Generalization of nonlinear control for nonlinear discrete systems
The problem of stabilization of unstable periodic orbits of discrete nonlinear systems is considered in the article. A new generalization of the delayed feedback, which solves the stabilization problem, is proposed. The feedback is represented as a convex combination of nonlinear feedback and semilinear feedback introduced by O. Morgul. In this article, the O. Morgul method was transferred from the scalar case to the vector one. It is shown that the additional introduction of the semilinear feedback into the equation makes it possible to significantly reduce the length of the prehistory used in the control and to increase the rate of convergence of the perturbed solutions to periodic ones. As an application of the proposed stabilization scheme, a possible computational algorithm for finding solutions of systems of algebraic equations is given. The numerical simulation results are presented.
By control of chaos we mean a small external influence on the system or a small change in the structure of the system in order to transform the chaotic behavior of the system into regular (or chaotic, but with different properties) . The problem of optimal influence on the chaotic regime is one of the fundamental problems in nonlinear dynamics [2, 3].
It is assumed that the dynamic system has a chaotic attractor, which contains a countable set of unstable cycles of different periods. If the control action locally stabilizes a cycle, then the trajectory of the system remains in its neighborhood, i.e. regular movements will be observed in the system. Hence, one of the ways to control chaos is the local stabilization of certain orbits from a chaotic attractor.
To solve the stabilization problem, various control schemes were proposed , among which controls based on the Delayed Feedback Control (DFC) principle are quite popular . Such controls, under certain conditions, allow local stabilization of equilibrium positions or cycles, which, generally speaking, are not known in advance. Among the DFC schemes, linear schemes are the simplest for physical implementation. However, they have significant limitations: they can be used only for a narrow area of the parameter space that enter the original nonlinear system. The necessary conditions for the applicability of the linear feedback are formulated more precisely in Section 2.1.
To extend the class of systems to which the DFC scheme applies, it is necessary to introduce non-linear elements into the control. For the first time, a nonlinear DFC with one delay was considered in , where the advantages of such a modification are also noted, in particular, the fact that the control becomes robust. In [7, 8] the concept of nonlinear control with one delay from  was extended: to the vector case; to manage with several delays; to the case of an arbitrary period . It is shown that the control allows to stabilize cycles of arbitrary lengths, unless the multipliers are real and greater than one. A relationship is established between the size of the localization set of multipliers and the amount of delay in the nonlinear feedback.
In [9, 10], a semilinear DFC scheme with linear and nonlinear elements was investigated. In spite of the fact that this scheme contains only one difference in control, nevertheless, it is possible to stabilize cycles with length under sufficiently general assumptions about cycle multipliers. For , the situation changes critically, and the stabilization of cycles is possible only if the rigid constraints on multipliers are met. The scheme of O. Morgul is considered in detail in Section 2.3. It will be generalized to the case of several differences in control, and transferred from scalar to vector case.
The purpose of the presented work is to improve the algorithms of M.Vieira de Souza, A.J. Lichtenberg, O. Morgul, and D. Dmitrishin: suppression of chaos in nonlinear discrete systems by local stabilization of cycles of a given length.
Accordingly, the problem consists of choosing the structure and parameters of the control system, in which beforehand unknown cycles of a given length would be locally asymptotically stable.
The paper considers delayed feedback in the form of a convex combination of nonlinear control and generalized control by O. Morgul. The characteristic polynomial of a closed system for a cycle of length is deduced and its structure has turned out to be quite simple. As a particular case, this polynomial contains the characteristic polynomials for non-linear control and generalized control of O. Morgul. The solution of the problem for stabilizing cycles of length one is given, i.e. equilibrium positions, and a theoretical basis is prepared for solving the problem in a general formulation for cycles of arbitrary length.
The special structure of the characteristic polynomial allows the use of methods of complex analysis. That is why the main method of constructing controls and investigating the conditions for their applicability is the geometric theory of functions of a complex variable. From the perspective of this theory, O. Morgul’s approach to stabilizing cycles and the conditions for its applicability are analyzed. The analysis of the influence of the control parameters on the quality of control is carried out and it is indicated why the combined control is better than the nonlinear or semi-linear control separately. Finally, applications of the proposed scheme of combined control to the improvement of iterative methods for solving algebraic equations are considered.
2. Review and preliminary results
We consider a nonlinear discrete system, which in the absence of control has the form
where is a differentiable vector function of the corresponding dimension. It is assumed that the system (1) has an invariant convex set , that is, if , then . It is also assumed that in this system there is one or more unstable -cycles , where all the vectors are distinct and belong to the invariant set , i.e. . The multipliers of the unstable cycles under consideration are defined as the eigenvalues of the products of the Jacobian matrices of dimensions . As a rule, the cycles of the system (1) are not a priori known as well as the spectrum of the matrix of the matrix .
It is required to describe the set in which it is possible to locally stabilize the -cycle of the system (1) by one control from the admissible control class for all multipliers localized in , ( is the extended complex plane). I.e., so that the system
would have a locally asymptotically stable -cycle with multipliers in , and on this cycle the control would vanish. In other words, we assume that for a given cycle length , we know the estimate of the localization set of the multipliers . In other words, we believe that the dynamic system is characterized not so much by the function (or a family of functions) as by the set of localization of multipliers of a cycle (or cycles) of known length.
2.1. Linear control
As a control, let us consider a law based on linear feedback
where the gain should be limited: , , . Accordingly, the system closed by such a control has the form
Note that when the state , is synchronized, the control (2) vanishes, i.e. the closed system (3) takes the form as in the absence of control. This means that the -cycles of the system (1) are -cycles of the system (3).
Consider the case . It is required to find the necessary conditions in terms of the localization set of the multipliers for which the equilibrium position of the system (3) is locally asymptotically stable (or sufficient conditions under which this equilibrium position is unstable). It is shown in  that the set of localization of multipliers of system (1) can not be arbitrarily large for any linear control of the form (2), more precisely, its diameter can not exceed sixteen, and the diameter of its each connected component is at most four and not depending on the dimension of the system neither on the number in the control (2).
This conclusion imposes significant limitations on the practical application of linear control. We also note one more drawback of linear control (2): the invariant convex set of system (1) will not be invariant for system (3).
2.2. Nonlinear control
Another type of feedback – nonlinear – has the form
and the corresponding closed system
where , , , , , . It is clear that . Only those controls of the form (4) for which , are considered admissible.
When the state , is synchronized, the control (4) vanishes, and the closed system (5) takes the form as in the absence of control. Therefore, the -cycles of the system (1) are -cycles of the system (5). In addition, the invariant convex set of system (1) remains invariant for system (5).
As shown in , for any set of localization of the multipliers of -cycles of system (1) that does not contain real numbers greater than one, there exists a control of the form (4) for which in system (5) these -cycles will be locally asymptotically stable. Thus, the specified control will have the property of robustness.
We give a solution of the problem of choosing the coefficients , , for special cases of the localization sets of multipliers,
case : , ,
case : , .
The algorithm for finding the minimal and the coefficients consists of the following steps :
a) nodes are calculated:
, if is even; , if is odd; in case , we should assume , and in case , assume ;
b) the following polynomials are constructed
if is even;
if is odd;
c) the coefficients of the polynomial are calculated (for example, by Vieta’s formulas);
d) the coefficients are computed:
e) in case , we introduce the quantities
if is even;
if is odd; the optimal value of is computed as a minimal natural number satisfying the inequality
f) in case we introduce the quantities
if is even;
if is odd; the optimal value of is computed as a minimal natural number satisfying the inequality
We note that for and , the polynomials are univalent in the central unit disk . Apparently, the univalence property of polynomials is true for and for all For different the set of localization of the multipliers of -cycles of the system (1) must lie in the half-plane (Figure 1-a, 1-b).
2.3. Semilinear control
To stabilize the cycle of the length O. Morgul [9, 10] proposed a feedback control that includes linear and nonlinear elements, i.e., a semilinear feedback control of the form
for which a corresponding closed-loop system is
where . On the cycle, the conditions are fulfilled, therefore, on the cycle .
We note that in  only the scalar case was considered. However, Morgul’s scheme can be generalized to the vector case, as will be shown below. In setting the invariant convex set of system (1) remains invariant for the system (7) too. If we assume that , then the convex invariant set can not be preserved, although in this case it is possible to stabilize the equilibrium positions with multipliers from the half-plane .
The characteristic equation for the -cycle, in the scalar case, has the form 
where is the multiplier of the cycle. Accordingly, in the vector case the characteristic equation takes the form
where are multipliers of the cycle (), in general, complex. Equation (9) is obtained as a special case of a more general characteristic equation, which we derive in Section 3.
If all the roots of equation (9) lie in an open central unit disc , then the -cycle is locally asymptotically stable [10, 13]. If the multipliers , are known exactly, then one can check whether the roots belong to the central unit circle by known criteria such as Schur-Cohn, Clark, Jury . However, cycles are not known, hence, multipliers are not known. In this case, the geometric criterion of A. Solyanik proved to be effective for the stability of cycles of discrete systems . Let us apply this criterion.
Making the change , we write equation (9) as a set of equations
where . The following observation is extremely useful in our settings.
All the roots of equation (9) lie in the central unit circle if and only if
where is a closed central unit disk, is an extended complex plane, the asterisk denotes the inversion . Here denotes the complex conjugated of .
Note that the set is the inverse of the set of exceptional values of the image of the disk under the mapping . According to Lemma 1, the -cycle will be locally asymptotically stable if the set covers the set of localization of multipliers. The condition (10) can be rewritten as , or in the equivalent form . This means that the set must be exceptional for the image of the disk under the mapping .
Let us consider some examples.
Example 1. Let . In this case the set
that is, this set is an open circle with center at the point and of radius
(Fig 2). If , the disk converges to the half-plane . If the disc converges to the half-plane . Therefore, if the set lies in the half-plane or , then the equilibrium position of the system (1) can be stabilized by the control of the form (6).
Example 2. Let . Then
that is, it is the interior of an ellipse with semiaxes and centered at the point (Fig 3). The ellipse foci are at the points and . Therefore, if the set lies in the half-plane , then the 2-cycle of the system (1) can be stabilized by the control of the form (6 ).
O. Morgul considered only the scalar case, and in the scalar case the set can consist only of real numbers. Then the condition of stabilization of the equilibrium is the following: or . Accordingly, for the 2-cycle, the stabilizability condition has the form: .
In the case , the function is univalent for all in the entire complex plane, with the exception of the point . For and , the function is univalent in the open central disk . In these cases, the functions for are univalent in the open central unit disc .
For the situation becomes different. The function will not be univalent in the disk for all , but only for . Since , then for , the condition for the stabilizability of the cycle in the scalar case takes the form . The function increases by , hence the maximum size for the multiplier localization set will be , i.e. . For , the function fails to be univalent, and the interval for the multiplier will decrease (Figure 4).
The function decreases as asymptotically tending to .
3. Generalized semilinear control
3.1. Formulation of the problem
A natural generalization of delayed feedback control is the joint use of the linear, nonlinear, and semilinear feedback considered in Section 2. Let us take into account that the linear feedback is ineffective, and we choose the control in the form of a convex combination of nonlinear and generalized semilinear feedbacks, specifically,
where . Note that the control (11) disappears on a cycle of length .
The motivation for using a control of the form (11) for is completely obvious. In the general case, the semilinear control does not allow stabilizing cycles of system (1) of length three or more. However, the combined use of semilinear and non-linear controls can allow reducing the necessary length of history used in the feedback.
For , we can also expect qualitatively new effects when the equilibrium position is stabilized due to a larger number of control parameters: an increase in the rate of convergence of the perturbed solutions to periodic ones, expansion of the basin of attraction of a locally stable periodic solution, and so on. In other words, combined control should improve the properties of both nonlinear and semilinear control.
Let us close the system (1) by the control (11), then we get
where the coefficients , are associated with the parameters , by a linear bijection
We request the invariant convex set of system (1) to be invariant for system (12). Therefore, we must require the following relations: , , , , . To this end, additional restrictions must be imposed on the control (11). Namely, ; , , , .
It is required to select the parameters , , satisfying the given constraints, so that the -cycle of the system (12) is locally asymptotically stable, and would be the smallest.
If we let in (12), then we obtain the system (5), i.e. the system (1), closed by nonlinear feedback. If we let in (12), then , therefore, we get the system (7), that is, as in the case of closure by the semilinear feedback. Thus, the system (12) contains the systems (5) and (7) as particular cases.
3.2. Construction of the characteristic polynomial
We begin the investigation of the stability of the -cycle of the system (12) with the derivation of the characteristic equation for this cycle. The classical way is the construction of the Jacobi matrix of a special mapping in the neighborhood of the cycle , and finding the characteristic polynomial of this matrix. As a result, this characteristic polynomial will have a cumbersome form, and the path of its simplification is not at all obvious .
The same polynomial can be constructed from a different mapping, and the polynomial is obtained in a very convenient form for further investigations [17, 18]. In  the equivalence of the classical O. Morgul method and the alternative method is proved, which will be applied below.
The solution of system (12) can be represented in the form
. We substitute solution (13) into (12), assuming that in the neighborhood of the cycle the norms of the vectors are small.
Let . Then
Selecting the linear part and taking into account that , we get
where , , are Jacobian matrices of dimension .
The system (14) is linear, so its solutions are represented in the form
Where is a complex number to be determined. Substituting (15) into (14), we obtain
Denote by , , . The system (16) considered with respect to the vectors , will have a nontrivial solution if and only if the determinant of the matrix
is non-zero, where is zero matrix of dimension , is the identity matrix of dimension . That is
Let the eigenvalues of the product of the Jacobi matrices be equal to . Then, replacing this product by Jordan’s canonical form, we obtain the final form of the characteristic equation
Hence, the desired characteristic polynomial has the form
The polynomial (18) contains, as a special case for , the polynomial (9).
3.3. Geometric criterion for local asymptotic stability of a cycle
The next step in the study of the stability of cycles is to analyze the location of the zeros of the characteristic polynomial (18) on the complex plane. Or, equivalently, the roots of equation (17). The local stability of cycles of difference systems is equivalent to the Schur stability of the characteristic polynomial corresponding to this cycle [cf. Ex. 13]. We present this fact in the following two Lemmas.
The -cycle of system (12) is locally asymptotically stable if and only if all zeros of polynomial (18) lie in the open central unit disk .
As noted in Section 2.3, it is not possible to apply the known criteria for testing the Schur stability of the polynomial (18), since the quantities are not known. Therefore, to verify the local stability of cycles of system (12), we apply the geometric criterion of stability suggested by A. Solyanik. Denote by .
All roots of the polynomial (18) lie in the open central unit disc if and only if
where is a closed central unit disk, is an extended complex plane, the asterisk denotes the inversion operation: .
The polynomial (18) is Shur-stable if and only if for all . This is equivalent to , , . Consequently, the necessary and sufficient conditions for the stability of the Schur polynomial (18) are the inclusions: , or , or , . ∎
In the general case, cycle multipliers are not known; hence, the -cycle is locally asymptotically stable if the set covers the set of localization of multipliers. This means that the set must be exceptional for the image of the disk under the mapping . This property will be the main one for constructing the control coefficients .
3.4. The design of controls that stabilize cycles
The next step is to construct a function so that the set covers the set of localization of multipliers. In this case, it is necessary to estimate the size of the set as a function of and . The function depends on the polynomials , , and the parameter , and , , , hence , .
To further advance the formulation of the problem, we impose an essential restriction on the function , namely, we know that the polynomial is calculated by the formulas indicated in Section 2.2 (for some ). That ensures that () or () for any admissible and , at least for a sufficiently large and . We want to choose the polynomial and the parameter so that a certain desired linear dimension of the set is maximal (or the set is minimal). The linear dimensions depend on the value . Since all , are not negative, then . Consequently, it is not possible to make small at the expense of the polynomial . The polynomial has a very specific role: this polynomial should be chosen so that the parameter can be varied within the widest limits.
We formulate this requirement. We consider the family of functions
where the polynomial is defined as above. It is required to find the polynomial (with a given degree and given normalization conditions) so that the family of functions (20) is univalent in the disc , and is maximal.
If it is necessary to maximize the linear dimension of the set in the direction of the negative real axis, then the requirement of univalence of the family (20) can be replaced by a weaker requirement to be typically real. We recall that an analytic function in is said to be typically real in the sense of Rogozinsky if real preimages correspond to real values of the function . In other words, a function that is typically real in must map an open upper semicircle to an open upper (or lower) half-plane.
We give a solution of this problem for . The function is univalent in for . The polynomial is also univalent for . Therefore, the function is univalent for and as a superposition of univalent functions.
In this case, the set . This set is obtained as a result of shifting the set by and the subsequent extension in times (Fig 5).
The system (12) takes the form
The role of the parameter is clearly visible in Fig 5. The value of plays a dual role with respect to the parameter . Increasing can reduce , leaving the linear dimensions of the set almost unchanged. We consider the system (12) for :
The boundary of the set is a circle passing through the point . Consider system (12) with other averaging parameters ,
For this system, the boundary of the set passes through the point , where , and the coefficients are calculated using the formulas of Section 2.2 for some . Let . This means that the linear dimension of the set for the system (22) is greater than for the system (23). In order for the linear dimensions of these sets to be almost equal, the second set must be stretched. The stretching coefficient is determined by the parameter . It is not difficult to establish a connection between the parameters and :
Let, for example, , , . Calculate . Then . This example shows how much the parameter can be made smaller compared to . The sets for systems (22), (23) are shown in Fig. 6.
Let us give some more examples for , , and .
If , then and . If , then and . If , then and .
Thus, the introduction of retardation in the feedback allows us to reduce the value of the averaging parameter . The advantages of this approach are discussed below.
3.5. On the rate of convergence of perturbed solutions to the equilibrium position
Consider again the system (22), and let be the localization set of the multipliers of the equilibrium position of this system. Let the control parameters be chosen so that the domain covers the set , where , and . In this case, the equilibrium position will be locally asymptotically stable. This means that for the initial vectors lying in a sufficiently small neighborhood of the equilibrium position, the solution of the system (22) determined by these initial vectors tends to the equilibrium position. Such a neighborhood is called the basin of attraction of the equilibrium position of the system (22) in the space of the initial vectors. Evaluation of the basin of attraction is, in general, a very complicated task, and it is not the subject of this article.
We note, however, that even when all the conditions for attraction of the perturbed solution to the equilibrium position are satisfied, the behavior of the perturbed solution may turn out to be complicated, and approach the equilibrium very slowly. This is the case when the multiplier of the system is near the boundary of the set . The rate at which the perturbed solution approaches the equilibrium is determined by the maximum among the moduli of zeros of the characteristic polynomial (18) (with