Simple polynomial classes of chaotic jerky dynamics

Abstract

Third-order explicit autonomous differential equations, commonly called jerky dynamics, constitute a powerful approach to understand the properties of functionally very simple but nonlinear three-dimensional dynamical systems that can exhibit chaotic long-time behavior. In this paper, we investigate the dynamics that can be generated by the two simplest polynomial jerky dynamics that, up to these days, are known to show chaotic behavior in some parameter range. After deriving several analytical properties of these systems, we systematically determine the dependence of the long-time dynamical behavior on the system parameters by numerical evaluation of Lyapunov spectra. Some features of the systems that are related to the dependence on initial conditions are also addressed. The observed dynamical complexity of the two systems is discussed in connection with the existence of homoclinic orbits.

Introduction

It is well known [1], [2], [3], [4], [5], [6], [7], [8] that chaotic behavior in time-continuous, autonomous dynamical systems requires a phase-space dimension which is equal to three or larger and nonlinearities in the model equations. It is, however, not definitely known, not even for the simplest case of three-dimensional phase-spaces, what degree of nonlinearity is necessary for the creation of chaos. So, it is tempting to pose the following question: What are the most elementary functional forms of three-dimensional dynamical systems that do exhibit chaotic behavior?

For polynomial vector fields, this problem has been attacked in several recent publications and seems to be related not only to the number and degree of nonlinear monomials but also to the total number of appearing terms. Zhang and Heidel [9], [10] have investigated the possibility of chaotic dynamics in systems with only quadratic nonlinearities. They found out that quadratic systems with a total of less than five terms on the right-hand side of the corresponding three first-order differential equations cannot exhibit chaotic behavior.¹ Their work was motivated by former studies of Sprott [11], [12]. Sprott has used a computer search procedure to find systems that are functionally as simple as possible but nevertheless chaotic [11]. He was able to identify 19 distinct chaotic models with vector fields that consist of five terms with two quadratic nonlinearities or of six terms with one quadratic nonlinearity. In subsequent studies [12], [13], Sprott even reported a chaotic model with a total of only five terms where one is a quadratic nonlinearity. All these models are algebraically simpler than the prominent models of Lorenz [14] and Rössler [15].

A further simplification was achieved in a recent study [16] by transforming the Sprott models to explicit third-order ordinary differential equations, so-called jerky dynamics [17], [18], [19], [20], [21], [22], [23], [24],

$$\text{x}\text{⃛}\text{=J(x,}\text{x}\text{̇}\text{,}\text{x}\text{̈}\text{),}$$

where the overdots represent the derivatives with respect to time t. This constitutes a simplification in the following sense. Defining $\text{x}\text{̇}\text{=y}$ and $\text{x}\text{̈}\text{=z}$, any jerky dynamics can obviously be rewritten as a dynamical system [25]

$$\text{x}\text{̇}\text{=y,}\text{y}\text{̇}\text{=z,}\text{z}\text{̇}\text{=J(x,y,z).}$$

The first and second component of this system are algebraically very simple (in fact are the simplest ones for an effectively three-dimensional dynamical system) and are identical for all jerky dynamics. The third component consists of the, in general, nonlinear jerk function J(x,y,z). After transforming a three-dimensional dynamical system to a jerky dynamics, solely the number and types of terms appearing in the jerk function reflect the complexity of the linear and nonlinear couplings between the different variables of the original dynamical system. In this sense, one can understand the result that different dynamical systems with the same total of terms and nonlinearities can lead to jerky dynamics with different numbers of terms and nonlinearities [16]. Therefore, the functional complexity of different dynamical systems can be compared more directly and more clearly if these systems are recast as jerky dynamics. From this point of view, functionally elementary jerky dynamics, that exhibit chaos, also constitute the simplest chaotic three-dimensional dynamical systems.

Apart from this, the representation of dynamical systems as jerky dynamics has another useful and important advantage: based on their jerky dynamics, a classification of different dynamical systems is possible, because the transformation of certain functionally different three-dimensional dynamical systems can lead to the same jerky dynamics. In [16], it has been shown that 16 of the 19 simple chaotic Sprott models [11] and the toroidal model of Rössler [26] can be classified into seven distinct basic classes of jerky dynamics (labeled by JD₁–JD₇) with different number and types of quadratic nonlinearities. The two simplest resulting classes of jerky dynamics are JD₁, i.e.,

$$\text{x}\text{⃛}\text{=k}{}_1\text{x}\text{̈}\text{+k}{}_2\text{x+x}\text{x}\text{̇}\text{+k}{}_3\text{,}$$

and JD₂, i.e.,

$$\text{x}\text{⃛}\text{=k}{}_1\text{x}\text{̈}\text{+k}{}_2\text{x}\text{̇}\text{+x}{}^2\text{+k}{}_3\text{,}$$

possessing only one constant, two linear and one quadratic term in their jerk functions $\text{J(x,}\text{x}\text{̇}\text{,}\text{x}\text{̈}\text{)}$. These two classes alone contain nine of the original models. Moreover, also Sprott's simplest model which has already been given as a jerky dynamics [12], [13] fits into these classes. It belongs to JD₁ as the special case where the constant term k₃ is zero. The model JD₂ has been introduced for the first time in the form $\text{x}\text{̈}\text{+β}\text{x}\text{̇}\text{+x=η}$, $\text{η}\text{̇}\text{=μx(1−x)}$ as a forced oscillator that displays a transition to chaotic behavior for appropriately chosen parameter values [27].

The jerk classes JD₁ and JD₂, , , are the algebraically simplest polynomial classes of dynamical systems that are known to exhibit chaotic behavior. Therefore, a detailed study of the dynamical properties of these two systems is interesting and important. In the slightly different functional form $\text{x}\text{⃛}\text{+}\text{x}\text{̈}\text{+b}\text{x}\text{̇}\text{−cx+x}{}^2\text{=0}$ that can be obtained from Eq. (4) by a translation of the origin, the class JD₂ has already been investigated as an example for the complicated local bifurcation structure that appears in the neighborhood of homoclinic orbits at a fixed point of saddle-focus type and its consequences for global bifurcation schemes near the homoclinicity including the occurrence of chaos [28], [29], [30]. Here, however, we are mainly interested in a global picture of the dynamical properties of the two classes JD₁ and JD₂, rather than in detailed local analysis of very small parameter ranges. Specifically, we want to detect the distinct regions of the parameter space with different character of the long-time dynamics, i.e., where the dynamics approaches a fixed point, a limit cycle, a chaotic strange attractor, or diverges. The interpretation of our results, though, will bring us back to the investigations of Glendinning and Sparrow [29] and Arnéodo et al. [30].

Several different points in the parameter space of JD₁ and JD₂ with chaotic dynamics (for specific initial values) are already known [16], [27], [28], [29], [30]. In [16], they were obtained from the original models [11], [12], [26] that belong to the corresponding basic class of jerky dynamics. The predominantly numerical study presented here will complete this very fragmentary picture. In this context, we also address the dependence of the dynamical behavior on initial conditions. Since the two classes JD₁ and JD₂ are derived from several different three-dimensional dynamical systems by globally invertible transformations [16], the understanding of the dynamical behavior of JD₁ or JD₂, respectively, simultaneously covers the corresponding dynamics of the original models by Sprott [11], [12] and Rössler [26].

This paper is organized as follows. Section 2 is devoted to a general discussion of several analytical methods that will be used later on to derive some properties of the investigated jerky dynamics. Then, these found properties are utilized to focus the numerics on the interesting regions in the parameter space. The reasoning behind the numerical methods are also described in Section 2. In Section 3, the jerk class JD₁ is studied in detail by applying the analytical and numerical tools of Section 2. The same procedure is applied to the class JD₂ in Section 4. Section 5 contains the summary and discussion of our results.