# Autonomous Dynamical System Approach for Gravity

###### Abstract

In this work we shall investigate the cosmological dynamical system of gravity, by constructing it in such a way so that it is rendered autonomous. We shall study the vacuum gravity case, but also the case that matter and radiation perfect fluids are present along with the gravity. The dynamical system is constructed in such a way so that the time-dependence of the system is contained in a single parameter which depends on the Hubble rate and it’s second derivative. The autonomous structure of the dynamical system is achieved when this parameter is constant, therefore we focus on these cases. For the vacuum case, we investigate two cases with the first leading to a stable de Sitter attractor fixed point but also to an unstable de Sitter fixed point, and the second is related to a matter dominated era. The stable de Sitter attractor is also found for the gravity in the presence of matter and radiation perfect fluids. In all the cases we performed a detailed numerical analysis of the dynamical system and we also investigate in detail the stability of the resulting fixed points. Also, we present an exceptional feature of the gravity model, in the absence of perfect fluids. Finally, we investigate what is the approximate form of the gravities near the stable and the unstable de Sitter fixed points.

###### pacs:

95.35.+d, 98.80.-k, 98.80.Cq, 95.36.+x## I Introduction

Since the observation of the late-time acceleration Riess:1998cb that our Universe experiences, the focus in modern theoretical cosmology is to find a theoretical framework that can consistently harbor this late-time acceleration era. Many theoretical proposals can potentially provide a consistent description of the so-called dark energy era, with the modified gravity being the main appealing approach, for reviews see reviews1 ; reviews2 ; reviews3 ; reviews4 . Among the various modified gravity descriptions, gravity is the most important and most studied approach, in the context of which it is possible to provide the unification of the early with the late-time acceleration of the Universe Nojiri:2003ft ; Nojiri:2006gh .

Due to the importance of gravity, it is compelling to know the general structure of the phase space of the cosmological dynamical system of the theory. Many studies already exist in the literature, studying various aspects of dynamical systems in cosmology of modified gravity Boehmer:2014vea ; Bohmer:2010re ; Goheer:2007wu ; Leon:2014yua ; Leon:2010pu ; deSouza:2007zpn ; Giacomini:2017yuk ; Kofinas:2014aka ; Leon:2012mt ; Gonzalez:2006cj ; Alho:2016gzi ; Biswas:2015cva ; Muller:2014qja ; Mirza:2014nfa ; Rippl:1995bg ; Ivanov:2011vy ; Khurshudyan:2016qox ; Boko:2016mwr ; Odintsov:2017icc , see also Odintsov:2015wwp , which focus on the existence of fixed points and also their stability. In this paper we shall attempt an alternative approach for the study of the gravity dynamical system case, in the presence or in the absence of perfect matter fluids. Particularly, we shall investigate the phase space in the limit that the corresponding dynamical system is strictly autonomous. The motivation for studying the autonomous limit of the cosmological dynamical system, comes mainly from the fact that if a dynamical system is non-autonomous, then the stability of the fixed points is not guaranteed by using the theorems which hold true in the autonomous case. Hence, the stability of a fixed point corresponding to a non-autonomous system is rather a problem without a solution, unless highly non-trivial techniques are employed. Let us explain our argument by exploiting a very simple example which can be found in dynsystemsbook , so consider the one dimensional dynamical system . The solution can be easily found to be , from which it is obvious that all the solutions asymptotically approach for . Also it is easy to see that the only fixed point is the time-dependent solution , which however is not a solution to the dynamical system. In addition, a standard analysis by using the fixed point theorems, shows that the vector field actually move away from the attractor , which is simply wrong. Therefore, it is conceivable that knowing the fixed points of a non-autonomous system, and then applying the Hartman-Grobman theorems, does not provide a consistent solution to the stability problem of the fixed points, and in effect, the structure of the cosmological phase space cannot be revealed in such a way.

The purpose of this paper is to construct a dynamical system for gravity, and we aim to study the cases for which the dynamical system is rendered autonomous. Particularly, we shall quantify the whole time-dependence of the dynamical system in terms of one single parameter , which turns out to be equal to , where is the Hubble rate of the Universe. Then we will focus on various values of this parameter with some physical significance, and specifically we focus on and . The first case turns out to describe an inflationary de Sitter final attractor which is stable, in both the vacuum and matter fluid gravity. We need to note that not only the de Sitter or a quasi de Sitter cosmology yield , so the general study we perform covers all the cosmologies which may yield . This means that the case may describe an approximate solution of some specific gravity, for example during the slow-roll era, or similar. In all the cases, we shall find the fixed points of the dynamical system, and the fact that the dynamical system is autonomous will enable us to use the fixed point theorems and thus reveal the stability structure of the system. Also we shall use various sets of initial conditions, emphasizing for -foldings numbers in the range , and we shall present in detail the stability properties of the fixed points in terms of the resulting trajectories. As a side study, we shall also consider in brief the case , which as we show, describes a matter dominated era, so we shall investigate the gravity phase space properties, and we also show that the stability of the matter domination final attractor can be achieved. The latter study clearly demonstrates that a vacuum gravity may successfully describe a matter dominated era, however caution is needed for the stability study of the fixed point. In principle there are other cases for which the gravity dynamical system can be rendered autonomous, but we do not cover all the different cases for brevity. We thoroughly discuss how an autonomous dynamical system can be obtained in the context of gravity in the presence of matter and radiation perfect fluids, and we mainly focus on de Sitter attractors. Also, along with the stable de Sitter attractor, in the vacuum gravity case, there exists also an unstable de Sitter fixed point. Intriguingly enough, as we explicitly demonstrate at a later section, when a vacuum gravity is considered, the dynamical system reaches the unstable de Sitter attractor. As we argue, this feature may be a sign of graceful exit from inflation. Also we shall investigate the approximate forms of the gravities near the stable and the unstable de Sitter fixed points, and also we discuss the role that play terms of the form, in the graceful exit issue.

This paper is organized as follows: In section II, we present the way to obtain an autonomous dynamical system from a general vacuum gravity. In section III, we study in detail the phase space of vacuum gravity, and we focus on the cases and . We find the fixed points in each case and we investigate the stability structure of the fixed points, by analyzing many possible initial conditions. Moreover, we investigate which are the approximate forms of the gravities, near the stable and the unstable de Sitter fixed points, and also we thoroughly discuss the graceful exit issue, in view of the presence of terms. Moreover, we present a specific example of an cosmology, which satisfies , and we examine the behavior of the dynamical variables which were introduced in the previous sections. In addition, other attractors apart from the de Sitter one, are also discussed. In section IV we thoroughly study the structure of the gravity autonomous dynamical system, in the presence of matter and radiation perfect fluids, emphasizing on the de Sitter attractors. Finally, the conclusions follow in the end of the paper.

Before starting our study, let us note that we will use a flat Friedmann-Robertson-Walker (FRW) metric, as a geometric background, the line element of which is,

(1) |

where is the scale factor. Also for a FRW metric, the Ricci scalar is,

(2) |

where , is the Hubble rate.

## Ii The Autonomous Dynamical System of Vacuum Gravity

In this section we shall present the general structure of the vacuum gravity, and we will discuss the cases that may render this an autonomous system. The vacuum gravity action is,

(3) |

where and also is the Planck mass scale. In the context of the metric formalism, the equations of motion are obtained by varying the gravitational action (3) with respect to the metric tensor , so by doing so we obtain the following equations of motion,

(4) |

which can be written as follows,

(5) |

with the prime indicating differentiation with respect to the Ricci scalar. For the FRW metric of Eq. (1), the cosmological equations of motion become,

(6) | ||||

(7) |

where , , and . In order to reveal the autonomous structure of the cosmological dynamical system described by the equations of motion (6), we shall introduce the following variables,

(8) |

In the following we shall use the -foldings number , instead of the cosmic time, so the derivative with respect to the -foldings number can be expressed as follows,

(9) |

which shall be useful. Hence, by using the variables (8) and also the equations of motion (6), after some algebra we obtain the following dynamical system,

(10) | ||||

where the parameter is equal to,

(11) |

By looking the dynamical system (10), it is obvious that the only N-dependence (or time dependence) is contained in the parameter . Also we did not expressed as a function of , since we shall assume that this parameter will take constant values. Hence if the parameter is a constant, the dynamical system is rendered autonomous since no explicit time dependence exist in it.

The effective equation of state (EoS) for a general gravity theory is,

(12) |

and it can be written in terms of the variable as follows,

(13) |

By using the dynamical system (10) and the EoS (13), given the value of the parameter , in the following sections we shall investigate the structure of the phase space corresponding to the vacuum gravity, and we shall discuss in detail the physical significance and implications of the results.

## Iii Study of the Vacuum gravity Phase Space

The parameter appearing in the non-linear dynamical system (10) plays an important role, since it is the only source of time-dependence in the dynamical system. Let us note that for certain cosmological evolutions this parameter is constant. For example, a quasi de Sitter evolution, in which case the scale factor is,

(14) |

the parameter is equal to zero, and the same applies for a de Sitter evolution. However, in this section we shall not assume that the scale factor has a specific form, but we shall study in general the cases and . As we shall see, these two cases have special physical significance. With regard to the case, this is easy to check, since if we solve the differential equation , this yields the solution,

(15) |

This means that we focus on cosmologies for which the approximate solution for the evolution is a quasi de Sitter evolution. This does not mean that the exact Hubble rate is a quasi-de Sitter evolution, but the approximate gravity which drives the evolution, leads to an approximate quasi-de Sitter evolution. Interestingly enough, for the quasi-de Sitter evolution (15), the following conditions hold true,

(16) |

which are the slow-roll conditions. Hence the case is related to the slow-roll condition on the inflationary era. In the following sections we shall perform an in depth analysis of the , since this is the most interest scenario.

Before discussing the details of the two cases and , let us present in brief the classical approach to the classification of fixed points. According to the classical approach, the linearization procedure applies dynsystemsbook , which can also be applied in our case, if the fixed point is hyperbolic. Also we briefly discuss some essential features of the stability theory for dynamical systems .

With the stability theory of dynamical systems, basically one reveals the stability of the solutions and of the trajectories of dynamical systems. Basically, the stability of the solutions-trajectories, addresses the behavior of these, if small perturbations of the initial conditions are performed. Also it is important to have a solid grasp of the asymptotic behavior of the solutions and trajectories of a dynamical system, with asymptotic meaning after a long period of time. The duration and determination of this long period of time is usually determined by the physical scales in the theory. For the purposes of this paper, since we are interested in inflationary dynamics, the asymptotic behavior corresponds to approximately -foldings. The simplest and most valuable behavior of the solutions-trajectories is exhibited by the fixed points of the dynamical system at hand, and also by periodic orbits. In all cases, how do orbits-trajectories or solutions behave near the fixed point, do these approach the fixed point, or alternatively said, are the trajectories attracted by the fixed point, or are these repelled from it? Also, if we perturb the orbit by using alternative initial conditions, how these perturbations affect the behavior of the trajectories near the fixed point? If the trajectory is always attracted by a fixed point, then this fixed point is called an attractor of the trajectories. Also if perturbed orbits tend asymptotically to a given orbit, then this orbit is called stable. The fixed points provide a characteristic picture of the structural stability of the dynamical system. The stability of a fixed point refers to the behavior of the trajectories near this point, if the trajectories are attracted to it, this is called stable (attractor), and if these are attracted to it asymptotically, this is called asymptotically stable (asymptotic attractor). In the case that trajectories are repelled from it, the fixed point is an unstable fixed point. One quantitative way to reveal the stability of a fixed point, is to apply the Hartman-Grobman theorem (see below), by linearizing the autonomous non-linear dynamical system. If the eigenvalues of the linearization matrix are negative real numbers, or even complex numbers with negative real parts, then the fixed point is stable (attractor). If none of the eigenvalues are purely imaginary, or equal to zero, then the fixed point can be an attractor or a repeller, and this is subject to the behavior of stable and unstable manifolds, see the book dynsystemsbook for more details on this. The important thing to note is that the Hartman-Grobman theorem applies only on hyperbolic fixed points, which means that only in the case that the eigenvalues of the linearization matrix have non-zero real parts the Hartman-Grobman theorem may apply. It is the purpose of this paper to examine what happens in the cases that the Hartman-Grobman theorem does not apply.

As it is well known, the Hartman-Grobman linearization theorem determines the stability and the structure of the phase space, when hyperbolic fixed points are taken into account. Let be the solution to the following flow,

(17) |

with being a locally Lipschitz continuous map . Let us denote with all the fixed points of the dynamical system (17), and the corresponding Jacobian matrix, which we denote as , is equal to,

(18) |

The Jacobian has to be calculated at the fixed points, and the eigenvalues must satisfy . Let denote the spectrum of the eigenvalues of , so a hyperbolic fixed point satisfies . The Hartman theorem ensures the existence of a homeomorphism , where is an open neighborhood of , such that . The homeomorphism generates a flow , which is,

(19) |

and in addition it is a topologically conjugate flow to the one appearing in Eq. (17). Sometimes the Hartman theorem holds true for non-autonomous system, but the general rule is that it does not apply to non-autonomous systems, or should be applied with caution. A direct implication of the theorem is that the dynamical system of Eq. (17) may be written as follows,

(20) |

where is a smooth map . In effect, if the Jacobian satisfies and also if

(21) |

the fixed point corresponding to the dynamical flow , is a fixed point of the flow in Eq. (20), and it is stable asymptotically. It is conceivable that if any of the conditions we presented above is violated, further study of the phase space is needed in order to see whether stability is ensure or not.

### iii.1 de Sitter Inflationary Attractors and their Stability

We start off with the case , which may possibly describe a quasi de Sitter evolution, however we shall analyze the dynamics of the system (10), for without specifying the Hubble rate.

In the case of the dynamical system of Eq. (10), the matrix is equal to,

(22) |

where the functions are,

(23) | ||||

The fixed points of the dynamical system (10), for general are the following,

(24) | ||||

In the case , the fixed points are,

(25) |

The eigenvalues for the fixed point are , while for the fixed point these are . Hence both equilibria are non-hyperbolic, but as we show the fixed point is stable and is unstable.

Before we proceed let us discuss the physical significance of the two fixed points, and this can easily be revealed by observing that in both the equilibria (25), we have . By substituting in Eq. (13), we get , so effectively we have two de Sitter equilibria.

Also it is worth to have a concrete idea on how the dynamical system behaves analytically. Actually, the third equation of the dynamical system (10) is decoupled, and the solution of it reads,

(26) |

where is an integration constant which can be fixed by the initial conditions. The asymptotic behavior of the solution (26), that is for large , is , which is exactly the behavior we indicated earlier. Also, by using the solution (26) and substituting it in the first two coupled differential equations of the dynamical system (10), we may obtain the analytic form of the solutions and , which read,

(27) | ||||

where and are integration constants and also and are the confluent hypergeometric function and the generalized Laguerre polynomial respectively. In order to have a direct command on the initial conditions, we shall proceed directly on the numerical analysis, but it can be shown that if the integration constants are appropriately chosen, the numerical and analytical results may coincide.

Now let us analyze the dynamics of the cosmological system, and for starters we numerically solve the dynamical system (10) for various initial conditions and with the -foldings number belonging to the interval . In Fig. (1) we present the numerical solutions for the dynamical system (10), for the initial conditions , and .

For the upper left plot of Fig. 1, the -foldings number is chosen in the interval and as it can be seen, the equilibrium is not reached yet, and the same applies for the upper right plot in Fig. 1, in which case , although the situation is a bit better. In the third plot of Fig. 1, the -foldings number is chosen in the interval , so by , the equilibrium is reached. The same can be seen in Fig. (2), where we plot , for various initial conditions.

From Figs. 1 and 2 it already seems that the fixed point is a stable de Sitter attractor. As we show by using further numerical analysis, indeed this de Sitter equilibrium is the final attractor of the phase space. In Fig. 3, we plot the three dimensional flow, and as it can be seen, for various initial conditions, the global attractor behavior of can be seen. Also the stability of the attractor can be seen in the reduced dynamical system for . Actually, the equation for can be solved directly since this is decoupled from the rest of the equations in (10).

The dynamics of the reduced system for are presented in Fig. 4, where we plot the vector field flow (left) and the trajectories in the plane (right), for various initial conditions. It is obvious that the attractor behavior of the de Sitter fixed point is further supported by these flows in the reduced system.

In conclusion, the case for the dynamical system (10) results to a stable de Sitter attractor and also to an unstable de Sitter fixed point. This result is particularly interesting since it dictates that a general gravity theory with Hubble rate that satisfies results to an asymptotic stable de Sitter attractor, which is reached at , as it can be seen in Fig. 1. This is quite appealing, if one considers an inflationary interpretation of the theory, for example if the scale factor is assumed to be a quasi-de Sitter one, namely . In the quasi-de Sitter case, the attractor property reveals that the phase space of a the gravity has an asymptotic de Sitter attractor, which is stable, which clearly may be interpreted having an inflationary evolution in mind. However, not only a quasi-de Sitter cosmology results to , consider for example the symmetric bounce cosmology Odintsov:2015ynk , in which case the scale factor is , with . In the symmetric bounce, the parameter is also zero, but in this case the physical interpretation of the de Sitter attractor is different, since it can be a late-time de Sitter attractor. However, this requires different initial conditions and perhaps a re-parametrization of the dynamical system (10), in terms of the cosmic time, in order to avoid inconsistencies. We leave this task as a possible future work. Also, it is possible that for certain cosmological models, we may have asymptotically , if the parameters of the theory are appropriately chosen. In such case, the interpretation of the fixed points needs special attention.

Let us note that there is also an unstable de Sitter fixed point, namely . So it may be possible that there are gravities which may lead the dynamical system directly on the unstable de Sitter attractor. This can be seen in the 3-dimensional flow of Fig. 3, since there are trajectories that move away from the stable de Sitter attractor . One example of this sort, we shall present at a later section.

Now the central question is, what is the physical interpretation of the above dynamics, and to which gravities the case corresponds. It is conceivable that we can find approximate expressions of the gravities near the fixed points, and . This is the subject of the next section.

#### iii.1.1 Approximate Form of the Gravities Near the de Sitter Attractors

In this section we shall analyze the physical implications of the dynamics we presented in the previous section. As we demonstrate, for the value of for which , there are two de Sitter fixed points, one stable, which is and also an unstable, which is . Now we shall investigate the behavior of the gravity near the fixed points and for . Recall that as we discussed in Eq. (16), the case is equivalent to the slow-roll approximation, so effectively what we seek for is the behavior of the gravities near the fixed points and with the slow-roll approximation holding true. Let us start with the first fixed point, namely , so the following differential equations must hold true simultaneously at the fixed point,

(28) |

which stem from the conditions and . Since (or equivalently since the slow-roll approximation holds true), the left differential equation can be written as follows,

(29) |

which can easily be solved and it yields,

(30) |

The gravity solution (30) is nothing but the approximate form of the gravity in the large curvature era, which generates the quasi-de Sitter evolution of Eq. (15) or equivalently, that yields . So since we are interested in the large curvature era, the exponential term is subdominant, so the cosmological constant leads to the stable de Sitter attractor . By using the solution (30), it is easy to show that the second condition in Eq. (28) can be written as follows,

(31) |

so it follows that in order the above condition is true, we must have and also . Notice that the final form of the gravity is very similar to the exponential gravity models of Ref. Elizalde:2010ts , in which case the full gravity is,

(32) |

Now let us consider the case of the second de Sitter fixed point, namely , and in this case the conditions and become,

(33) |

By using the fact that , when the quasi-de Sitter evolution is taken into account, the second differential equation can be written,

(34) |

which can be solved to yield,

(35) |

The solution (35) is not the exact form of the gravity which leads the cosmological system to the fixed point, but it is the approximate form of the gravity near the fixed point which corresponds to the case . The approximate gravity of Eq. (35) is very similar to the model. Finally, the first condition in Eq. (33), for the gravity being of the form (35), becomes,

(36) |

which holds true if .

Thus by taking into account the resulting forms of the gravities we discussed in this section, we may conclude that the terms related to quasi-de Sitter solutions always lead to an unstable de Sitter fixed attractor, while terms containing cosmological constants and exponentials, lead to a stable de Sitter attractor. This result is interesting, since it is well known Bamba:2014jia that corrections to viable gravities, like the exponential, always trigger graceful exit from inflation, see the well-known viable Starobinsky inflation model starobinsky . Hence, the presence of an term leads to instability, which may be viewed as an indication of the graceful exit from the inflationary era. In the following section we shall analyze in detail the case of the gravity, and we will investigate whether the predicted behavior we found in the previous sections, holds true in this case.

#### iii.1.2 Specific Example I: Gravity and Graceful Exit from Inflation

An intriguing example that leads to the picture we discussed in the previous section, occurs for the gravity, with regards to the final unstable de Sitter fixed point. As we will show, this gravity makes the dynamical variables to end up to the unstable de Sitter fixed point. The functional form of the gravity is,

(37) |

where the parameter has dimensions of mass. We shall focus on the vacuum gravity case in the rest of this section. In order to investigate the behavior of the dynamical variables (8), we shall need the exact functional form of the Hubble rate, and also this technique will hold true only if the specific Hubble rate yields . This issue needs to be further explained at this point. The behavior of the dynamical variables we described in the previous sections, holds true for all the cosmologies which yield , without specifying the exact form of the gravity, however the result shows us that there are gravities, actually those of Eq. (30), which yield a Hubble rate for which and also the variables , and approach the values , which correspond to the fixed point . Also there are gravities which lead to an unstable de Sitter fixed point, which can be approximated near the fixed point as in Eq. (35). However, finding the Hubble rate for a general gravity functional form, is quite a formidable task. Nevertheless, for the gravity (37) this is easy when the slow-roll approximation is used. The cosmological equations for the gravity in a FRW background metric, are,

(38) |

so in the slow-roll case, since and , the second equation of Eq. (38) can be solved to yield the following Hubble rate,

(39) |

which is a quasi-de Sitter evolution. As it can be easily checked, and we also discussed this issue in the previous sections, the Hubble rate (39) yields . By using the Hubble rate (39), the functional form of the gravity (37) and also by expressing the cosmic time as a function of the -foldings number (recall that ), the dynamical variables (8) as a function of the -foldings number become,

(40) | ||||

As it was shown in Odintsov:2015gba , for the choice sec and sec, a viable inflationary cosmology can be generated, so by using the aforementioned values, in Fig. 5 we plot the behavior of the variables (40) as a function of the -foldings number. The red curve corresponds to , the black to and finally the blue dashed curve corresponds to the variable .

As it can be seen, the variables , and approach quite fast the fixed point values , and therefore the gravity is an example of an gravity which leads to an unstable de Sitter fixed point. Actually this is quite intriguing, since the unstable de Sitter fixed point is reached for the gravity, and not the stable de Sitter point. This feature could be seen as an indication that the graceful exit from inflation occurs for the gravity. In principle, the graceful exit issue cannot be addressed by studying solely the variables , , since this may occur due to curvature perturbations, so it is an effect which can be seen in higher orders of curvature. For a detailed analysis on this, see Odintsov:2015wwp . However, the gravity shows an exceptional behavior, since for this case the unstable de Sitter fixed point is reached, and this is not accidental, since as we discussed earlier, terms are always related to the graceful exit from inflation, see Bamba:2014jia ; starobinsky .

Let us briefly discuss our claim that connects the graceful exit and the gravity. In order to be more concise, there is indication for the graceful exit from inflation in the gravity case, thus our claim is more mild in rigidity. According to Refs. Bamba:2014jia ; Odintsov:2015wwp , where a deeper analysis was performed, the graceful exit from inflation may be triggered by growing curvature perturbations around an unstable de Sitter solution of the theory. In this case, the de Sitter solution ceases to be the final attractor of the theory, and thus inflation ends as a consequence of that, since the de Sitter solution is not the final attractor of the theory. What follows is part of the subsequent physical evolution, for example the reheating era, since the slow-roll era ends. In order to investigate the perturbations of the curvature, we perturb the de Sitter solution as follows,

(41) |

and we plug the perturbation in the first equation of motion of Eq. (38), and by keeping linear terms of the derivatives of , we obtain the following differential equation,

(42) |

which can be solved analytically and the solution is,

(43) | ||||

where and are integration constants. As it can be seen from the solution of Eq. (43), the term proportional to is growing exponentially as a function of the cosmic time, therefore this supports our claim that the de Sitter point of the gravity is unstable, which we also demonstrated earlier by using the fixed point analysis. Basically, this unstable de Sitter point is the fixed point . We need to stress once more that this is an indication that graceful exit is triggered by curvature perturbations, however this indication cannot be considered as a proof of the graceful exit from inflation.

### iii.2 Other Possible Attractors and their Stability

Let us now briefly discuss the case , which as we show corresponds to a matter dominated Universe. In the case at hand, the matrix is equal to,

(44) |

where the functions are in this case,

(45) | ||||

The fixed points in this case are,

(46) |

For the first fixed point the eigenvalues are , while for the second fixed point these are . Both equilibria are non-hyperbolic and the first seems to be stable. However as we show, if the initial conditions on the variable deviate from the equilibrium value , the dynamical system solutions simply blow-up.

The fixed points have some physical interpretation, since for , the EoS in Eq. (13) becomes , which perfectly describes a matter dominated era. What now remains is to investigate the dynamics of the cosmological system.

As in the de Sitter case, it is possible to solve analytically the dynamical system, and in this case the solution for is,

(47) |

where is again an integration constant. The solutions for and can also be found analytically, but the general solutions are particularly complicated and lengthy to include them at this point. From the solution (47) it is obvious that asymptotically, which does not correspond to the matter dominated era value of . Actually, this corresponds to an additional fixed point, which we shall consider below, however it has no physical significance, since it corresponds to a phantom evolution.

Now we proceed to the dynamical system analysis, and as we now show, if the initial conditions on deviate from the equilibrium value, the solutions tend to infinity. In Fig. 6, we plotted the behavior of the solutions for the initial conditions , and (left plot) and for , and (right plot).

As it can be seen in Fig. 6, if the variable deviates from it’s equilibrium value, the solutions blow up and the fixed point is not a stable attractor. Hence, the stability of the system is ensured only if , and only for this case . Now we need to interpret this physically, and the interpretation is simple, since the behavior of the dynamical system indicates that when deviates from it’s equilibrium value , which is equivalent to say when , then the vacuum gravity does not approach the first fixed point . Actually, this is reasonable, since it seems that the vacuum gravity fails to describe a matter dominated era unless . To our opinion, the matter dominated attractor is reached only if , and if this condition is not met, instability occurs and the system ceases to describe a matter dominated era.

Let us now demonstrate that the reduced dynamical system by fixing , is indeed stable. In Fig. 7 we plot the vector field flow in the plane by fixing (left plot), and various trajectories in the plane. As it can be seen in both plots, the fixed point is reached asymptotically.

Finally, the strong dependence of the three dimensional system can be seen in the left and right plots of Fig. 9. As it can be seen, depending on the initial conditions, the stable fixed point may or may not be reached.

Having discussed the vacuum case, what now remains for completing the study is to add matter and radiation perfect fluids, and this is the subject of the next section.

Before closing, we need to discuss another interesting point, having to do with the existence of two more fixed points of the dynamical system (10) for , which are,

(48) |

The form of the new fixed points renders them physically unappealing, since the variables and take complex values. Actually, this means that the gravity is complex, and this is connected with phantom behavior, as was shown in Ref. phantom . Actually, the complex gravity is usually connected with a phantom scalar theory which develops a Big Rip singularity, and particularly, the complex gravity corresponds to the region, where the phantom scalar theory develops the crushing type Big Rip singularity. This cannot be accidental, since, as we already mentioned below Eq. (47) for , the EoS becomes , which clearly describes a phantom evolution. We shall not extend the analysis further, since this theory can be useful for a dark energy description, however from an inflationary point of view, it is not so appealing having a phantom theory at hand.

## Iv The non-vacuum Gravity Autonomous Dynamical System: The de Sitter Attractor Case

In the previous sections we discussed the vacuum gravity case, and in this section we shall incorporate the radiation and matter domination perfect fluids for completeness. We shall focus on the case , since the other cases can be easily studied. So for the case let us analyze the behavior of the phase space.

Let us first construct the dynamical system in the case that matter and radiation perfect fluids are considered. The equations of motion in this case become,

(49) | ||||

(50) |

where and are the total effective energy density and the effective pressure of the radiation and matter perfect fluids. In order to reveal the autonomous structure of the gravity cosmological system, we introduce the following variables,

(51) |

where and are the radiation and matter energy densities. In terms of the -foldings number , the dynamical system in the case at hand reads,

(52) | ||||

where the parameter is again as it appears in Eq. (11). The matrix in the case at hand is equal to,