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Estimation of the world economic system stability from 1963 to 2013 by using a discrete dynamic model.

1. Introduction

Discrete dynamic model (DDM) allows estimating the world economic system stability, which is the defining characteristic of the latter. Loss of stability leads the world economic system to crises, to prediction of which is an important task for policy-makers. DDM can be used to describe the state of the world economy (whether it is sable, potentially stable, or unstable) at different time intervals by means of a generalized image, a "pictogram". These pictograms are radii of convergence (Julia sets) of DDM approximating polynomials.

The main DDM assumption is that the world economy has inertia. Chaldaeva & Kilyachkov (2012, 2014) took into account that the assumption of the world economy inertia can be formalized in an assertion that the rate of world GDP changes in the following year [X.sub.n+1] depending on its value in the previous year [X.sub.n]. Therefore, the following expression holds: [X.sub.n+1] = F([X.sub.n]).

Although the specific form of the function F(X) is unknown, it can be formally expanded in a Taylor series up to the m-th order inclusive:

[mathematical expression not reproducible] (1)

Where, [X.sub.n+1] - the world GDP annual change rate in year (n + 1); [X.sub.n] - the world GDP annual change rate in the previous n-th year; [a.sub.k] - the coefficients of the expansion of the function F ([X.sub.n]) in a Taylor series. These coefficients can be determined by the method of least squares, by approximating the actual rates of world GDP change, represented in coordinates ([X.sub.n+1], [X.sub.n]), by a polynomial of the form (1).

Expansion of the function F([X.sub.n]) into a Taylor series up to the polynomial of degree 3 (Kilyachkov, Chaldaeva, & Kilyachkov, 2015; 2016; 2017a; 2017b) demonstrated the necessity of studying attractors of approximating polynomials (1). Each attractor is characterized by a radius of convergence, which is a Julia set possessing a unique structure. This allows us to describe the state of the world economy at different time intervals not by a group of numbers or a set of graphs, but in the form of a generalized image, a "pictogram". This methodology for analyzing the state of the world economic system is fundamentally different from the approaches currently applied in this area. However, it requires a detailed study of the basins of attraction of DDM and the pattern of its stability and makes it necessary to involve a large amount of experimental data. We used World Bank Open Data site as the source of such information. It contains information on the rates of world GDP change for 1961-2015 and prepared with it the analysis of the world economic system in the interval from 1963 to 2013.

The obtained results confirm the conclusion that the development of the world economy is largely unstable. But the stability of the world economic system has increased over the period under review (1963-2013). Moreover, the latest world economic crisis (2010 - 2013) was not accompanied by a loss of stability. Hence, potential stability of the development of the world economic system and economic crises are not unambiguously related concepts. It is necessary to distinguish between the situations of stable development of the economy and the crisis in the economy. The economy in a stable state can show low and even negative growth rates, i.e. its condition can represent a steady crisis. It was also shown that in 1971 and 1983 there were objective prerequisites for implementing various scenarios for the development of the world economic system.

Organization of the paper is as follows. Section 2 contains the Literature review. Section 3 states the course of the conducted studies, the following interrelated tasks and methods of solving them. Section 4 sets forth obtained results, describing the development of the world economic system in the interval 1963-2013 and specific cases of stability. Section 5 states about research limitations. Section 6 sets forth conclusions.

2. Literature review

The fundamental task of economic science is the development of models describing the functioning of the world economic system. Scientists have proposed both general theories describing the economy as a whole (see Blaug, 1985) for a review of these theories) and specific models that explain individual economic processes (Allais, 1977; Arrow, 1963; Debreu, 1952, 1962, 1970; Friedman, 1968; Harsanyi & Selten, 1988; Kantorovich, 1939; Koopmans, 1960; Koopmans & Montias, 1971; Lucas & Prescott, 1971; McFadden, 1986; Nash, 1951; Simon, 1959; Tinbergen, 1956; Tobin, 1955). In any case, there is no doubt that the world economy depends on many factors that are of different nature and interact with each other in a complex and, often, non-obvious way. In other words, the dependence of some economic factors on others can be described on the level of "influence", but not on the level of "functional dependence".

Chaldaeva & Kilyachkov (2012, 2014) proposed representation of such an influence for describing the rates of world GDP. They took into account that the assumption of the world economy inertia results in the fact that the rate of world GDP change in the following year depends on its value in the previous year (see equation 1). The proposed model was called the discrete dynamic model (DDM). The expansion of function (1) up to the polynomial of degree two (Chaldaeva & Kilyachkov, 2012) allowed to describe the emergence of all known economic cycles (Kitchin, 1923; Juglar, 1862; Kuznets, 1930; Kondratieff, 1922, 1925, 1926, 1928, 1935) in a unified manner as a bifurcation of some basic cycle (T[approximately equal to] 3 years).

Expansion of the function F ([X.sub.n]) into a Taylor series up to the polynomial of degree 3 (Kilyachkov & Chaldaeva, 2013, Chaldayeva & Kilyachkov, 2014), along with the bifurcation effect, explained some other subtle phenomena found in the works of Korotayev & Tsirel (2010). In addition, Kilyachkov, Chaldaeva, & Kilyachkov (2015, 2016, 2017a, 2017b) demonstrated the necessity of studying attractors of approximating polynomials (1). Using current terminology (Arnold, 2009; Arrowsmith & Place, 1982; Bezruchko, Koronovskiy, Trubetskov, & Khramov, 2010; Danilov, 2010; Haken, 1978, 2004; Malinetsky, 2009; Sekovanov, 2013; Trubetskov, 2010), we shall call attractive fixed points and periodic points as attractors, and local complicated sets - as strange attractors.

Kilyachkov, Chaldaeva, & Kilyachkov (2018) suggested another method, based on analyzing the structure of the Mandelbrot set of DDM, but there is a need for more careful research of how one can use it.

3. Statement of the problem and method of solving it

In the course of the conducted studies, the following interrelated tasks were solved:

1. Determining the coefficients of the approximating polynomial for actual values of the rates of world GDP change;

2. Finding basins of attraction of this polynomial for different values of the coefficients corresponding to different years;

3. Determining the pattern of convergence of the basins of attraction thus found;

4. The construction of the Julia sets (the radii of convergence) corresponding to them, for a detailed study of the type of convergence.

In this paper, we used the expansion of expression (1) to polynomials of degree 3 which has the form:

[mathematical expression not reproducible] (2)

Determination of the coefficients of the approximating polynomial (2) for actual values of the rates of world GDP change was carried out using the method of least squares on a sliding interval. The minimum duration of the interval, which is necessary for calculating the four coefficients of equation (2), is four years. In order to compensate the influence of random factors, it is advisable to increase this interval. However, if it is increased excessively, the economic situation can significantly change during the interval, which will lead to a decrease in the accuracy of determining the coefficients. The carried out researches have shown that for definition of the coefficients of approximating polynomial (2) an optimum duration of the interval is 5 years.

The specific feature of the sliding interval method is that the coefficients ([a.sub.0], [a.sub.1], [a.sub.2], [a.sub.3]) are determined on an interval having a fixed duration (5 years), but whose position shifts from the initial value for the considered period of time to the final one. Thus, the coefficients ([a.sub.0], [a.sub.1], [a.sub.2], [a.sub.3]) are first determined on the interval from 1961 to 1965, where the obtained values are matched to the middle of the approximation interval, i.e. to 1963. The same procedure is performed for the interval from 1962 to 1966, and the result is matched to 1964. This procedure is carried out until the interval from 2011 to 2015, the result of which is matched to 2013. Note that the polynomial with coefficients, which are corresponded to the [n.sup.th] year, approximates the behavior of the rates of world GDP change on the interval (n [+ or -] 22) years, and its coefficients ([a.sub.0], [a.sub.1], [a.sub.2], [a.sub.3]) are corresponded to some average state of the economy for this time interval.

To determine the dynamics of change of the coefficients ([a.sub.0], [a.sub.1], [a.sub.2], [a.sub.3]), it was assumed that their values change linearly during the year from the values corresponding to the previous year to the values corresponding to the next one. Such interpolation is justified, because the coefficients {([a.sub.0], [a.sub.1], [a.sub.2], [a.sub.3])} strongly correlate with one another (*). The number of interpolation points between years was chosen to be 10. Such a choice (10, and not 12) was made specially in order to be sure that the interpolated points would not be compared with the respective months.

Basins of attraction of the approximating polynomial were defined for obtained coefficient groups {([a.sub.0], [a.sub.1], [a.sub.2], [a.sub.3])}. For this, expression (2) was used as an iterative equation, i.e. [X.sub.0] was used to determine [X.sub.1]; then [X.sub.1] was used to determine [X.sub.2] and so on up to some final value of [X.sub.n]. At each step, the convergence of the iteration expression (2) was controlled. The initial values of [X.sub.0] were set on the interval [-0.1, 0.1] with a step of 0.001. The convergence condition for the iterative expression (2) had the form |[X.sub.0]| < 2, and its execution was checked at each step up to n = 100. Since the coefficients ([a.sub.0], [a.sub.1], [a.sub.2], [a.sub.3]) were constant in these calculations, the fulfillment of the convergence condition indicated the fundamental stability of the corresponding state of the world economy regardless of whether this stable state was a fixed stable point or it was dynamically stable (a limit cycle or a strange attractor). Accordingly, failure to meet the convergence condition indicated a fundamental instability of the state of the world economy that corresponded to these values of the coefficients.

It should be noted that even in the presence of a basin of attraction for some state of the economy, the actual indicator of GDP growth was situated outside this area in some cases. It indicates that for these cases a possibility of a sustainable development was not realized. Thus, DDM shows the presence or absence of potential - not actual sustainability. Further on, in order not to complicate the text, we will use the term "sustainability" instead of "potential sustainability" unless otherwise specified.

The type of the convergence of the basins found was determined using Lamerey diagram. This diagram is constructed in coordinates (x, y) [right arrow] ([X.sub.n], [X.sub.n+1]) as a sequence of linear segments parallel to the axes 0[X.sub.n] and 0[X.sub.n+1]. The coordinates of the beginning and the end of these segments are given as follows: [([X.sub.0], [X.sub.1]);([X.sub.1], [X.sub.1])], [([X.sub.1], [X.sub.1]);([X.sub.1], [X.sub.2])], [([X.sub.1], [X.sub.2]);([X.sub.2], [X.sub.2])] ... etc. For a fixed stable point, Lamerey diagram is contracted to a point (Figure 1a). For limit cycle, Lamerey diagram forms a rectangle (Figure 1b). In the case of a strange attractor, Lamerey diagram forms a complex structure represented by a set of embedded rectangles (Figure 1c).

The radius of convergence (Julia set) is an important characteristic of the corresponding basin of attraction. Julia set is the boundary of the area of complex variable values of [X.sub.n], belonging to which ensures the convergence of the iteration expression (2) to the attractor. To construct Julia sets, we used the standard method (Sekovanov, 2013).

4. Results and discussion

4.1. Analysis of the development of the world economic system in the interval 1963-2013

The study made it possible to construct basins of attraction, i.e. regions of initial values ([X.sub.0]) for which the polynomial (2) converges, and the ranges of final values [X.sub.final] to which this polynomial converges as time goes to infinity (n [right arrow] [infinity]). In calculations, n = 100 was taken. The results obtained are shown in Figure 2. This figure also shows actual rates of world GDP change for corresponding years. Table 1 and Figure 2 provide the information on the type of convergence for the years taken.

Comparison of basins of attraction to actual values of the rate of world GDP change has shown that the development of the world economic system in the interval from 1963 to 2013 in approximately half of the cases was characterized by a lack of convergence. In 27 cases (53%) of basins of attraction of equation (2) either were absent (see Table 1), or the actual rates of world GDP change were outside of them (1968, 2010, see Figure 2). For another 14 cases, world economic system demonstrated dynamic stability (limit cycles and strange attractors). Moreover, the actual rates of world GDP change only belonged to the final values range ([X.sub.final], Figure 2) in 6 cases, which corresponded to 1970, 1996-1999 and 2002.

It should be noticed that if we break down the time interval under consideration (from 1963 to 2013) into decades, then for the interval 1963-1972 convergence will be absent in 60% of cases and for the interval 2003-2013 - only in 36%. Thus, the stability of the world economic system has increased over the time period under review.

Julia sets (convergence radii) in the interval from 1963 to 2013 are shown in Figure 3. Since these sets can only be constructed under the condition of convergence of equation (2), then in Figure 3 there are no images of convergence radii for those years when convergence was absent.

It is worth noting a long period of continuous sustainable development of the world economic system from 1995 to 2004 (2001 being an exception) which we will consider in more detail.

To study the dynamics of the change of Julia sets, we use coefficients calculated for interpolation points located between consecutive years. The Julia sets for such interpolation points (where they exist) are represented below by integers with a decimal part.

For three years (from 1995 to 1997), the world economic system was passing from one strange attractor to another, being in a dynamic stable state. Julia sets corresponding to this time period are shown in Figure 4.

It can be seen from Figure 4 that the world economic system remained stable throughout the considered interval, but the type of this stability was changing significantly. Various types of dynamic stability were identified, such as strange attractors (1995.2-1996.1), limit cycles (1996.2-1996.9) and also a combined stable state, which is a set of a fixed stable point, limit cycle and strange attractor (1995.1). In the latter case, different stable states of DDM are realized under different initial conditions [X.sub.0].

In the period from 1997 to 1998, a transformation of the character of stability of the world economic system takes place from a strange attractor to a limit cycle. Julia sets that reflect this transition are shown in Figure 5.

Then for three years from 1998 to 2000, the development of the world economic system was characterized by limit cycles (Table 1). Figure 6 shows the corresponding Julia sets.

It is noteworthy that Julia sets for 1968, 1989, 1998-2000, and 2007 are very close to one another by the structure of the dominant area of the figure. This assertion is also valid for Julia sets corresponding to the years 1984, 1985, 2011, and for 1966, 2003, 2004. These regions are similar in form to one another but differ in size and position along the actual value axis. This circumstance gives grounds to assume that in the specified years the state of the world economic system as comparable in terms of some essential parameters.

In the period from 2000 to 2004, the development of the world economic system has undergone very dramatic changes (Figure 7).

During 2000, the world economic system was losing its stability by degrading through a limit cycle (2000.1-2000.3) and a strange attractor (2000.4-2000.9) to a total loss of stability in 2001. Then the stability recovered first in the form of a strange attractor (2001.1-2001.4), in the form of limit cycles (2001.5-2001.8), after which the stability degraded into a strange attractor (2001.9-2002.1), and through limit cycles (2002.2-2002.6) it transformed into a stable fixed point (2002.7-2004). The following circumstance attracts attention: the pattern of the world economic system stability and the corresponding Julia sets were changing dramatically in 2000-2002.1 and smoothly in 2002.2-2004.

Sharp changes in the pattern of the stability of the world economy from 2000 to 2002.1 showed that in fact the economy did not move toward a state of stability, although during almost all this period it was possible. It appears that only from 2002.2 to 2004 it is possible to speak about the transition of the economy into a state of actual rather than potential sustainability.

4.2. Specific cases of stability of the world economic system

In this part of analysis we examine some cases of the stability of the world economic system development.

4.2.1. World economic crisis (2010-2013)

Consider the transformation of Julia set (Figure 8) which describes the behavior of the world economic system in the period of the world economic crisis (2010-2013).

The world economy in 2010-2013 evolved very dramatically as can be seen from Figure 8. At first, its development was characterized by stable fixed points (2010-2010.2). Then a disruption of stability and the lack of convergence occurred (2010.3), although the Julia set existed. That means that the loss of stability was very close to a strange attractor. This suggestion is supported by the fact that at the next step convergence was restored in the form of a strange attractor (2010.4). After that (2010.5-2013), the pattern of sustainability of the world economic development takes the form of a stable fixed point the value of which changes smoothly with time (Figure 9).

The results indicate that from 2010 to 2013 the world economic system was developing quite steadily. However, it made a qualitative leap in 2010 moving from one type of sustainability to another. This becomes apparent from a comparison of the form of the Julia set describing the state of the world economy in 2010 and the forms of the Julia sets in 2011-2013, which have the same topology but differ in the shape of the border.

The striking result is that despite the generally accepted opinion about the crisis of the world economic system in the period 2010-2013, the system during this period of time was almost always in a steady state. Therefore, it seems more correct to speak not about a crisis but about slowing down growth rates of the sustainable development of the world economic system in the process of transition to a qualitatively different state.

Therefore, a detailed analysis of each type of sustainability appears appropriate. To do this, we will group the obtained results by the type of convergence.

4.2.2. Strange attractors

The world economy had a sustainability pattern of the type of a strange attractor in 1970, 1975, 1995-1997 and 2002. The corresponding Julia sets are shown on Figure 10.

It is noteworthy that Julia sets and Lamerey diagrams corresponding to 1970 and 2002 differ from the others in the following characteristics:

- Lamerey diagram completely covers the area including two local extrema of the approximating polynomial. For other years, this area either overlaps only the value of one local extremum or a part of it;

- The range of changes in the imaginary values of Julia set is insignificant and amounts to < 0.0002. For other years it is an order of magnitude larger;

- For these years, it was only possible to obtain a more or less obvious form of the Julia set when estimating its convergence for 1.2 billion points. For other years, it was sufficient to make such calculations for no more than 3 million points.

From our point of view, the obtained results show that stability of strange attractors in 1970 and 2002 was low, i.e. in these years the sustainability of global economic processes was precarious.

4.2.3. Limit cycles

In considering cases characterized by this type of stability (limit cycle), the Julia set for 1971 is of interest (Figure 11).

Figure 11 shows the Julia set describing the state of the world economy in 1971 which was characterized by three stable states: a multi-cycle, varying in the range from 0.0447 to 0.049 (Figure 11b); a strange attractor varying within the same limits (Figure 11c), and a stable cycle that varied between 0.0573 - 0.06 (Figure 11d). The multicyclic state was formed as a result of at least 3 bifurcations and was a set of 8 stable cycles. Depending on the initial conditions, the world economic system described by DDM could transfer to one of these three states. Taking into account the value of the actual rate of world GDP change in 1971 (World Bank Open Data), equal to [X.sub.real] = 0.04312, we can conclude that the world economy in 1971 could have moved into a stable multicyclic state.

4.2.4. Stable fixed points

For this type of stability, the Julia set which corresponds to 1983 stands out for its unusual shape. The peculiarity of the behavior of the world economic system, described by DDM for this year, is that it corresponds to a state with two different fixed stable points. Moreover, depending on the initial value of [X.sub.0], DDM converges either to one final value ([X.sub.final] = 0.014) or to the other ([X.sub.final] = 0.041) - see Figure 12.

Comparing the Julia set which corresponds to 1983 and Figure 12(a), it can be seen that the different regions of convergence are determined by the different domains of connection of the Julia set. Taking into account the value of the actual rate of the world GDP change in 1983 (World Bank Open Data) equal to [X.sub.real] = 0.024, we can conclude that in this year there were conditions for transition of the world economy to a stable growth at a rate of [approximately equal to] 4.1% per annum.

5. Research limitations

The coefficients {[a.sub.k]} from formula (1) reflect the influence of all factors that determine the rate of world GDP change. These factors are of a different nature (resource, technological, financial, etc.) and in the general case are functions of time. It is impossible to mark out the effect of any particular factor in the current version of the model.

The coefficients change as time passes. We need to get an interval to calculate the coefficients of the polynomial, during which they may have altered. We assume that these coefficients can be considered as constant for small time intervals not accompanied by economic shocks.

DDM treats economy as an isolated system. This is plausible for the world economy considered in this paper but may not work for national economies. DDM may also work inappropriately for the periods when large investments are made on the world scale (i.e. there is flow of economic resources between the periods more distant than adjacent years).

6. Conclusions

The obtained results confirm the earlier conclusion that the development of the world economy is largely unstable (Kilyachkov, Chaldaeva, & Kilyachkov, 2018). In 53% of cases, the convergence region of DDM which describes the world economic system is either absent or the actual rates of world GDP change are outside the area of convergence (1968, 2010). Furthermore, for two years (1970 and 2002) one can conclude that stability of the world economic processes was precarious. For the remaining cases, the state of the world economic system corresponds to fixed stable points or is characterized by dynamic stability (limit cycles and strange attractors).

Attention should be drawn to the fact that if we consider the period from 1963 to 2013 by decades, the proportion of years when there was no convergence decreased from 60% (1963-1972) to 36% (2003-2013), i.e. the stability of the world economic system has increased over the period under review.

For us, it was unexpected that the last world economic crisis was not accompanied by a loss of stability. The results indicate that from 2010 to 2013, the global economic system was mainly in a stable state. Moreover, this type of stability corresponded to fixed stable points. However, the system made a qualitative leap in 2010. Therefore, it is correct to speak not about the crisis, but about a slowing growth of the sustainable development of the economic system in the process of transition to a qualitatively different state.

The obtained results make us ponder over the question whether the sustainable development of the economy characterized by fixed stable points is a positive circumstance. Perhaps, it is preferable to maintain (if at all possible) the dynamic stability of the world economic system.

Based on the obtained results, it can be concluded that the potential sustainability of the development of the world economic system, analyzed with the help of DDM, and economic crises are not unambiguously related concepts. It is necessary to distinguish between the situations of sustainable development of the economy and the crisis in the economy. In particular, the following differences are seen between them:

1. Under a potentially stable state of the economy, sustainable growth is possible only within a certain range of rate of world GDP change. If the actual growth rates due to various reasons (as a result of an incorrect economic policy, by coincidence, etc.) is outside this range, then the possibility of sustainable development will not be realized, and the economy will experience a crisis.

2. On the contrary, the economy in a stable state can show low and even negative growth rates, i.e. its condition can represent a steady crisis.

The conducted research also showed that in addition to "pure" stable states (point, limit cycle, strange attractor) there are more complicated situations. The state of the economy in 1971 was characterized by three synchronous stable states: a multicyclic, a strange attractor and a limit cycle. Considering the value of the actual rate of world GDP change in 1971 equal to [X.sub.real] = 0.04312, it can be concluded that the world economy in 1971 could move into a stable multicyclic state with range value of GDP growth rates from 4.7 % to 4.9%. In 1983 the world economic system was in a state with two different fixed stable points. Since the actual rate of world GDP change in 1983 was [X.sub.real] = 0.024, we can conclude that in this year there were conditions for the transition of the world economic system to a state of sustainable growth at a rate of about 4.1%. Thus, in 1971 and 1983, there were objective prerequisites for implementing various scenarios for the development of the world economic system.

The conducted research also allowed revealing that in 1968, 1989, 1998-2000, and 2007, Julia sets had a similar structure corresponding to limit cycles. The same situation was observed in 1984, 1985, and 2011, and also in 1966, 2003, and 2004, for which Julia sets corresponded to fixed stable points. Differing in magnitude and position along the actual value axis, the dominant regions of Julia sets were similar in shape to one another. Therefore, it can be assumed that in these years the state of the world economic system was comparable by some essential parameters. The study of this issue should be the subject of a separate research.

The obtained results are largely correlated with those we obtained earlier using another methodological approach (Kilyachkov, Chaldaeva, & Kilyachkov, 2018). The time block of 1998-2003 for which the Mandelbrot set was constructed, suits to the range of 1998-2000 identified in this paper as corresponding to limit cycles. A time block of 1982-1986 suits to the interval 1983-1985 corresponding to fixed stable points.

However, there are discrepancies between the two approaches (the method of analyzing the structure of the Mandelbrot set and the method of least squares on the sliding interval). The time block 2004-2009 described in the paper of Kilyachkov, Chaldaeva, & Kilyachkov (2018) as stable, is not stable by the results of the present work. In 2005, 2006, 2008, and 2009 the world economic system did not experience any stability. Although the previous study (Kilyachkov, Chaldaeva, & Kilyachkov, 2018) and this study used different research methods, one can nevertheless expect that the results of two approaches must correspond to each other in a greater degree. The revealed discrepancies indicate the need for more careful research of how we can use the method, based on the consideration of the structure of the Mandelbrot set, to analyze the stability of the world economic system.

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Anatoly Kilyachkov, (1) Larisa Chaldaeva, (2) Nikolay Kilyachkov

(1) Expert, freelancer

(2) Department of Economics of Organizations, Financial University at RF Government, Russia

corresponding e-mail: AAKil[at]mail(dot)ru

address: Kilyachkov A. A., flat 16, building 21, Malysheva Str., Moscow, 109263, Russia

(*) Correlation coefficients between different pairs of coefficients ([a.sub.i], [a.sub.j]) vary from 0.58 to 0.97.

DOI: http://dx.doi.org/10.15208/beh.2019.9
TABLE 1. THE TYPE OF STABILITY OF THE WORLD ECONOMIC SYSTEM, DESCRIBED
BY THE DISCRETE DYNAMIC MODEL IN THE INTERVAL 1963-2013

THE TYPE OF STABILITY                      YEARS

Strange attractors              1970, 1975, 1995-1997, 2002
Limit cycles              1968, 1971, 1989, 1990, 1998-2000, 2007
Stable fixed points       1966, 1974, 1976, 1983-1985, 2003, 2004,
                                         2010-2013
Basin of attraction
is absent              1963-1965, 1967, 1969, 1972, 1973, 1977-1982,
                                   1986-1988, 1991-1994,
                                2001, 2005, 2006, 2008, 2009

Source: Own elaboration.
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Author:Kilyachkov, Anatoly; Chaldaeva, Larisa; Kilyachkov, Nikolay
Publication:Business and Economic Horizons
Geographic Code:4EXRU
Date:Jan 1, 2019
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