lemma 25. *Let be a bounded function on , and let be a decreasing sequence of consecutive zeros of such that . Let , and let be an index function satisfying
**
Then
*

*Proof. *It is exactly the same as [3, pp. 2350].

Let us remark that for some , we say that a function is an index function on ifâ€‰â€‰â€‰â€‰and .

Lemma 26. *Let . Then,
*

*Proof. *Let . Set , and set
for . Then, there exists such that . Set . We see that ,
and if , then
We will show that
where
Let . Then there exists such that . Because of the definition of , we find that for some , so that
Hence, it follows that
which means that . Therefore, we obtain (38). By (38), we conclude that
When , from (42), it follows that
Now, we assume that . We observe that
that is,
Combining (42) with (45), we obtain

Lemma 27. *Let be bounded on , and let . Then, there exists such that, for every and ,
*

*Proof. *Let , and let . From the geometry point of view, it is clear that
where is a rectangle in defined by
Hence, we have
where . Lemma 26 implies that
Consequently, we see that
where .

Lemma 28. *Let be bounded on . Assume that
**
for some . Then, there exists such that
*

*Proof. *Let and let be satisfy (54). By (53), there exists , which depends on and , such that
Therefore, we have
By (54), there exists such that
which implies that
that is,
where . From Lemma 27, (57), and (60), it follows that

In order to show Theorems 8 and 11, we need the following two geometric lemmas.

Lemma 29 (see [1]). * If is a simple curve (i.e., its parameterization is a bijection) andâ€‰â€‰, then
**
where denotes the -neighborhood of the graph .*

Now, we are able to prove Theorem 8.

*Proof of Theorem 8. *Let , and let be a chirp function given by (1). We note here that it is enough to show that satisfies (31).

At the first, let be a sequence defined by for all sufficiently large . From (4), it follows that is decreasing. Hence, is decreasing as well as as because of (see (4)). We note that and on for all sufficiently large . Also, is an increasing function because of (16). The mean value theorem shows that
Now, let . Let be the smallest natural number satisfying
Such exists for every , since as and (this equality is true because since ). Moreover, since is decreasing and is increasing, we obtain
Combining (63) and (65), it is easy to deduce that such defined satisfies condition (32).

By (16), there exists such that for , which means that
Integrating (66) on , we have
that is,
By the definition of and (64), we see that
Hence, from (68), it follows that
which implies that
where .

Next, it is clear that the assumption (5) ensures a real number such that
Hence, from (15), we have
Now, from (63) and (73), it follows that
Since is decreasing as well as is increasing and is nondecreasing and positive, we conclude that the function
Hence, we observe that
Next, from assumption (17), we get some such that
Now, from (76), (77), and (71), we obtain
Thus, using Lemma 25, we have
for some .

Next we prove that satisfies (53). Assume to the contrary that
Then, we have
From Lemma 29, it follows that