Bifurcation Diagrams of Nonlinear Oscillatory Dynamical Systems: A Brief Review in 1D, 2D and 3D

Abstract

We discuss 1D, 2D and 3D bifurcation diagrams of two nonlinear dynamical systems: an electric arc system having both chaotic and periodic steady-state responses and a cytosolic calcium system with both periodic/chaotic and constant steady-state outputs. The diagrams are mostly obtained by using the 0–1 test for chaos, but other types of diagrams are also mentioned; for example, typical 1D diagrams with local maxiumum values of oscillatory responses (periodic and chaotic), the entropy method and the largest Lyapunov exponent approach. Important features and properties of each of the three classes of diagrams with one, two and three varying parameters in the 1D, 2D and 3D cases, respectively, are presented and illustrated via certain diagrams of the K values, $-1\le K\le 1$, from the 0–1 test and the sample entropy values $SaEn>0$. The K values close to 0 indicate periodic and quasi-periodic responses, while those close to 1 are for chaotic ones. The sample entropy 3D diagrams for an electric arc system are also provided to illustrate the variety of possible bifurcation diagrams available. We also provide a comparative study of the diagrams obtained using different methods with the goal of obtaining diagrams that appear similar (or close to each other) for the same dynamical system. Three examples of such comparisons are provided, each in the 1D, 2D and 3D cases. Additionally, this paper serves as a brief review of the many possible types of diagrams one can employ to identify and classify time-series obtained either as numerical solutions of models of nonlinear dynamical systems or recorded in a laboratory environment when a mathematical model is unknown. In the concluding section, we present a brief overview of the advantages and disadvantages of using the 1D, 2D and 3D diagrams. Several illustrative examples are included.

1. Introduction

Bifurcation diagrams of nonlinear dynamical systems can be of different natures, and various tools can be applied to create such diagrams. Typical diagrams showing oscillatory responses may use the largest Lyapunov expenent (LLE) method [1], 0–1 test for chaos [2,3,4,5,6] (see Appendix A) various definitions of entropy [7,8,9,10,11,12], statistical quantities and hypothesis testing [13,14,15,16,17] or machine learning methods [18]. These methods, in spite of the fact that they are based on different mathematical concepts, can be used to identify and classify the oscillatory behavior of nonlinear systems described by nonlinear ordinary differential equations (ODEs) or, sometimes, of the time series recorded in a laboratory setting when a mathematical model is unknown. The LLE method has been known for a long period of time and is well-researched. Entropy methods have been applied for the analysis and identification of time series, followed by the 0–1 test approach and hypothesis testing and machine learning approaches. Obvious questions arise when applying such different methods to a particular nonlinear system: do such methods provide the same (or similar) results and conclusions about the examined system? What internal parameters (or constants) should be chosen in these methods to derive and obtain an acceptable outcome about the system under investigation? For example, what parameters $n_{cut}$ and $\overline{N}$ (see Appendix A) should be chosen in the 0–1 test method to obtain the same conclusion when certain m, r and N values are used in the sample entropy (SaEn) method? Are the illustrative diagrams representing the behavior of nonlinear dynamical systems close to each other when different methods are used? This paper examines such issues, and, at least partially, attempts to provide a comparative analysis of the above-mentioned methods. This is one of the goals of the current analysis. Another goal is to look at those methods when we move from one varying parameter in a dynamical system to two, and further to three parameters, thus analyzing the same dynamical system in our notation of the 1D, 2D and 3D bifuraction diagrams.

Although the analyzed methods can be used in many nonlinear dynamical systems in engineering, science and economics, we decided to focus on two dynamical systems that differ significantly from each other. First, they come from different engineering and science areas (electrical and biochemical). Second, the mathematical models have different numbers of parameters that may vary. The electrical arc system (see Equation (A6) in Appendix B) has only three parameters, R, L and C, when m is kept constant, while the calcium oscillating system (see Appendix C) has seventeen parameters. The nonlinear systems under consideration are as follows:

• the electric arc RLC system described in ref. [18,19,20,21,22],
• the calcium system given in ref. [8,9,23,24,25,26].

We present five different diagrams in the 1D case in Figure 1, while applying the 0–1 test and sample entropy methods in these systems to the 2D and 3D cases in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7.

An exploration of diverse types of trajectories in nonlinear dynamical systems using the concepts of persistency, regularity, intermittency and transiency (in addition to chaos) is presented in a recent paper [27].

2. One-Dimensional Bifurcation Diagrams

Five typical 1D bifurcation diagrams are shown in Figure 1. These are different from the 1D diagrams for the nonlinear system presented in Appendix B, in which resistance R is treated as a variable parameter changing in the range $R\in [5,25]$ with 1000 discrete values (step size $\Delta R=0.02$). Two other parameters, the capacitance C and inductance L, were kept constant at $C=3.14$ and $L=1$. The vertical axis in Figure 1a shows the local maximum values of the current $i_{\theta }$ with the solutions (for each discrete value of R) obtained for $t\in [0,2000]$, but with the period $[0,500]$ being discarded. Thus, the identification of the maximum values was performed in the period $t\in [500,2000]$. The Runge–Kutta IV method was used to integrate the nonlinear system of ODEs describing the arc circuit. Figure 1b shows the corresponding LLE, while Figure 1c shows the diagram obtained using the 0–1 test with $-1\le K\le 1$ and $K\approx 0$ indicating periodic and quasi-periodic responses, while $K\approx 1$ stands for chaotic ones. Next, Figure 1d depicts the sample entropy diagram. Finally, Figure 1e shows the LD diagram in which the red horizontal segments correspond to the chaotic intervals with the LD close to but greater than 2. Note that each of the above diagrams provides slightly different details about the obtained responses. For example, it is clear from Figure 1a that increasing R from the value of 5 to the middle of the interval $[\alpha ,\beta ]$ gives periodic responses with the period doubling bifurcation; that is, the period-1 response changes to period-2, then period-4, etc., eventually becoming a chaotic one. The period-5 response occurs around a narrow window with $R=\delta$. Then, the period-3 responses occur for the values $R>\zeta$, changing further to period-6, period-12, etc., with increased values of R. Analyzing the LLE diagram in Figure 1b, we obtain periodicity at the above-mentioned values of R without the period-doubling phenomenon. Moreover, based on Figure 1a, one can easily identify the maximum values of the periodic and chaotic responses, while such an identification based on the diagram in Figure 1b is not possible. For example, for $R=7$, we obtain the period-2 response with two local maximum values at around $i_{\theta ,max}$ equal 4.4 and 3.5. For R values slightly greater than $\zeta$, we have a period-3 response with three local maximum values at around $i_{\theta ,max}$ equal to 5, 3.5 and 2.4. Overall, the diagram in Figure 1a seems to convey more information about the system under investigation than the diagram in Figure 1b.

The five types of 1D diagrams are not the only ones that can be drawn. Other possibilities are the bifurcation diagrams showing the changing frequency of oscillations, numbers (positive integers) of local maximum values in one period, or certain statistical parameters [6,14,15,16,17].

3. Moving from 1D to 2D Bifurcation Diagrams

Going from 1D to 2D diagrams with two changing parameters is often challenging computationally. Such calculations often require dedicated accelerators working in hierarchical environments and a hybrid programming model; e.g., the MPI+OpenMP [6]. When one parameter changes (as in Figure 1) with 1000 discrete values (note that the step size $\Delta R=0.02$ in Figure 1) with two changing parameters, we obtain ${10}^6$ discrete points in some rectangular area of the changing parameters. For each discrete point (out of one million of them), we have to solve our nonlinear system, perform identification of the types of steady-state solutions and draw a color diagram representing these solutions. In this environment, the use of the diagram of the type shown in Figure 1a is problematic, as, in addition to the periodic/chaotic representation, we would have to add another dimension in which the local maxiumum values should be stored. This is impossible to do in the 2D case. Thus, one should rather use other types of diagrams; for example, the sample entropy method (see Appendix D) or the 0–1 test, as shown in Figure 2 and Figure 3. Figure 2 is interesting because it shows relatively similar (to the naked eye) diagrams obtained using conceptually different methods, namely the sample entropy and 0–1 test methods. Note that further, somewhat artificial adjustment (or rescaling) of the $SaEn$ values could bring the sample entropy diagram very close to the 0–1 test diagram. The brighter areas in the sample entropy diagram would become close to the white areas in the 0–1 test diagram, while the darker areas in the sample entropy diagram would become close to the black areas in the 0–1 test diagram.

Figure 3 shows a series of 2D bifurcation diagrams of the nonlinear calcium model considered in ref. [8] (see Appendix C), which is quite interesting with a possibility of changing seventeen parameters. The two parameters used to obtain the diagrams in Figure 3 were the pairs of $K_{ch}$, $K_{ER,ch}$ and $k_{ER,pump}$ parameters, with various ranges of changes. Thus, the rectangular areas have different sizes, but each of the six diagrams in Figure 3 contains the K values from the 0–1 test for ${10}^6$ points, where $0\le K\le 1$. These values are represented by various shades of the grey color, as given by the vertical bars on the right-hand sides of each diagram. Other details of the computations are given in the captions in Figure 3. The blue color is used in Figure 3 to represent non-oscillatory solutions (constant steady-state values)—the solutions converge to constant stable equilibria for the parameters in the blue areas.

Note that having obtained the 2D diagrams in Figure 3, one can easily create 1D diagrams; for example, along line a in Figure 3a with fixed $K_{ch}$ and varying $1500\le K_{ER,ch}\le 4500.$ The obtained 1D diagram along the line a will be of the type shown in Figure 1b. Thus, each 2D diagram in Figure 3 corresponds to 1000 1D diagrams of the type shown in Figure 1b, as 1000 discrete points were used for the variable $K_{ch}$ between its lower value 2.0 and the highest value of 5.5.

As mentioned before in the Introduction, one can also create 2D bifurcation diagrams using other measures,;for example, the frequency diagrams or 2D arrays (with positive integers) representing the numbers of local maximum values in one period [9]. Other types of 2D bifurcation diagrams are also possible; for example, as an extension of the concept of the pseudo-periodic surrogate series to 2D and the hypothesis testing diagrams presented in ref. [14].

4. Three-Dimensional Bifurcation Diagrams

Moving one step further to three varying parameters, one can obtain 3D bifurcation diagrams. For example, if we simultaneously vary the three parameters used to obtain the diagrams in Figure 3 in some hypothetical cube of $K_{ch}\times k_{ER,ch}\times k_{ER,pump}$ with 1000 discrete points for each of the three parameters, then the 3D cube will contain ${10}^9$ discrete points. For each of these billion points, one has to solve the underlying nonlinear ODE system and identify the type of response (periodic, chaotic or constant). Computational effort increases exponentially.

Two 3D bifurcation diagrams (for the 0–1 test) for the nonlinear RLC arc system are shown in Figure 4, and three 3D diagrams for the nonlinear calcium system are shown in Figure 5. These diagrams were created using either $100\times 100\times 100$ or $200\times 200\times 200$ discrete points (see the captions of both figures). The diagram in Figure 4b of size $200\times 200\times 200$ is the one for the range of the three parameters R, L and C determined by the green cube in Figure 4a. The entire diagram in Figure 4a of size $100\times 100\times 100$ is of a poorer resolution compared to that in Figure 4b, which is clearly noticeable by comparing the quality of the front wall of the two diagrams (for $L=0.2$). Making suitable cuts along chosen planes in both 3D diagrams in Figure 4, one can obtain 2D diagrams in the same way one obtains a 1D diagram (of any type shown in Figure 1) from a respective 2D diagram.

The 3D diagrams in Figure 5 are for the nonlinear calcium system. Figure 5a,b show the responses in the same rectangular cube, but with different resolutions, while Figure 5c is the diagram in the rectangular green cube marked in Figure 5b. Note that the eight-fold increase in the number of discrete points in the cube in Figure 5b in comparison to the number of discrete points in the cube in Figure 5a results in an approximately eight-fold increase in the computation time, but it also provides better-quality results.

Next, the four diagrams in Figure 6 show only selected results from those shown in Figure 4a. Namely, Figure 6a includes only those points for which strongly chaotic responses were obtained with $K\in [0.9,1]$, while Figure 6c includes only those points for which periodic responses with $K\in [0,0.1]$ were detected. Similarly, Figure 6b includes only those points for which strongly chaotic responses were obtained with $K\in [0.99,1]$, while Figure 6d includes only those points for which periodic responses with $K\in [0,0.1]$ were detected. It is easy to notice that the points in both Figure 6a,c, when combined together, fill up almost the entire 3D cube. Thus, the test 0–1 indicates that other responses, those with $K\in [0.1,0.9]$, are almost non-existent in the chosen 3D cube.

Finally, Figure 7 shows 3D bifurcation diagrams for the 0–1 test and sample entropy methods applied to the electric arc circuit. The two resolutions used clearly indicate that the eight-fold increase in the number of discrete points in the assumed cube leads to a better-quality diagram with smooth transitions between the sample entropy values. However, as shown in Appendix E, the computation time to obtain the diagram in Figure 7c is about 7.3 times greater than that in Figure 7b. On the other hand, with the same number of discrete points in the diagrams in Figure 7a,b, the sample entropy method requires about twice as much time as the 0–1 test method. Going one step further to see the inside of any 3D diagram (with millions of discrete points), one can design a relatively simple code to obtain several—say, 5, 10 or 20—2D crosscuts inside a cube (that is, 5, 10 or 20 two-parameter diagrams), rotate them at various angles to align them into preferred perspectives, and further zoom in on parts of the cube (3D diagram).

5. Conclusions

Bifurcation diagrams show different features of analyzed dynamical systems depending on the number of parameters being varied—one, two or three—as well as of the type of diagram, as illustrated in Figure 1 and Figure 2. Is it also possible to create other types of diagrams; for example, using the pseudo-periodic surrogates and hypothesis testing parameters from statistics or machine learning-based methods [13,14,18]. In certain cases, using two different types of diagrams, one can obtain almost identical results; for example, considering the periodicity–chaoticity, as shown in Figure 1. In other cases, it is difficult to compare the diagrams when using parameters of different natures and mathematical foundations; for example, the 0–1 test and sample entropy. One can find an interesting comparison of the diagrams obtained using the 0–1 test and sample entropy in ref. [7,9]. The difficulty in comparing is due to the uncertainty of the correctness of the chosen parameters in both methods; for example, the choice of the m, r and N parameters in the sample entropy method [7].

Another important issue is the computational effort needed to create the diagram. The effort (time of computation, storage requirement) increases exponentially with moving from 1D to 2D and then further to 3D. With increased resolutions (decreased step size of the changing parameters), in 2D and 3D, one should consider parallelization of computation to decrease the computational effort; preliminary results have been reported in [20,21]. The computational times for the 3D diagrams presented in this paper are shown in Appendix E. When choosing a particular method for analyzing and identifying time series, one needs to consider the computational time and memory requirements, the availability of computational codes and types of machines, as well as what information should be derived regarding the analyzed time series. Notice, for example, that the parallel computation system needed approximately 477,600 s (=132.7 h = 5.5 days) to create just one $200\times 200\times 200$ diagram, shown in Figure 7c.