On the Dynamics of an Oil Price Model

We address the dynamic of an energy price model in a more general approach; to our knowledge, in previous works, different conditions are imposed on the energy price function at a particular value in order to characterize stability and unstability for real or complex eigenvalues. Nonexistence of periodic orbits is shown by means of Dulac’s criterion; we also exhibit some pictures of solutions. Finally we modify this model and apply it to some particular cases in Venezuela economy.


Introduction
Energy is a kind of natural resource that can be utilized to obtain power by mankind.It is an important basic resource for human survival, economical development, and social progress.It plays an important role in both the social and economic development and the structure of modernization.It is also an important lever in guiding and promoting the rational development, effective utilization, and optimal allocation of energy resources [1].In the case of crude oil it is a key commodity for global economy.As a matter of fact, it is a vital component for the economic development and growth for industrialized and developing countries in a likely manner.Moreover, political events, extreme weather, and speculation in financial market, amongst others, are major characteristics of crude oil market which increase the level of price volatility in the oil markets.The effect of oil price fluctuation extends to reach large number of goods and services which have direct impact on the economy as well as the communities.Therefore, to reduce the negative impact of the price fluctuations, it is very important to forecast the price direction.Unfortunately, fundamental variables such as oil supply, demand inventory, and gross domestic product (GDP) are not available on daily frequency which adds additional difficultly to the prediction.Some studies have indicated that the global demand will continue to rise for the long term despite the fact that oil demands from Organization for Economic Cooperative Development (OECD) countries have decreased.However, the overall demand for oil has increased and this to a large extent is due to the increasing demands of non-OECD countries, especially China.
The recent surge in the price of oil has created concern in both theory and practice.The reasons for this development can be premised on the following theoretical grounds: (i) oil price data are available at a high frequency and, therefore, there is increasing evidence of the presence of statistically significant correlations between observations that are large distance apart and (ii) in connection with the high frequency of oil price data, there is possibility of conditional heteroscedasticity, that is, time varying volatility [2][3][4].
Energy producers and consumers regularly attempt to forecast prices of oil, coal, and other resources over time horizons as long as twenty or thirty years.Producers make these forecasts for general purposes of strategic planning and for specific purposes of evaluating investment decisions, for example, related to resource exploration, reserve development, and production.Industrial consumers, such as petrochemical companies or electric utilities, make these forecasts for the same kinds of reasons-oil, coal, and natural gas are important input costs that can affect investment decisions (e.g., an oil-versus coal-fired power plant for an electric utility), or even the choice of products to produce (e.g., a set of chemicals or the processes used to produce those chemicals).Ideally, we would like to be able to explain energy prices in structural terms, that is, in terms of movements in supply and demand, and variables that determine supply and demand.However, structural models are not always useful for long-run forecasting, in part because it is difficult to forecast the explanatory variables in such models (like investment and production capacity, inventory levels, and determinants of demand) over long horizons [2,5,6].
Theoretical models are based on ideals relations between factors that influence the market according to economic and financial theories.The immediate practical application of these models is usually limited, given an assumed ideal behavior, which very often does not obviate the consideration of multiple real factors that show up in practice.These models attempt to simulate the behavior of the market under ideal theoretical conditions, to try to asses, for instance, macroeconomic consequences arising from oil prices shocks, or the nature of the behavior of market factors.Most economic models of the Global Oil Market (GOM) are linear forms reduced to a single equation with some exceptions [7][8][9][10][11].
Nonlinear dynamics theory is introduced into energy demand-supply economic system and scores great successes in model of energy demand-supply and energy price and its analysis [12][13][14][15][16]. Based on the researches above, the main question is that net import has not been taken into account in the study of the energy price changes.
In this paper we address the local and global dynamic of a model considered in [1,14,15], described by a new differential equation of energy price in a region of China (Jiangsu province) with huge consumption but with a little production so its energy mostly comes from import.We consider the possible existence of periodic orbits and different types of bifurcations and the interpretation of this dynamics in terms of the model under consideration.

Model Formulation
The model we consider is established and investigated in [1,14,15]; however we consider a more detailed analysis and formulate it for a more general function of energy price.Let us denote by (), (), (), (), and () quantity supplied of energy, quantity demanded of energy, net quantity of energy (import minus export, denoted by  and , resp.), energy price, and quantity stocked of energy (neglecting net import), respectively, at  ≥ 0. It is important to point out that even when we mention energy instead of oil, the model works in the same way for different types of energy like gas, electricity, oil, and so on.Seeking for simplicity we drop the variable  in foregoing functions, and we will use it again only if necessary.As in [1], the rising of price / is assumed proportional to the gap between quantity stocked  and the determined quantity stocked  plus net quantity imported; that is, where  > 0,  > 0 are constants and their size determines adjustment velocity of energy price.
Remark 1.No import ( = 0) means that we only have export , so [ − ] < 0 and ,  depend only on production, but then if  −  < 0, there is  −  for export.
Remark 2. In case of no import and taking into account foregoing remark, the model given in (1) now becomes In the same token,  is equivalent to initial energy stock  0 plus total volume of gap between current energy supply and energy demand, so When energy does not surpass the threshold that purchasing power can bear, energy demand stimulates increasing of energy prices and energy prices are positively correlated with energy demand; when the increasing prices surpass the threshold, purchasing power may decrease with the increasing energy prices; in other words, where  0 > 0 may be considered as initial demand,  > 0 is the so-called marginal energy demand quantity, and () is known as function of energy price; its size reflects the dependence of energy demand on the increase rate of energy price and will be assumed to be () ≥ 0. Energy supply is energy quantity that energy producing enterprize is willing to sell and can be sold under the given energy price level; indeed with  ≥ 0,  > 0;  means marginal energy supply quantity [12].Finally, energy net import quantity  is assumed as From [1,14,15] the following nonlinear second-order differential equation comes out: with  ≥ 0,  > 0 being constant and representing adjustment velocity of energy prices;  0 ≥ 0 may be assumed as initially supplied quantity of energy,  > 0 is marginal energy supply quantity, and  > 0 is a constant that shows up when  is assumed as a linear function of  [1].
Remark 3. In [1] function of price which is given as () = + with ,  being real numbers, here we consider a general () and impose some conditions on it shortly.
If we call  =  and  = /, then (7) becomes It is easy to show that the only critical point of ( 8) is given as ( 0 ,  0 ) = (( 0 − 0 )/(+), 0), so the condition  0 − 0 > 0 is a must in order for the model to make sense (price has to be positive) [1].Now we perform the following transformation of coordinates,  =  −  0 ,  = , and use that ( 0 ,  0 ) is critical point, so (8) Notice that (10) has the origin as unique critical point.

Stability
We study local and global stability of (10) at the origin and begin by showing no existence of periodic orbits in by means of Dulac's criterion [17,18]. Then The conclusion now follows from Dulac's criterion.
In order to calculate Jacobian for (8) at the origin we assume that  is differentiable at  0 =  0 −  0 ; then it is given by Consequently, each eigenvalue  associated with  satisfies the equation Roots of foregoing equation are given as We always may assume  > 0 because otherwise eigenvalues of  are  1 = 0,  2 = − < 0, and in this case a saddle node bifurcation shows up.
Remark 7. The number  2  2 /4( + ) is a threshold value for the above case.
The following result gives a characterization of eigenvalues of  in terms of ( 0 ).Theorem 8.If ( 0 ) ̸ = 0, then the following statements take place.
(1) Δ = 0 if and only if ( here  1,2 are different but negative. ( that is,  1,2 are complex conjugate with negative real part.
Stability now can be summarized, from the above results, in the following.

Corollary 9. Equilibrium of system (8) is
(1) locally asymptotically stable (stable node) if (2) unstable (unstable node) if ( 0 ) = 0 and Δ > 0; (3) stable focus if It is worth mentioning here that price function  at  0 plays a key role in studying stability properties of critical point as previous results show.Example 11.We realize that a simple curve like affine function is not the best approximation to the price function; instead we may consider quadratic or cubic ones in order to get an improved approximation.In the next two examples a quadratic and cubic expression for  and conditions for real roots at initial  0 are given.

Some Numerical
Next we perform some numerical implementation in order to show pictures of trajectories for particular functions like the ones given in previous example (Figures 1, 2, and 3).
It is important to point out that we have chosen cubic and quadratic expressions for ; our numerical analysis shows no presence of periodic orbits which is something we know from the previous analysis.Figures 4, 5, and 6 are included only to demonstrate that, in the case when  may be zero at some point, the result is no realist since a kind of attractor may show up (Figure 6), and that is impossible to happen.
Finally we would like to state that for other types of  like () = 1/ and () =  2 +  + 1 the general behavior is similar to the case shown in Figures 4, 5 and 6.

An Application to Local Economy
In a country like Venezuela where most of cereals, grains, and vegetables which constitute important part of Venezuelan diet are grown in its fields and do not need to be imported, so the country stands up on local production, the model we have studied can be reformulated in order to describe variations of where  is quantity stocked of item and is given by (3),  is quantity needed to satisfy local necessities of consumption (country needs),  > 0,  > 0 are constants, and their size determines adjustment velocity of item price.We use the word "item" to indicate one of the above mentioned, that is, cereals, grains or vegetables.In this context some words need to be said;  and  are quantity supplied and quantity demanded of item, respectively, and given by ( 4) and (5) and  is the price.At the same time, (), the function of item price reflects the dependence of item demand on the increase rate of item price and will be assumed to be () ≥ 0.

1 Figure 6 :
Figure 6: A quadratic being null at some point and orbits.