Dynamical System of Scalar Field from 2-Dimension to 3-D and its Cosmological Implication

We give the three-dimensional dynamical autonomous systems for most of the popular scalar field dark energy models including (phantom) quintessence, (phantom) tachyon, k-essence and general non-canonical scalar field models, change the dynamical variables from variables $(x, y, \lambda)$ to observable related variables $(w_{\phi}, \Omega_{\phi}, \lambda)$, and show the intimate relationships between those scalar fields that the three-dimensional system of k-essence can reduce to (phantom) tachyon, general non-canonical scalar field can reduce to (phantom) quintessence and k-essence can also reduce to (phantom) quintessence for some special cases. For the applications of the three-dimensional dynamical systems, we investigate several special cases and give the exactly dynamical solutions in detail. In the end of this paper, we argue that, it is more convenient and also has more physical meaning to express the differential equations of dynamical systems in $(w_{\phi}, \Omega_{\phi}, \lambda)$ instead of variables $(x, y, \lambda)$ and to investigate the dynamical system in 3-Dimension instead of 2-Dimension. We also raise a question about the possibility of the chaotic behavior in the spatially flat single scalar field FRW cosmological models in the presence of ordinary matter.


Introduction
Scalar field models have played a vital role in cosmological theoretical studies in nearly half a century. Those assumed scalar fields appeared in different cosmological research aspects to settle different cosmological problems [1], such as to drive inflation, to explain a time variable cosmological ′ constant ′ and so on. The scalar fields have played another important essential role in the past nearly fifteen years as a candidate of dark energy after the discovery of the accelerating expansion of universe. There are so many phenomenological dark energy models of scalar fields, such as quintessence, phantom, quintom and the scalar fields with non-canonical kinetic energy term (for a review, see [2,3]).
To study the dynamical evolution of those scalar fields models and their cosmological implications with a phase-plane analysis is a very useful and common method [4]. However, most of those works only focus on the quintessence models(including phantom quintessence and quintom) with unique exponential potential and tachyon models(including phantom tachyon) with inverse square potential, and correspondingly, the dynamical systems are two dimensional autonomous system with those special form of potentials(see the references cited in [7,23]). Using a method which considers the potential related variable Γ as a function of another potential related variable λ (see Eq. (10) for the definition of Γ and λ) [7,23,38], we are able to analyze the phase-plane of the dynamical systems of the quintessence and tachyon models with many different potentials.
When the potentials are beyond the special type such as exponential or inverse square potentials, the dynamical systems consequently become a three-dimensional autonomous systems. This method is quite effective and powerful, it therefore has been generalized to several other cosmological contexts [8,9,10,11,12,13,14,15,16,37,17]. However, there is very few work focusing on the dynamical behavior of the scalar field with a general modified kinetic term, such as k-essence (L = V (φ)F (X)) and general non-canonical scalar field (L = F (X) − V (φ)). Recently, Josue De-Santiago et. al analyzed the dynamical system of general non-canonical scalar field with the lagrangian L = F (X)−V (φ) and studied the phase plane after a suitable choice of variables [32]. They obtained the three-dimensional autonomous system of this non-canonical scalar field after specifying the kinetic term as F (X) = AX η and choosing the potential as V (φ) = V 0 (φ − φ 0 ) 1/(1−Γ) (i.e., the special case that potential related parameter Γ is a constant) and studied the critical points as well as their stability.
Motivated by the work, in this paper we try to extend our works in [7,23] to give the threedimensional autonomous dynamical systems for most of the popular scalar field dark energy models including (phantom) quintessence, (phantom) tachyon, k-essence and general non-canonical scalar field models. We will show that the three-dimensional autonomous systems of general non-canonical scalar field and k-essence will reduce to the quintessence and tachyon scalar field respectively. Not like the previous works, here we express the three dimensional autonomous systems from the trivial variables (x, y, λ) to the observable related variables (w φ , Ω φ , λ). It will be very convenient to investigate the dynamical properties of the autonomous system based on the observable related variables w φ and Ω φ (see [18,19,20,21,22,23,24] and a recent paper about the general property of dynamical quintessence field [25]). Since the definition of the trivial variables x and y could vary with different scalar field models, while the observable related variables such as the equation of state w φ and the dark energy density parameter Ω φ are the same for different dark energy models.
The paper is organized as follows. We firstly present the basic theoretical framework for (phantom) quintessence, (phantom) tachyon, k-essence and general non-canonical scalar field models in section 2, and try to give the relationships between those different scalar fields in this section. We then give the three dimensional autonomous dynamical systems for those scalar fields and switch the dynamical variables from (x, y) to (w φ , Ω φ ) in each subsection of section 3. Additionally, using the dynamical systems, we give the exact solution of w φ and Ω φ for a special case of tachyon model when the potential is chosen to be a constant in subsection 3.2. We show that the dynamical autonomous system of k-essence can reduce to tachyon model, and investigate another special case called kinetically driven quintessence with the lagrangian p(X, φ) = f (φ)(−X + X 2 ) detailedly in subsection 3.3. In subsection 3.4, we show that the dynamical autonomous system of general noncanonical scalar field can reduce to quintessence and tachyon model respectively for some special cases. We also studied two special cases of purely kinetic united model L = F (X) in detailed in this subsection. We try to give the cosmological implications of the three-dimensional dynamical autonomous system and present the conclusion in section 4. We proved in section 4 that the dark energy density parameter Ω φ would obey the same differential equation for all the non-coupled dark energy model under the GR frame. We also raise a question about the possibility of the chaotic behavior in the spatially flat single scalar field FRW cosmological models in the presence of ordinary matter.

Basic Framework for various Scalar Fields
Let us restrict ourselves to a flat universe described by the FRW metric, and consider a spatially homogeneous real scalar field φ with non-canonical kinetic energy term. The lagrangian density is given as where L is a function of X and potential V (φ), X = 1 2 ∇ µ φ∇ µ φ = 1 2φ 2 for a spatially homogeneous scalar field. The pressure, energy density and sound speed of the scalar field could be easily obtained as following: where 8πG = κ 2 = 1/M 2 pl , ρ b is the density of a barotropic fluid component with the equation For quintessence, general non-canonical scalar field, tachyon and K-essence model, the pressure p and energy density ρ are respectively: If ς = 1, Eq.(6) and Eq.(8) correspond to the quintessence and tachyon scalar field. If ς = −1, Eq.(6) and Eq.(8) correspond to the phantom quintessence and phantom tachyon scalar field.
General non-canonical scalar field (Eq. (7)) can recover to (phantom) quintessence (Eq.(6)) if F (X) = ςφ 2 = 2ςX. K-essence model (Eq.(9)) can recover to (phantom) tachyon (Eq.(8)) if Moreover, if redefined the scalar field, it is demonstrated that K-essence model described by Eq.(9) with a linear kinetic function F (X) = X + 1 can reduce to any quintessence model described by Eq.(6) [5]. It means that any quintessence can be contained into K-essence frame, so each quintessence model is kinematically equivalent to a k-essence model.
The authors also give the relationship between the potentials of the two models. For example, the exponential potential V (φ) = V 0 e −λφ in quintessence model plays the similar role as the inverse We will also show that the role of inverse square potential in k-essence model is very similar with exponential potential in quintessence in the next section.

Dynamical System of various Scalar Fields
In this section, we will give the dynamical system for all the quintessence, tachyon, K-essence and general non-canonical scalar field model. We will summarize the dynamical system analysis and give our comments.

Dynamical System for Quintessence and Phantom Quintessence Scalar Field
For the (phantom) quintessence scalar field with lagrangian L = ς 1 2φ 2 − V (φ), we can define the following dimensionless variables: Where The parameters Γ(φ) and λ(φ) of the potentials can be related with the famous slow roll parameters ǫ V and η V (e.g., see [6]): Using Eq.(4), Eq.(5),Eq.(6)and Eq. (7), We can write down the following equations for the evolution of the (phantom) quintessence: where N = ln(a), a is the scale factor. ς = 1 or −1 for quintessence and phantom quintessence model. Here we should emphasize that Eqs. (13)(14)(15) is not a dynamical autonomous system since the parameter Γ(φ) is unknown. The energy density fraction of dark energy scalar field is while the equation of state of the dark energy scalar field is In the other hand, it is more convenient to rewrite the dynamical system Eqs. (13)(14) from the dependent variables (x, y, λ) directly to the observable quantities (Ω φ , γ φ , λ) [18]: Eqs. (18)(19)(20) can reduce to quintessence when ς = 1 (Eqs (3)(4)(5) in paper [19] or Eqs. (17)(18) in paper [18]) and reduce to phantom quintessence when ς = −1 [20]. Eqs. (18)(19)(20) are very useful to study the cosmological implication of the evolutional behavior of the dynamical system because the dynamical variables Ω φ and γ φ are the observable quantities. For example, dγ φ /dN > 0 corresponds to the thawing model and dγ φ /dN < 0 corresponds to the freezing model of the evolution of equation of state of dark energy [21,22].
If Γ = 1, Eq.(15)(or Eqs. (20)) will disappear, Eqs.(13-14)(or Eqs. (18)(19)) will become a twodimensional dynamical autonomous system. Γ = 1 corresponds to the exponential potential V 0 e −λφ , which has been studied in many literatures(e.g., see the references of paper [7]). However, the system described by Eqs.(13-15)(or Eqs. (18)(19)(20)) are not a dynamical autonomous system for other potentials because the potential related parameter Γ is unknown. Since λ is a function of quintessence scalar field φ and Γ is also a function of φ, then Γ can generally be expressed as a function of λ. So if we consider Γ as a function of λ, namely Γ = Γ(λ), then Eqs.(13-15)(or Eqs. (18)(19)(20)) are definitely a dynamical autonomous system, we therefore can study its properties and dynamical evolution using the phase plane and critical points analysis. Moreover, considering Γ as a function of λ can cover many potentials beyond the exponential potential [7].
If Γ = 3/2, Eq.(24)(or Eqs.(29)) will disappear, Eqs.(22-23)(or Eqs. (27)(28)) will become a two-dimensional dynamical autonomous system. For (phantom) tachyon scalar field, Γ = 3/2 corresponds to the case that the form of potential is inverse square potential. But for other potentials, the system described by Eqs.(22-24)(or Eqs. (27)(28)(29)) will not be a dynamical autonomous system any more since the potential related parameter Γ is unknown, and we can not exactly analyze the evolution of universe like the inverse square potential any more. However, since λ is the function of tachyon field φ and Γ is also the function of φ, Γ can be generally expressed as a function of λ.
So as the method used in [7], we can consider Γ as a function of λ, Γ = Γ(λ), then Eqs.(22-24)(or Eqs. (27)(28)(29)) will become a dynamical autonomous system, then we can study its properties and dynamical evolution using the phase plane and critical points analysis. For each form of function Γ(λ), we can figure out the detailed form of potential, so this method can cover many potentials beyond the inverse square potential [23].
For the most simple case of potential V (φ) = V 0 , λ = 0, Eqs.(27-29) become a very simple differential equation: We can get the exact solution for the above equations: Tachyon scalar field with this constant potential had been studied in [26]. According to the best-fit values of the parameters they obtained, we can obtain the value of the integral constant c 1 = 0.0082 and c 2 = 0.59 here. We know from Eq. (31), when N → +∞, we have γ φ → 0 and Ω φ → 1, the universe will be a de Sitter like universe filled with the tachyon scalar field.

Dynamical System for K-essence Scalar Field
For the k-essence scalar field with lagrangian L = −V (φ)F (X), we define the following dimensionless variables as the same as Eq. (21): Using Eq.(4), Eq. (5),Eq. (6)and Eq.(32), we can get following dynamical system: The density parameter of tachyon scalar field Ω φ , the equation of state w φ are: Where F ′ = dF (X)/dx and F ′′ = d 2 F (X)/dx 2 , they both are the functions of x. Using Eq. (36) and Eq. (37), we can also rewrite the dynamical system Eq.(33), Eq.(34) and Eq.(35) from the dependent variables (x, y, λ) directly to the observable quantities (Ω φ , γ φ , λ): where Ξ = XF XX /F X = (xF ′′ − F ′ )/2F ′ . We have pointed out the relationship between quintessence and K-essence in previous section. k-essence model described by Eq.(9) with a linear kinetic function F (X) = X + 1 can reduce to any quintessence model described by Eq.(6) [27]. So all quintessence model with any potentials can be considered as the special cases of k-essence model. Moreover, it proved that the correspondence of the exponential potential V (φ) = V 0 e −λφ in quintessence model is exactly the inverse square potential V (φ) = ( 1 2 κλφ−c 1 ) −2 in k-essence model. We think this is the reason that the dynamical system Eqs.(13-15)(or Eqs. (18)(19)(20)) of quintessence reduces to two-dimensional autonomous system for exponential potential(Γ = 1) while the same situation happens for inverse square potential (Γ = 3/2) in tachyon and k-essence model.
Let us discuss whether Eqs. (38)(39)(40) could be considered as an autonomous system. Firstly, we realized that the variables x and y still appear in Eq.(39) because we do not know the detailed form of function F (X) and can not figure out the solution of x and y. However, from Eqs. (36)(37), we are sure that variables x and y are the functions of variables Ω φ and γ φ . Since it is also a function of Ω φ and γ φ . Secondly for the potential related parameter Γ, if we consider Γ as a function of λ just similar with tachyon scalar field in subsection 3.2, Eqs. (38)(39)(40) can eventually become a three-dimensional dynamical autonomous system for any k-essence models, and then we can easily study the critical points and the dynamical evolution beyond the inverse square potential.
Eq.(45) will vanish when Γ = 3/2, then Eqs.(43-44) is a two-dimensional autonomous system which describes the dynamical evolution of the kinetically driven quintessence with the lagrangian In [29], authors obtained the twodimensional dynamical autonomous system with the dimensionless variables (x, y) for this type of k-essence model. They studied the phase-space properties and the cosmological implications of the critical points in detail. However, here we give the two-dimensional autonomous system Eqs. (43)(44) with the observational quantities (Ω φ , γ φ ) instead of the variables (x, y). We can obtain the critical points of the observational quantities (Ω φ , γ φ ) directly, so it will be more convenient to study the properties of the critical points and their cosmological implication with these new variables. Furthermore, If we consider Γ as a function of λ, we can study the critical points and the evolution of the universe beyond the inverse square potential, just like the method used in [23].  Figure 2: The evolution of c 2 s with respect to N . Potentials and initial conditions are the same as Fig.1.
It is interesting that w φ could be larger or less than −1 for this kind of k-essence model (Fig.1).
We have solved numerically Eqs. (43)(44)(45) for several different potentials and plot the evolution of w φ crossing the Phantom Line in Fig.1. However, when the equation of state w φ crosses the Phantom Line, the effective sound speed of perturbations c 2 s will change its sign from positive to negative simultaneously (Fig.2). For the stability with respect to the general metric and matter perturbation, the condition c 2 s ≥ 0 is necessary, so the background models with c 2 s < 0 are violently unstable and do not have any physical significance. Therefore this model of transition is not realistic [30,31].
The curves in yellow, blue, green and red in Fig.1 and Fig.2 are for Γ = 0, 1, 3/2, 2, corresponding to the potentials being the form of The initial conditions for these four potentials are the same, γ φ0 = 0.1, Ω φ0 = 0.7 and λ = 0.1 when N = 0(at present time).
For the case that the equation of state w φ is near −1(so γ φ ∼ 0), we can drop terms of higher order in γ φ , then we get a simple differential equation for γ φ with the dependent variable from N to Ω φ from Eqs. (43)(44): We assume that λ is approximately constant(= λ 0 ) when γ φ is near 0, so that the above equation can be solved exactly: to Ω φ in Fig.3 and Fig.4.
The solid curves(red and black) in Fig.3 and Fig.4 show the general relationship(Eq. and 0.4(black) in Fig.4. We find that the red solid curve and the dashed color curves around it in both Fig.3 and Fig.4 are more close to the Phantom Line(w φ = −1). So the smaller the initial value of |λ| is(i.e., the more flat the potential is), the less deviation the equation of state w φ is from −1. It is interesting that, for different potentials V 0 e cφ , V 0 (φ + c) −2 and V 0 (φ + c) −1 , the state of equation w φ can larger or less than −1, and also can increase or decrease with respect to Ω φ , only depending on the initial value of λ(determined by the initial value of φ). Unlike the evolution in Fig.1 and Fig.2, w φ plotted in Fig.3 and Fig.4 does not cross Phantom Line due to the different choice of the initial conditions.

Dynamical System for general non-canonical scalar field
In the last subsection of section 3, we consider the general non-canonical scalar field with a lagrangian L = F (X)−V (φ) filled in a spatially flat Fridmann-Lemaitre-Robertson-Walker(FLRW) cosmology. The motion of this scalar field and the evolution of the universe are described by Eqs. (4)(5) and Eq. (7).
In order to obtain the autonomous system we define the variables as follows [32] † : Using Eq.(4-5), Eq. (7)and Eq.(49), we get the following dynamical system: where From Eq.(53), we can get following relationships: We think the definition of variables in [32] can not work if 2XFX − F < 0. Take a very simple example of phantom quintessence with F = −X, 2XFX − F = −X < 0, then the definition of variables in [32] is undefined.
We take two simple examples to check the correctness of Eqs. (55)(56)(57) and study its properties.
Another example is that, if the potential V (φ) = −1 in K-essence and the Potential V (φ) = 0 in general non-canonical scalar field, these two scalar field models will have the same form of lagrangian L = F (X), then both will reduce to the so-call purely kinetic united model [33,34]. In this case, dλ/dN = 0 and dσ/dN = 0, then both Eqs. (38)(39)(40) and Eqs.(55-57) will reduce to a two-dimensional dynamical system as follows: For the case of purely kinetic united model L = F (X), we know from Eq. (49) and Eq.(53) that γ φ is a function of X, γ φ = γ k = F 2XF X −F . In the meantime, Ξ is also a function of X because Ξ = XF XX /F X = (xF ′′ − F ′ )/2F ′ . So generally speaking, Ξ can be expressed as a function of γ φ .
Then the system of Eqs.(58-59) could be a two-dimensional autonomous dynamical system. For some special cases of F (X), we can even get the exact solution for Ω φ and γ φ .
we take two examples to illustrate our viewpoints. The most simple is the case that Ξ is a constant. We know that Ξ = 2−α 2(α−1) = XF XX F X , we can integrate and get the form for F (X): We can also get the exact solution for Ω φ and γ φ from Eqs.(58-59): where α = 1 + 1 2Ξ+1 , C, X 0 , c 3 and c 4 are the integral constants. If we set the equation of state of dark energy γ φ (0) = γ 0 and energy density of dark energy Ω φ (0) = Ω 0 at present N = 0, we then get c 3 = α−γ 0 αγ 0 and c 4 = α(1−Ω 0 ) Ω 0 γ 0 . For the cosmic evolution in very early time, N is negative and |N | is very large, we get γ φ ≈ α from Eq.(61), and then energy density of scalar field will behave as ρ φ ∼ a −3α . For the cosmic evolution in late time, N is positive and very large, γ φ ≈ 0 and Ω φ ≈ 1, scalar field behaves as the cosmological constant. Noted that there is only kinetic term in the lagrangian which gives the cosmological constant solution. Moreover, we know from Eq. (61) that Ω φ might not be 0 in the very early time. Its value depends on the value of α comparing with the value of γ b (γ b = 1 for matter and γ b = 4/3 for radiation). We have plotted the evolution of Ω φ with respect to N for different α and different γ b in Fig.5 and Fig.6, and it is shown that the value of Ω φ in the very early time could be 1, 0 or a positive constant which is less then 1.
The second case is the lagrangian as follows: where A 1 , A 2 , and β are constants. This form of F (X) was proposed in [33,35]. From Eq.(53), we can get that Ξ = (2β − γ φ )/2γ φ , then dynamical system Eqs.(58-59) becomes the following equations: Solving above differential equations, we obtain the following exact solution for γ φ and Ω φ : where c 5 and c 6 are the integral constants, η = 2β 2β−1 , is also a constant. Eq.(65) is very similar but a little different with the evolution described in Eq.(61): For the cosmic evolution in very early time, N is negative and |N | is very large, we get γ φ ≈ η = 2β/(2β − 1) from Eq.(65). The scalar field will mimic the evolution of matter with zero pressure ρ ∼ a −3 in the limit of β → ∞. For the cosmic evolution in late time, N is very large, γ φ ≈ 0 and Ω φ ≈ 1, scalar field behaves as the cosmological constant. So it is the second case that a lagrangian without a potential term gives the cosmological constant solution. Moreover, we know from Eq.(65) that Ω φ might not be 0 in the very early time. It depends on the value of η comparing with the value of γ b (γ b = 1 for matter and γ b = 4/3 for radiation). When N → −∞, we can get and Ω φ → 0 for η < γ b . In this case, it will be interesting to investigate the impact of the scalar field on the evolutional history of the early universe. We have plotted the evolution of Ω φ with respect to N for different η and different γ b in Fig.7 and Fig.8, and it is shown that the value of Ω φ in the very early time can actually be 1, 0 or a positive constant which is less then 1.

Cosmological Implications and Conclusion
The main purpose of the paper is not to analyze the dynamical behavior about different scalar fields in detail, so we will not investigate the detailed critical points and their stable properties for The only difference are the form of function dγ φ /dN . In fact, we can prove that Eq.(66) holds for all non-coupled dark energy models as long as they satisfy the following equations: where subscript i denotes each energy component such as dark energy, matter or radiation. If we set γ de = w de + 1 = p de /ρ de + 1 and Ω de = ρ de /3M 2 pl H 2 , we can obtain following equation from Eqs.(67): Eq.(68) holds for the different scalar field models mentioned in this paper and even all the non-coupled dark energy models which satisfy Eq.(67), so we can conclude that there are only three possible destinies(three types of critical points) for Ω de to be in these models. The value of Ω de will be determined by the stable properties of these three types of critical points, and the stable properties depend on different models and the value of parameters in different models.
Two of them are the cases that Ω de = 0 or Ω de = 1 which are completely opposite destinies.
These two solutions correspond to the universe completely dominated by the scalar field or by the barotropic fluid. The last destiny corresponds to the case of γ de = γ b , and the value of Ω de will vary from different models and be determined by other equations in the dynamical system.
Generally we can obtain 0 ≤ Ω de ≤ 1 and so this solution is called the scaling solution. However, for the scaling solution, the equation of state of dark energy w de is the same as the equation of state of barotropic fluid w b , so there is no accelerating expansion. Since the observation suggested that we are living in a accelerated expanding universe with Ω de ∼ 0.7, none of these three destinies correspond to the present universe we observed. If we try to construct a successful theoretical model which has a dynamical stable solution corresponding to present observational universe(namely a scaling attractor with Ω de ∼ 0.70 and γ de ∼ 0.1 in agreement with observations) to solve or at least alleviate the cosmological coincidence problem without fine-tunings, we must consider the interaction between dark energy and other barotropic fluids(see [37] for such model). This result is valid for not only all the non-coupled dark energy models, but also for many modified gravity models as long as the energy density and the pressure of dark energy or effective dark energy satisfy the continuity equation Eq.(67).
Another important thing we want to emphasize is that, it is more reasonable and more scientific to investigate the dynamical behaviors of a dark energy model under the three-dimensional autonomous system rather than the two-dimensional system. Firstly, the two-dimensional dynamical autonomous system is just a specific case when the potential takes a special form. if we want to completely study the general dynamical properties of a dark energy model, we need to study the system beyond a special potential. Then we can find more critical points than the ones found in a two-dimensional system. We therefore are able to analyze which critical points are possessed by a class of dark energy models and which ones exist only due to the concrete potentials. The method studying the three-dimensional dynamical autonomous system beyond one special potential is originated for the quintessence [7,38] and then developed to other dark energy models [8,9,10,11,12,13,14,15,16,37,17]. Here we extend this method to the more general scalar field models in Section 3. Secondly, from the viewpoint of chaos theory, the dynamical properties of three-dimensional autonomous system is more fruitful than the two-dimensional system. According to the Poincaré -Bendixson theorem, chaos does not exist in any two-dimensional autonomous dynamical system [39,40] but could be possible in three-dimensional autonomous dynamical system.
For a number of three-dimensional systems, such as the famous three-dimensional Lorenz equations which is a model describing the atmospheric convection [41], there exist chaos for certain values of the parameters.
The studies of chaotic dynamics in cosmological models has a long story. Chaotic properties had reported in spatially closed scalar field FRW cosmological models [42,43,44,45,46,47], spatially flat FRW cosmological model with two or more scalar fields [48,49], Bianchi IX universe [50,51], Bianchi I universe [52] and the mixmaster universe [53]. It would be very interesting and also a big challenge for the theoretical study of dark energy if the dynamical systems we consider here(i.e., Eqs. (18)(19)(20),Eqs. (27)(28)(29),Eqs. (38)(39)(40) and Eqs.(55-57) ) exist the chaotic properties. Then the evolution of Ω φ and γ φ will be very sensitive to the initial condition, and therefore predicting their evolution in future becomes totally impossible. However, it is proved that there is no chaotic behavior in spatially flat single scalar field FRW cosmological models [54,55]. Since for the spatially flat case with k = 0, the dynamical system can be described by a three-dimensional autonomous system with a set of variables (H, φ,φ) under a Hamiltonian constraint, so the dynamical system is actually a two-dimensional autonomous system(a andȧ appear only in the combination H = a/a) [56]. We know that for the two-dimensional autonomous systems, there are no enough degrees of freedom to exist chaos, so this proved no-chaotic dynamics in the spatially flat scalar field FRW cosmological model. However, we noted that this result is obtained in the absence of matter and radiation. This may be the case in the very early time when our universe is undergoing an inflation era and completely dominated by the scalar field. However, for the study of dark energy of late-time cosmic acceleration, the component of matter is comparable with the density of dark energy and should not be ignored when we investigate the dynamical behavior of scalar field. In the presence of matter, scale factor a will reappear in the dynamical system beside the variables (H, φ,φ), and then the system can not be reduced to two-dimensional autonomous dynamical system any more. So here we argue that it is still possible for the chaotic behavior in spatially flat single scalar field FRW cosmological models in the presence of matter. It is very like the case of spatially non-flat(k = 0) single scalar field FRW cosmological models where the dynamical system can not be reduced to two-dimensional autonomous dynamical system too. What we argued here is also supported by the equations in section 3( i.e., Eqs. (18)(19)(20), Eqs. (27)(28)(29), Eqs. (38)(39)(40) and Eqs.(55-57)), which described three-dimensional autonomous dynamical systems. However, we are not sure whether there truly exist the chaotic behavior in spatially flat scalar field FRW cosmological models now, to find the chaotic behavior is beyond the scope of this paper, we will investigate it in future.

Acknowledgement
This work is partly supported by National Nature