Global Stability of an Economic Model with a Continuous Delay of Kaldor Type Modified

This paper studies continuous nonlinear economic dynamics with a continuous delay of a Kaldor type modified in dimension two. The important results are, on the one hand, the boundedness of solutions, the existence of an attractive set, and the permanence of the system and, on the other hand, the local and global stability of equilibrium points.


Model
The complex nonlinear dynamic has been introduced into the analysis of economic phenomena to explain not only the fluctuations observed in the series studies but also the economic crisis in the capitalist system. Thus, economists such as Goodwin (1967) and Kaldor (1955Kaldor ( -1956 have employed dynamic models to explain that the cyclic and chaotic growth curves are the economic phenomena endogenous to the capitalist system itself. Within the framework of structural reforms for economic dynamics, NINDJIN et al. in [1] suggest a proven model originating from ecology which could have a certain range in analysis and regulation of financial systems. This model of dimension two describes how the GDP and an economic capital interact in accordance with the model of Kaldor modified in order to increase the resilience of these dynamics against possible disruptions. Mathematical analysis of this model (see [1]) demonstrates that it is bounded and permanent and admits under certain circumstances an attractive set. On the one hand, this permanence manifests itself in the form of stationary growth of capital stock and product (stable interior equilibrium point). On the other hand, it appears in the capital cyclic growth and the product (limit cyclic). So, a capital stock rupture or a long-term production is prevented. It is also shown that the financial system stability (relative to the capital and the product) is overall; i.e., it does not depend on either stock level or production at the initial time. Facing a possible disruption of one of the control parameters of the economic system, we have analyzed, in [2], the model bifurcations. We have shown that the model admits a transcritical bifurcation, a pitchfork bifurcation or a Hopf bifurcation. In the last one, when the model is disrupted, it changes from a stationary balanced growth to a cyclic growth by preventing a GDP or capital crisis. Interactions between the GDP and the economy's capital could not be clearly and definitively understood or explained regardless of past situations which may affect the present or the future. In this paper, we are going to focus on the way in which investments are evolved and savings are established. Indeed, economies finance their investments through their own savings or those of other economies via financial structures with an interest rate. Regarding the savings, it is known that they are made up of a portion of profits or wages. Thus, when the economy is able to be selffunding by its savings, then the net profit increases = − (i.e., profit, deprived of the portion set aside to pay interest, debt, ). So, saving at a time depends on the net profit; consequently, the GDP from a time 0 = − where > 0. Let us suppose that is the deadline needed by this saving to reach a certain threshold likely to ensure the selffinancing of the investments of the economy. Let us consider the dynamics with no delay of the modified Kaldor type and the following assumption: with ( 0 , 0 , 1 , 2 , 1 , 2 , ) ∈ (R * + ) 7 and ( 1 , 2 , 1 , 2 , ) ∈ (R + ) 5 .
denotes the product, denotes the stock of capital, anḋ anḋindicate, respectively, the growth rate of the product and the stock of capital depending on the following economic parameters: (i) 0 the trend of rate of increase in GDP for a given (future) period in absence (or in neglect) of the losses, (ii) 0 the maximum (monetary) value of GDP we can get from this economy for that given period, (iii) the currency adjustment factor, (iv) 1 the maximum (monetary) value of the genuine saving of this economy for the given period, (v) 1 the maximum (monetary) value of the saving supported by the economy in the given period, (vi) 2 the maximum value of the investment rates losses for the given period, (vii) 2 the maximum capital stock for the given period, (viii) 1 the derivative relative to the capital of the investment rate in the absence (or in the neglect) of the losses (when 2 = 0) for the given period, (ix) 1 the investment rate when the capital is null ( = 0) and this rate has suffered no loss ( 2 = 0) for the given period, (x) 2 the share of the GDP converted into stock of capital for the given period, (xi) 2 the accumulation rate of capital when the product is null ( = 0) and that the investment rate has suffered no loss ( 2 = 0) for the given period.
International Journal of Differential Equations
. So, we can adjust the minimum value of lim inf →+∞ [ ( )] by choosing the value of .

Border Dynamics on the Plan
The equilibria points of model (3.1) are International Journal of Differential Equations where 0 ( ) = 03 3 + 02 2 + 01 + 00 = 0. (20) Let us give below the results on the permanence of the model. Therefore, let us consider the notations of Theorem 4. If 2 > and (0) > 0 then model (3.1) is permanent. Among others, the set defined by which is a bounded set, positively invariant for model (3.1).

Local Stability of the Equilibria Points of Model (3.1).
Performing the spectral study of Jacobian matrix of the system linearized around each of the points of equilibrium, one obtains, classically, the following conclusions: (1) Stability of (1) 0 = (0; 0): 0 is a point unstable saddle repulsive following direction and attractive following direction V if 2 < .
With ( ) the trace and det( ) determinant of J, vectors (1) 3 = ( ; V). Now let us define the conditions for which the stability of the product and the stock of capital of the economy is global; i.e., it does not depend on the produced quantities and level of stock at the initial moment. For this study, we define an appropriate Lyapunov function.
Remark 7. As 1 = 0, the unique interior equilibrium point of model (3.1) is globally and asymptotically stable if the following assumptions are verified: with constant and V defined in Theorem 4. In fact, it is worth taking the Lyapunov functional :  with ( 1 , , ) ∈ (R * + ) 3 . The equilibria of model (4.1) are
The global stability of equilibrium (2) 1 is given in the following theorem.
Meanwhile, 2 > and, then, Let us consider the function defined on R by the following: Then, (72) Given = ( * ) positive root of ( ) and * the associated delay certifying the following relation: Let us pose ( * ) = sgn{( Re( )/ )| = ( * ) }. Then, Proof. Given = ( * ) positive root of ( ). Let us assume that there is a delay * so that * = 1 It is noticed that the analysis of the local stability of the system with delay when ( , ) ̸ = 0 depends on the existence of positive root for function ( ). The coefficients 0 and 1 , of ( ), depend on . Therefore, its positive root depends on . Using the Cartan method and Viet formulas, one proves on the one hand that ( ) admits at least a positive root if 0 < 0. On the other hand, if 0 ≥ 0, then, ( ) admits either of two positive roots whereas ( ) does not admit any positive root.
The condition 0 < 0 is equivalent to the following assumption: Now, let us use the results of Theorem 4 from [5] to study the stability of these points of equilibriums * 2 and * 3 .
Theorem 12. One poses Let us consider one of these assumptions (82)-(85) and one of the following conditions: With ( ) defined in formula (23), consider the following.

Numerical Simulations
(i) Interpretation. The trajectories of GDP and of capital V and (the effect of delay) are periodic. The orbit starting from the initial condition ( 0 , V 0 , 0 ) gravitates around the equilibrium point * without never reaching it. There is, therefore, a limit cycle around this equilibrium point * . The orbit revolves around the equilibrium point. They are moving away from trivial equilibrium points and converge towards the limit cycle around * . Indeed, one has = 0.0000123 < (3) = 0.2302 and = 45 < (3) = 305733.267595647. According to Proposition 11, = 0.000686044679105366 is the positive root of ( ). One has * 3 = 2383.38276771532 and ( * 3 ) < 0. So, * is unstable according to Theorem 12 because = 3.255 > 0 for = 0. The trivial equilibrium points of the model are all unstable for < * 3 < * 2 = 4437.41337922542. Hence, we have the graphic illustration. (ii) Interpretation. The trajectories of GDP and capital V and (the effect of delay) stabilize, respectively, around * = 1796.505, V * = 777.6445, and * = 526.184 when is greater than . The border dynamics show that when the orbit arrives in a border plan, it remains and converges towards a corresponding equilibrium point (see Orbit.2 to Orbit.4 of Figure 2). Moreover, the trivial equilibrium points of the model are all unstable. So, the orbits, are moving away from the borders plans and converge towards * . Hence, * is globally stable, which is illustrated by the figure "phase portrait" of Figure 2.

Conclusion
The economic basic model of our work was one of dynamics Kaldor at an effective growth rate whose rate of saving is the Holling-II type and investment rate is from Leslie and Gower type modified in dimension two. Taking into account the time required for the saving to ensure its self-financing of all the investments, one obtains a model with a delay of the Kaldor type modified. This model is bounded and admits an attractor unit (set). Thus, under certain conditions, this model with delay is permanent. On the one hand, this permanence appears in the form of stationary growth of the capital stock and the product (stable interior equilibrium point) and, on the other hand, in the form of cyclic growth of the capital and the product (limiting cycle). In other words, this permanence avoids a shortage of the capital stock or the long-term production. In front of some other conditions, the dynamic stability with delay is also global; i.e., it does not depend on the level of the capital stock and the level of the production at the initial time. The consideration of the delay (due to the time for the economy to finance investments) can justify the bifurcation of an economic model of the stationary growth towards cyclic growth (see Theorem 12, (3)(a)(i)). However, this delay in the model can stabilize an initially unstable equilibrium of the system (see Theorem 12, (3)(b)(ii)), which represents a major economic interest since it allows the saving to get rid of risks from the cyclic growth.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that there are no conflicts of interest for this paper.