Abstract

New results about the existence of chaotic dynamics in the quadratic 3D systems are derived. These results are based on the method allowing studying dynamics of 3D system of autonomous quadratic differential equations with the help of reduction of this system to the special 2D quadratic system of differential equations.

1. Introduction

Chaos theory has successfully explained various phenomena from natural science to engineering and is being applied in many fields. In mathematics, chaos has become an important branch of dynamical systems. In physics, it has been used to illustrate complex behaviors, such as planetary orbits and fluid motion. In chemistry, it has been used to analyze the amounts of chemical compounds. In engineering, chaotic systems provide many potential applications, such as the secure communication, the chaos control, the information encryption, and the synchronization. In economics, it has been used to develop new types of econometric models and to analyze job selection. In psychology, chaos has been applied to cognitive and clinical psychology. In biology, it has been used to study insect populations.

There are two basic methods of search of chaotic systems. They are based either on establishment of the existence fact of a homoclinic orbit or on construction for the given system of a discrete map and proof of its state of chaos.

The construction of discrete maps for continuous dynamical systems is still small studied. Here basic results are contained, for example, in [111]. The main idea of these papers is that properties of the being created discrete maps describing the dynamics continuous dynamical systems are based on the well-known properties of the Ricker map and the logistic map . Our approach to research of chaos in 3D autonomous quadratic systems is also based on this idea.

There is a huge number of papers devoted to the search of homoclinic orbits in 3D systems of differential equations (see, e.g., [1228]). Here, in our opinion, one of the most essential results was got in [17]. In this article the Fishing Principle allowing deciding a question about the existence of homoclinic orbits for large class of nonlinear systems was offered.

The most general approach at the study of chaos in the continuous 3D system consists in finding of a basin of attraction for this system. The simplest situation arises up then, when the basin of attraction is whole space . In other words, for existence of the basin it is sufficient that all solutions of the system were bounded at any initial data.

Notice that in the theory of dynamic systems, besides the methods mentioned above, methods connected with the invariant analysis of differential equations are also widespread (see, e.g., [29]). In [29] mentioned methods were applied for the study of dynamics of the Yang-Chen system. A construction of an invariant algebraic surface, which was introduced for research of the Lorenz system, is basis of these methods. Due to the concept invariant algebraic surface some new results on a behavior at infinity of the Yang-Chen system were got.

The present paper is a continuation of work [7]. It is needed to say that the basic idea of the represented work and article [7] consists in the reduction of 3D dynamic system to some 2D dynamic system. Howewer, in the present paper this idea was extended on other classes of quadratic 3D systems, which were not considered in [7] (see Section 4). In addition, the primary purpose of work [7] is a construction of implicit discrete mappings generating chaos in 3D dynamic systems. On the contrary, the obtaining of new existence conditions of chaos in 3D quadratic systems, which were not presented in work [7], is the main purpose of this paper.

At first we consider the following system:where are real numbers and is a parameter. (If for some system in form (1) the summands and look like and , then by suitable linear real replacement of variables and it is always possible to obtain that, in the new variables , we will have .)

In paper [7] the theorem about boundedness of solutions of the general quadratic 3D systems of autonomous differential equations was proved. In application to system (1) this theorem looks as follows.

Theorem 1 (see [7, Theorem  3]). Suppose that for system (1) the following conditions are fulfilled: (i);(ii)the quadratic form is positive definite (negative definite);(iii)the quadratic form is negative definite (positive definite).
Then for any initial values and the solutions of system (1) are bounded.

Without loss of generality, one will consider that the quadratic form is positive definite and the quadratic form is negative definite.

2. On Existence of Limit Cycles in System (1)

Letbe a system of ordinary autonomous differential equations and let be a trajectory of this system with initial data . Here is a continuous vector-function; .

The trajectory of system (2) is called periodic if there exists a constant such that

Let () be another trajectory of system (2) such that , where the symbol means the Euclidean norm of the vector ; is any positive small enough number.

The periodic trajectory of system (2) is called isolated if for any positive small enough number there does not exist the periodic trajectory such that . The isolated periodic trajectory of system (2) is called a limit cycle.

A set is said to be a positively invariant set with respect to (2) if from it follows that    .

A point is said to be a positive limit point of if there is a sequence , with as , such that as . The set of all positive limit points of is called the positive limit set of .

Let be a compact set.

Lemma 2 (see [30, Lemma  3.1]). If solution is bounded and belongs to , then its positive limit set is a nonempty, compact, invariant set. Moreover, as .

Let one for system (1) define new variables and by the following formulas: and . Then we obtain the new following system:

Further, one assumes in system (4) , , , , and . Then we get

In order to determine the chaotic properties of system (1) system (5) will be used.

Theorem 3. Let system (5) be , . Assume also that the quadratic function and the conditions of Theorem 1 are valid. Then in system (5) there exists either a limit cycle or a limit torus.

Proof. (A) Let us calculate Lyapunov’s exponent for a real function [7]: We take advantage of the following properties Lyapunov’s exponents:(a1);(a2)if , then ;(a3)if , then ;(a4);(a5).
Taking into account the boundedness of all solutions of system (5) we have , , and .
Consider the function where It is clear that , if In this case for the function Lyapunov’s exponent . If then there exists the point such that . In this case .
We write the first equation of system (5) in the following integral form: From here and (a1) it follows that or . Thus, from (a2) and (a3) it follows that Therefore, from boundedness of and (a4), we have .
From the third equation of system (5), we get Then from (a3)–(a5) it follows that From the condition it follows that the origin is a unique equilibrium of system (5). As , then the origin is a saddle-focus, and we have , , and .
(B) According to the conditions of Theorem 3 solutions and of system (5) are bounded at any initial data. Consequently for the concrete solution (in polar coordinates it is ) there is a nonempty compact invariant set (see Lemma 2).
Assume that the vector solution . By virtue of boundedness of any solutions and of system (5) there must be points such that , . Therefore, from the first equation of system (5) and condition , we have .
We will consider that . Then from this condition and the first equation of system (5) it follows that (see the proof of Theorem 3 [7]). From the condition and -periodicity of the nonnegative function it follows that the relation is also -periodic, .
There are only two possibilities:(b1)the functions and are periodic;(b2)the functions and are not periodic.
Assume that condition (b2) takes place. Then there does not exist a number such that . Consequently the inequality follows from the second equation of system (5). It means that the function is unbounded. We derived the contradiction. Consequently the condition (b1) must be valid.
In (A) for any solutions () the conditions , , and were got. In addition, we know that the periodic solution exists. From here it follows that in system (5) there is either a limit cycle or a limit torus (if ). This completes the proof of Theorem 3.

Theorem 4. Let system (1) be . Assume that all conditions of Theorem 3 are valid. If the parameter is small enough, then in system (1) there exists a limit cycle.

Proof. (C) According to Theorem 3 at in system (1) there is a limit cycle. We designate this cycle as . Let be a period of the cycle .
Compute Lyapunov’s exponents , , and on cycle . From Theorems 1 and 3 it follows that(c1)at and any initial values all solutions of system (1) are bounded;(c2)system (1) at has a unique equilibrium point; it is a saddle-focus (an unstable point);(c3)system (1) at has a unique stable limit cycle ; all solutions of system (1) at any initial values are attracted to this cycle.
Thus, from item (c1) it follows that Lyapunov’s exponents of cycle are as follows: , , and .
Now let be the sufficiently small positive parameter. Then, by virtue of continuity of Lyapunov’s exponents for the small enough , it is possible to consider that , . Consequently for the exponent such following suppositions take place.
; it contradicts with item (c1) (indeed, the presence of positive Lyapunov’s exponent will result in unlimited growth of solutions of system (1)).
; in this case all solutions must converge to the origin (it must be a stable node); we get the contradiction with item (c2).
Thus, there is only the case ; it means that for the small enough in system (1) there is a stable limit cycle different from . The period of this cycle is equal to , where is integer.

3. Chaotic Behavior of System (4)

In this section we will consider that all conditions of Theorems 3 and 4 are valid.

Let system (4) be , , , , and . Then we derive the following system:where initial values , , and are given.

Theorem 5. Let be the maximal positive parameter such that for system (16) and any all conditions of Theorems 3 and 4 are valid. Assume that there exists the parameter such that the conditiontakes place. Then in system (16) there is chaotic dynamics.

Proof. (D) Let the condition (17) be valid. Investigate a behavior of the function at the increase of .
(d1) At first we show that if , then . Assume that there exists such that .
Compute the second derivative of the function in the point . We get where is a continuous function such that is a finite number. Thus, if and , then and the function in the point is concave. Therefore, is a minimum point, , and . In addition, from second equation of system (16) we have .
(d2) The solution of the second equation of system (16) can be represented in the following integral form: where .
Suppose that in formula (19) the variable takes two values: and , . Then we can define the numbers , , , and . Introduce the following designation: Then from formula (19) it follows that The first equation of system (16) on interval may be also written in the integral form as (d3) Let , , and , where are roots of the first equation of system (16), [7]. We can consider that and .
We can also assume that the variants taking place, , , , , are sequential maximum (or a point of inflection), minimum (or a point of inflection), and maximum (or a point of inflection) of the function . As the function is -periodic, then .
According to Theorems 3 and 4 for sufficiently small solutions and of system (16) are bounded. Therefore, we can consider that the integer is large enough in order that the trajectory of system (16) got on an attractor (it is a limit cycle). Thus, we haveTaking into account formula (24) and the known first Theorem about Mean Value we can rewrite formula (22) as where and , we have .
(d4) It is clear that in (25) we have , , and . In this case we can introduce the following designations:where the functions , and do not depend on . The last inequality is stipulated to those in formula (25). We obtain at some , if and only if the second summand of this formula is negative. It is obvious that at the implementation of condition (17) this restriction will be satisfied. In addition, as shown in [31] (Lemma  10) the condition is necessary for existence in the system of a separatrix loop. This peculiarity of system (27) will be used in future for research of generalizations of system (16).
Further, process (25) may be represented in the following form:Introduce the new variable , where . Then process (28) will be generated by known logistic map as follows:where .
Let be the minimal fixed point of mapping . It is known that if , then the logistic map is chaotic and . Hence, from condition (17) it follows that there exists the parameter such that at process (28) generates the subsequence , for which , . It means that in system (16) there is chaotic dynamics.

4. Generalizations

Consider 3D autonomous systemwhere and , , is a real -matrix. Consider are real quadratic polynomials.

Suppose that the matrix has rank or . Then by suitable linear transformations of variables (), , and system (30) can be represented in the same form (30), where , , and and or .

Thus, without loss of generality, we will study system (30) under the following conditions:

Introduce into system (30) (taking into account (34)) new variables and under the formulas: , , where . Then, after replacement of variables and multiplication of the second and third equations of system (30) on the matrix we get

Consider the first and second equations of system (36). One haswhere is a real parameter. Consider (System (27) is a special case of system (37).)

Let , · , and be the bounded functions. In addition, we will consider that and are nonsingular forms of the variables and .

Theorem 6. Let . Suppose also that for system (37) with the following conditions:(i)either or is periodic alternating in sign on function;(ii)either and or and are periodic nonpositive functions are fulfilled.
Assume that the conditionis also valid. (From this condition it follows that such that .)
Then in system (36) (and system (30)) there is chaotic dynamics.

Proof. (E) Above it was shown that for system (30) can be always found the set of parameters such that at some values of these parameters in system (30) there is a limit cycle (see Theorem 4). Suppose that the parameters of system (30) have values from the set . Then from third equation of system (36) it follows that the right part of this equation is a periodic function (let it be ).
We take advantage of the next known result. Let be a period of the continuous periodic function . Then from third equation of system (36), we get where is -periodic function, is a linear function, and and are constants.
Therefore, in system (36) we can consider that (in other words ). Thus, in system (37) it is possible to do the replacement . The coefficients of system (37) are periodic functions and therefore these coefficients are bounded. Thus, the conditions of Theorem 6 coincide with the conditions of Theorem 1 [31]. From here it follows that for any initial values solutions of system (37) are uniformly bounded.
It is clear that without loss of generality we can consider that . Then we must consider two cases: (e1) and (e2) .
(e1) In this case the proof of Theorem 6 almost fully repeats the proof of Theorem 5. This process amounts to construction of the logistic map and proof of its state of chaos.
(e2) In this case the proof of Theorem 6 (without loss of generality!) is very convenient to show on a concrete example.
Consider the well-known Lorenz systemwhere parameters , , and . (It is the classic values of parameters at which system (41) demonstrates a chaotic behavior.)
We pass in system (41) to the coordinates under the formulas: , , and , where . Then we obtainBy definition, we have . Let and be new variables. Then from the first and second equations of system (42) it follows that Consider the systemIt is clear that the function is Lyapunov’s function for system (44). (Really, and .) Thus, at the solutions and of system (44) are bounded. Since and , then with the help of Comparison Principle [30] it is easy to check that solutions and of system (42) are also bounded at .
As well as in Section 3, we can assume that , , , are sequential maximum (or a point of inflection), minimum (or a point of inflection), and maximum (or a point of inflection) of the function . Thus, we have As (), then it is clear that , .
The second equation of system (42) on interval may be also written in the integral form as Introduce the designations and Then taking account of (46) formula (47) may be transformed aswhere .
Introduce the variable By virtue of Theorem 4 for the large enough we can consider that , , and .
Then from (49) it follows thatAs it is shown in [6] the iterated process (51) at suitable is chaotic. Thus, this process defines a chaotic behavior of system (42) and (41).

A next obvious corollary of Theorem 6 has a large practical application.

Corollary 7. Let be a vector of parameters of system (36) for which the conditions of Theorem 6 are valid. Then for any sufficiently small -perturbation of the vector in system (36) the perturbed system (36) with vector parameters has the same type of chaos.

Consider instead of system (30) more simple system

Taking into account the formulas and , we calculate the following functions:

Corollary 8. Assume that for system (52) the following conditions,(i)the forms and are nonsingular;(ii);(iii)either the quadratic form is an alternating function or is negative definite and ;(iv) is nonpositive definite; it means that either = or one of the quadratic forms or is nonpositive and other form is nonnegative, are fulfilled. Then under condition (39) where in system (52) there is chaotic dynamics.

At research of Lorenz-like and Chen-like attractors there is a situation when one of equations of describing the dynamics of the corresponding system is linear (see [32, 33]). This system can be represented in the following way:

Introduce the matrix

Let . By means of approaching a linear invertible real replacement of variables and the matrix can be reduced to the normal Frobenius form as follows:where and .

Thus, system (54) may be transformed to the following system: (For simplicity we have left the former designations of variables and and corresponding coefficients.)

For system (57) we suppose and . Then we get

Note that the structure of the first two equations of system (58) is the same as that in system (37). Therefore, we can introduce the following functions: (It is clear that if , then the form is nonpositive definite.)

Corollary 9. Assume that for system (57) the following conditions,(i)the form is nonsingular;(ii);(iii)either the quadratic form is an alternating function or is negative definite and ;(iv),are fulfilled. Then under condition (39), (where ) in system (57) there is chaotic dynamics.

System (54) may be also transformed to the following system:

If for system (60) we suppose and , then system (37) takes the following form:

Taking advantage of method of computation of functions and , we can compute similar functions for system (61) as follows:

If we change , , , , , and , then for system (60) the conditions of Corollary 9 remain valid.

5. Examples

(1) For Lorenz’s system (41) represented as (57) we have , , and and the form is an alternating function. Thus, the conditions (i)–(iv) of Corollary 9 are fulfilled. (For system (60) we obtain , , and and the form is an alternating function.) On Figure 1 the dynamics of polar radius for system (42) is shown.

It is easy to check that conditions (i)-(ii) of Theorem 6 and also condition (39) are valid. (Indeed, for system (42) on Figure 1 at , we have .)

(2) Consider the system [12]where , and are real numbers.

Let be an equilibrium point of system (63). (The real numbers , , and are determined from the system of the following equations: .)

Introduce in system (63) new coordinates under the formulas , , and . Then the system (63) can be transformed to the following system:with origin of coordinates in the point . (For simplicity we saved former designations of system (63) in system (64).)

Now we introduce in system (64) polar coordinates under the formulas and . Then we have

System (65) is a special case of system (36). Besides, the subsystem consisting of the first two equations of system (65) is a special case of system (37).

Let , , and . Then we have , , and . It is easy to check that conditions (i)-(ii) of Theorem 6 are valid. It is yet necessary to check up condition (39).

Consider the dynamic change of polar radius for system (65) (see Figure 2). Then we have that dynamical behavior in Figure 2.

It is clear that for , , and condition (39) () is fulfilled. On Figure 3 the chaotic attractor of systems (64) is shown.

(3) Consider the following system:

In the polar coordinates system (66) takes the following form (see (16)):

Let initial data be , , and . It is easy to check that the conditions of Theorems 5 and 6 at and , , , and are fulfilled. Thus, .

On Figures 410 different characteristics of system (66) are shown.

From Figures 610 it follows that in domain of chaos one of the Lyapunov exponents is positive, another is zero, and third is negative. If two exponents are negative and one exponent is zero, then in system (66) there is a limit cycle.

(4) Consider the Liu system [13]

In the polar coordinates system (68) takes the following form:

It is easy to check that the conditions of Theorem 6 at , and are fulfilled. The behavior of the attractor of system (68) is shown on Figures 11 and 12.

(5) Consider the Chen system [13]

In the polar coordinates system (70) takes the following form:

The conditions of Theorem 6 at , , and are fulfilled. The behavior of radius of system (70) is shown on Figure 13.

6. Conclusion

In [31] (Theorem  1) the necessary and sufficient conditions of boundedness of solutions for system (37) with constant coefficients were found. Then from these conditions in [31] (Lemma  10) the existence condition of the loop of separatrix (homoclinic orbit) was got. It is easy to show that the presence of homoclinic orbit in system (37) is a necessary condition for the existence of chaotic dynamics in system (30).

A search of the existence sufficient conditions of homoclinic orbits for system (30) is a difficult problem. Therefore another idea was realized. For the quadratic 3D system (16) the discrete 1D mapping was built; its state of chaos is then proved. On system (30) this approach was extended. As a result in Theorem 6 the existence conditions of chaos unknown earlier were got.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author is grateful to the anonymous referee for the comments which led to an improvement of the presentation.