A New Generalized Financial Model in the Presence of Price Change Delay and Delayed Feedback on Investment Demand

1. Introduction

Complexity has become a fundamental concept in various disciplines, providing a robust framework for understanding systems composed of a large number of interacting elements whose collective behavior leads to emerging phenomena that cannot be fully explained by the properties of individual components alone. In physics, complexity is associated with nonlinear interactions, self-organization and emerging phenomena observed in systems like turbulent fluids, climate dynamics and complex networks [1,2,3]. In biology, complexity refers to the nonlinear interactions among interconnected components across multiple levels of intricate organization, from genes to ecosystems. Biological phenomena such as morphogenesis, immune responses, ecological dynamics, and evolution emerge from these interactions across different spatial and temporal scales [4]. In sociology, complexity is viewed as a mode of thinking that aims to understand reality through the interconnectedness, integration and recurring interactions of its components. According to Morin [5,6], social phenomena cannot be adequately explained by reductionist and fragmented approaches; rather, they must be analyzed as interconnected systems in which order and disorder, organization and uncertainty, and emergent properties coexist. His theory of complex thought emphasizes the need to connect disciplines and integrate multiple perspectives to understand the multidimensional nature of human and social reality.

In the context of economics, the proliferation of hyperconnected societies, together with the intricate and multilayered interdependencies among diverse economic agents, contributes to a significant increase in complexity. In [7], Rosser examined the concept of complexity in economics and distinguished several forms of complexity, including computational complexity, which relates to the difficulty of solving certain economic problems; dynamic complexity, which arises from nonlinear and sometimes chaotic behavior of economic systems; and hierarchical complexity, which results from the organization of economic phenomena at different interconnected levels. Building on this perspective, Arthur [8] proposed a conceptual framework for complexity economics, which is based on the proposition that the economy is not necessarily in equilibrium. Instead, economic agents (firms, consumers and investors) continually modify their actions and strategies according to the results of their collective interactions. Furthermore, the complexity in economics can be viewed as complex system composed of heterogeneous agents whose interactions lead to the emergence of macroeconomic phenomena such as business cycles and financial crises. This is fully consistent with the principles of complexity in diverse disciplines of science, including emergence, adaptation, feedback and nonlinearity interactions.

A financial system is composed of numerous interconnected elements and influenced by a variety of economic factors, including interest rates, investment demand and price indices. Due to its complex and nonlinear dynamics, a wide range of mathematical models has been proposed to understand and analyze the behavior of financial systems. In 1993, Huang and Li [9] proposed a financial model to describe the interactions between interest rate, investment demand, and price index. A simplified version this model was rigorously studied in 2001 by Ma and Chen [10,11]. Inspired by the idea of the chaotic Lorenz model [12], Liping et al. [13] formulated a new financial chaotic model. Further, they extended this model into fractional and stochastic versions using the Atangana-Baleanu (AB) fractional derivative. More recently, the modeling of the dynamics of nonlinear financial systems through the Hattaf fractional derivative, which generalizes the AB fractional derivative and others, has been fully studied in [14].

Delay differential equations (DDEs) are a generalization of ordinary differential equations (ODEs) and constitute a powerful mathematical tool for modeling the time delays that naturally occur between actions and responses, which arise in numerous processes across many disciplines. For instance, DDEs have been used in virology to model several time delays such as the time needed for infected cells to produce new virions after viral entry and the time necessary for the newly produced virions to become mature and infectious [15], as well as the immunological delay that refers to the time needed for immune response to recognize a viral antigen and become activated [16]. In epidemiology, DDEs permit to model the incubation period [17]. In macroeconomics, Kalecki [18] introduced the idea that there is a time delay between the decision of investment and its implementation. In 1999, Krawiec and Szydlowski [19] incorporated the idea of Kalecki into Kaldor model [20]. In 2017, Hattaf et al. [21] introduced a second time delay needed for the investment to be productive.

In the field of financial systems, Gao and Ma [22] incorporated a delayed feedback term into the second equation of the simplified financial model of Ma and Chen [10,11]. They investigated the impact of the delay on the financial system by considering it as a bifurcation parameter, and they proved that when this delay exceeds certain critical values, the equilibrium loses its stability and a Hopf bifurcation occurs. In 2019, Zhang and Zhu [23] pointed out that changes in prices do not immediately influence interest rates and that a time lag is often present in this relationship. Consequently, they introduced a time delay into the first equation of the simplified financial model of Ma and Chen in order to account for this price change delay. Additionally, a recent study on government debt and delayed feedback on the investment demand was investigated in 2026 by Phukan et al. [24].

The present study adopts an interdisciplinary approach and aims to understand and describe the complex dynamics of financial systems by developing a general model that encompasses all the previously mentioned financial models, both with and without delays. To this end, the organisation of the remainder of this paper is outlined as follows. The next section presents the new generalized financial model and its particular cases. Section 3 analyzes the dynamical behavior of the model. Finally, the last section is devoted to sensitivity analysis and numerical simulations.

2. New Generalized Financial Model and Its Special Cases

This section begins by developing a new generalized financial model that describes the interactions among various financial factors. It subsequently presents several particular cases of this model that have been introduced in previous studies.

The developed model is formulated by a nonlinear system of DDEs as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{dR}{dt}={\alpha }_{1}P(t-{\tau }_1)+\left(I\left(t\right)-{\mu }_1\right)R\left(t\right)+\gamma R\left(t\right),}\hfill \\ {\displaystyle \frac{dI}{dt}={\alpha }_2-{\mu }_{2}I\left(t\right)-\beta R^2\left(t\right)+\kappa \left(I\left(t\right)-I(t-{\tau }_2)\right),}\hfill \\ {\displaystyle \frac{dP}{dt}={\alpha }_{3}I\left(t\right)R\left(t\right)-{\mu }_{3}P\left(t\right)-\rho R\left(t\right)+\gamma P\left(t\right),}\hfill \end{array}\right.\end{array}$$

(1)

where $R\left(t\right)$, $I\left(t\right)$ and $P\left(t\right)$ are the interest rate, the investment demand and the price index at time t, respectively. The parameters ${\mu }_1$, ${\mu }_2$ and ${\mu }_3$ describe respectively the total saving amount, the cost per investment and the elasticity of the demand of the commercial markets. The interest rate increases at a rate of ${\alpha }_1$ due to an increase in the price index, while ${\alpha }_2$ denotes the natural growth rate of investment. In other words, ${\alpha }_1$ captures the responsiveness of the interest rate to inflation and ${\alpha }_2$ is interpreted as the rate of return on investment. The parameter ${\alpha }_3$ represents the elasticity of investment, indicating that economic expansion and rising production costs prompt firms to raise prices, thereby increasing the rate of change of the price index. In addition, we assume that investment demand is negatively correlated with the square of the interest rate, represented by the term $-\beta R^2$. This assumption models the idea that investment is highly sensitive to high interest rates, as well as the quadratic term emphasizes that the effect accelerates as rates increase. Furthermore, an increase in interest rates leads to a decrease in the price index at a rate of $\rho$. This reflects that higher interest rates increase borrowing costs for firms and households, reducing spending and investment, which lowers overall demand for goods and services and puts downward pressure on prices. Finally, $\gamma$ denotes the amount of debt borrowed from national or international institutions, ${\tau }_1$ represents the delay in price adjustment, and ${\tau }_2$ denotes the time lag in investment demand with feedback factor $\kappa$.

It is important to note that our financial model described by system (1) generalizes and improves various financial models existing in the literature. For example,

• When ${\alpha }_1=1$, ${\alpha }_2=1$, $\beta =1$, ${\alpha }_3=0$, $\rho =1$, $\gamma =0$ and ${\tau }_1={\tau }_2=0$, we obtain the financial model proposed by Huang and Li [9] and investigated by Ma and Chen [10,11].
• When ${\alpha }_1=1$, ${\alpha }_2=2$, $\beta =1$, ${\alpha }_3=1$, $\rho =1$, $\gamma =0$ and ${\tau }_1={\tau }_2=0$, we get the chaotic model in finance presented by Liping et al. [13].
• When ${\alpha }_1=1$, ${\alpha }_2=1$, $\beta =1$, ${\alpha }_3=0$, $\rho =1$, $\gamma =0$ and ${\tau }_2=0$, we obtain the delayed finance model of Zhang and Zhu [23].
• When ${\alpha }_1=1$, ${\alpha }_2=1$, $\beta =1$, ${\alpha }_3=0$, $\rho =1$, $\gamma =0$ and ${\tau }_1=0$,, we get the financial model of Gao and Ma [22].
• When ${\alpha }_1=1$, ${\alpha }_2=1$, $\beta =1$, ${\alpha }_3=0$, $\rho =1$ and ${\tau }_1=0$, we obtain the recent financial model of Phukan et al. [24].

3. Dynamical Behavior of the New Financial Model

This section analyzes the dynamical behavior of our model by establishing the local stability of equilibria and the existence of Hopf bifurcation.

3.1. Equilibria

Now, we discuss the equilibria of our model by resolving the following system of algebraic equations:

$$\begin{array}{ccc}\hfill {\alpha }_{1}P+\left(I-{\mu }_1\right)R+\gamma R& =& 0,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill {\alpha }_2-{\mu }_{2}I-\beta R^2& =& 0,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill {\alpha }_{3}IR-{\mu }_{3}P-\rho R+\gamma P& =& 0.\hfill \end{array}$$

(4)

From (2) and (4), we have

$$[({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})I-(\widetilde{{\mu }_1}\widetilde{{\mu }_3}-{\alpha }_1\rho ]R=0,$$

where $\widetilde{{\mu }_1}={\mu }_1-\gamma$ and $\widetilde{{\mu }_3}={\mu }_3-\gamma$.

• If $R=0$, then $P=0$ and $I={\displaystyle \frac{{\alpha }_2}{{\mu }_2}}$. Hence, model (1) has an equilibrium point of the form $E_0\left(0,{\displaystyle \frac{{\alpha }_2}{{\mu }_2}},0\right)$.
• If $R\ne 0$, then $I={\displaystyle \frac{\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho }{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}}$ provided that ${\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}\ne 0$. According to (3), we have

  $$R^2={\displaystyle \frac{{\alpha }_2({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})-{\mu }_2(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )}{\beta ({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})}}.$$
  Thus, model (1) has two other equilibrium points when ${\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}\ne 0$ and ${\alpha }_2-{\displaystyle \frac{{\mu }_2(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )}{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}}>0$, which are

  $$E_1\left(\sqrt{{\displaystyle \frac{{\alpha }_2({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})-{\mu }_2(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )}{\beta ({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})}}},{\displaystyle \frac{\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho }{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}},P_1\right),$$

  and

  $$E_2\left(-\sqrt{{\displaystyle \frac{{\alpha }_2({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})-{\mu }_2(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )}{\beta ({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})}}},{\displaystyle \frac{\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho }{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}},P_2\right),$$

  with $P_i={\displaystyle \frac{{\alpha }_3\widetilde{{\mu }_1}-\rho }{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}}{R}_i$, for $i=1,2$.

Therefore, we have the following result.

Theorem 3.1.

Assume that ${\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}\ne 0$.

(i)

If ${\alpha }_2-{\displaystyle \frac{{\mu }_2(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )}{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}}\le 0$, then model (1) has a unique equilibrium $E_0\left(0,{\displaystyle \frac{{\alpha }_2}{{\mu }_2}},0\right)$.

(ii)

If ${\alpha }_2-{\displaystyle \frac{{\mu }_2(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )}{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}}>0$, then model (1) has two additional equilibria besides $E_0$, namely $E_1$ and $E_2$, where

$$E_1\left(\sqrt{{\displaystyle \frac{{\alpha }_2}{\beta }}-{\displaystyle \frac{{\mu }_2(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )}{\beta ({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})}}},{\displaystyle \frac{\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho }{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}},{\displaystyle \frac{{\alpha }_3\widetilde{{\mu }_1}-\rho }{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}}\sqrt{{\displaystyle \frac{{\alpha }_2}{\beta }}-{\displaystyle \frac{{\mu }_2(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )}{\beta ({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})}}}\right),$$

and

$$E_2\left(-\sqrt{{\displaystyle \frac{{\alpha }_2}{\beta }}-{\displaystyle \frac{{\mu }_2(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )}{\beta ({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})}}},{\displaystyle \frac{\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho }{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}},{\displaystyle \frac{\rho -{\alpha }_3\widetilde{{\mu }_1}}{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}}\sqrt{{\displaystyle \frac{{\alpha }_2}{\beta }}-{\displaystyle \frac{{\mu }_2(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )}{\beta ({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})}}}\right).$$

Remark 3.2.

Theorem 3.1 generalizes and extends the recent results presented in Propositions 1 and 2 of [24]. Indeed, these propositions are obtained as a special case by choosing ${\alpha }_1={\alpha }_2=\beta =\rho =1$ and ${\alpha }_3=0$.

In the following, we assume that $\gamma <min\{{\mu }_1,{\mu }_3\}$. This assumption means that the level of indebtedness is relatively low compared to two fundamental economic forces that are saving and demand elasticity. In practical terms, this indicates that the economy generates sufficient savings to sustain its activity and limit reliance on debt, while demand remains sufficiently elastic to adjust efficiently to changes in prices or output. Consequently, debt does not constitute a binding constraint in the system, as it is dominated both by internal financing capacity and by the flexibility of demand behavior. On the other hand, it follows from $\gamma <min\{{\mu }_1,{\mu }_3\}$ that ${\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}>0$ and $\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho >0$. Hence, we introduce the following threshold parameter given by

$${\mathcal{T}}_0={\displaystyle \frac{{\alpha }_2({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})}{{\mu }_2(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )}}.$$

(5)

Based on Theorem 3.1, we get the following result.

Corollary 3.3.

(i)

If ${\mathcal{T}}_0\le 1$, then model (1) has a unique equilibrium $E_0\left(0,{\displaystyle \frac{{\alpha }_2}{{\mu }_2}},0\right)$.

(ii)

If ${\mathcal{T}}_0>1$, then model (1) has two additional equilibria $E_1$ and $E_2$ besides $E_0$, where

$$E_1\left(\sqrt{{\displaystyle \frac{1}{\beta }}\left(1-{\displaystyle \frac{1}{{\mathcal{T}}_0}}\right)},{\displaystyle \frac{{\alpha }_2}{{\mu }_2{\mathcal{T}}_0}},{\displaystyle \frac{{\alpha }_3\widetilde{{\mu }_1}-\rho }{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}}\sqrt{{\displaystyle \frac{1}{\beta }}\left(1-{\displaystyle \frac{1}{{\mathcal{T}}_0}}\right)}\right),$$

and

$$E_2\left(-\sqrt{{\displaystyle \frac{1}{\beta }}\left(1-{\displaystyle \frac{1}{{\mathcal{T}}_0}}\right)},{\displaystyle \frac{{\alpha }_2}{{\mu }_2{\mathcal{T}}_0}},{\displaystyle \frac{\rho -{\alpha }_3\widetilde{{\mu }_1}}{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}}\sqrt{{\displaystyle \frac{1}{\beta }}\left(1-{\displaystyle \frac{1}{{\mathcal{T}}_0}}\right)}\right).$$

3.2. Stability Analysis and Hopf Bifurcation

Obviously, the characteristic equation at $E_i$ ($i=0,1,2$) of model (1) is given by

$$\left|\begin{array}{ccc}{I}_i-\widetilde{{\mu }_1}-\lambda & R_i& {\alpha }_1{e}^{-\lambda {\tau }_1}\\ 2\beta R_i& -{\mu }_2+\kappa -\kappa e^{-\lambda {\tau }_2}-\lambda & 0\\ {\alpha }_3{I}_i-\rho & {\alpha }_3{R}_i& -\widetilde{{\mu }_3}-\lambda \end{array}\right|=0.$$

(6)

Here, we investigate only the stability of $E_0$, as the other equilibria can be analyzed similarly.

At $E_0\left(0,{\displaystyle \frac{{\alpha }_2}{{\mu }_2}},0\right)$, (6) becomes

$$(\lambda +{\mu }_2-\kappa +\kappa e^{-\lambda {\tau }_2})({\lambda }^2+a_1\lambda +a_2+b_1{e}^{-\lambda {\tau }_1})=0,$$

(7)

where

$$\begin{array}{ccc}\hfill a_1& =& \widetilde{{\mu }_1}+\widetilde{{\mu }_3}-{\displaystyle \frac{{\alpha }_2}{{\mu }_2}},\hfill \\ \hfill a_2& =& \widetilde{{\mu }_3}(\widetilde{{\mu }_1}-{\displaystyle \frac{{\alpha }_2}{{\mu }_2}}),\hfill \\ \hfill b_1& =& {\alpha }_1(\rho -{\displaystyle \frac{{\alpha }_2{\alpha }_3}{{\mu }_2}}).\hfill \end{array}$$

Lemma 3.4.

If ${\mathcal{T}}_0>1$, then the equilibrium $E_0$ is unstable for all ${\tau }_1,{\tau }_2\ge 0$.

Proof. 

Let $Q_1\left(\lambda \right)={\lambda }^2+a_1\lambda +a_2+b_1{e}^{-\lambda {\tau }_1}$. We have ${\displaystyle \underset{\lambda \to +\infty }{lim}{Q}_1\left(\lambda \right)=+\infty }$ and $Q_1\left(0\right)=a_2+b_1=(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )(1-{\mathcal{T}}_0)<0$. By the intermediate value theorem, there exists a ${\lambda }_0\in (0,+\infty )$ such that $Q_1\left({\lambda }_0\right)=0$. Consequently, (7) admits at least one positive eigenvalue, implying that $E_0$ is unstable when ${\mathcal{T}}_0>1$. ■

For the case ${\mathcal{T}}_0<1$, we use a similar technique as in [21,25] by distinguishing three cases.

The case ${\tau }_1={\tau }_2=0$. When ${\tau }_1={\tau }_2=0$, then (7) becomes

$$(\lambda +{\mu }_2)[{\lambda }^2+(\widetilde{{\mu }_1}+\widetilde{{\mu }_3}-{\displaystyle \frac{{\alpha }_2}{{\mu }_2}})\lambda +(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )(1-{\mathcal{T}}_0)]=0.$$

(8)

Hence, the roots of (8) are: ${\lambda }_1=-{\mu }_2<0$, ${\lambda }_2$ and ${\lambda }_3$ are determined by

$${\lambda }^2+(\widetilde{{\mu }_1}+\widetilde{{\mu }_3}-{\displaystyle \frac{{\alpha }_2}{{\mu }_2}})\lambda +(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )(1-{\mathcal{T}}_0)=0.$$

(9)

Since ${\mathcal{T}}_0<1$, we deduce that all roots of (9) have negative real parts if and only if

$${\tilde{\mu }}_1+{\tilde{\mu }}_3-{\displaystyle \frac{{\alpha }_2}{{\mu }_2}}>0.$$

(H1)

Therefore, we have the following result.

Lemma 3.5.

If ${\mathcal{T}}_0<1$ and ($H_1$) holds, then the equilibrium $E_0$ is locally asymptotically stable for ${\tau }_1={\tau }_2=0$.

Remark 3.6.

Lemma 3.5 provides a generalization of Lemma 2 in [23] and Proposition 3 in [24], both of which are recovered as particular cases of the present result within a more general framework.

The case ${\tau }_1\ne 0$ and ${\tau }_2=0$. Under this case, (7) can be written in the following form

$$\left(\lambda +{\mu }_2\right)({\lambda }^2+a_1\lambda +a_2+b_1{e}^{-\lambda {\tau }_1})=0.$$

(10)

It obvious that ${\lambda }_1=-{\mu }_2<0$ or

$${\lambda }^2+a_1\lambda +a_2+b_1{e}^{-\lambda {\tau }_1}=0.$$

(11)

Let $i\varpi$ ($\varpi >0$) be a root of (11). Hence,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-{\varpi }^2+a_2=-b_{1}cos\left(\varpi {\tau }_1\right),\hfill \\ \varpi a_1=b_{1}sin\left(\varpi {\tau }_1\right),\hfill \end{array}\right.\end{array}$$

(12)

which leads to

$${\varpi }^4+(a_1^2-2a_2){\varpi }^2+a_2^2-b_1^2=0.$$

(13)

Let $z={\varpi }^2$. Then (13) becomes

$$z^2+(a_1^2-2a_2)z+a_2^2-b_1^2=0.$$

Since $a_1^2-2a_2={(\widetilde{{\mu }_1}-{\displaystyle \frac{{\alpha }_2}{{\mu }_2}})}^2+\widetilde{{\mu }_3}>0$ and $a_2+b_1=(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )(1-{\mathcal{T}}_0)>0$ for ${\mathcal{T}}_0<1$, we easily get the following result.

Lemma 3.7.

Assume that ${\mathcal{T}}_0<1$.

(i)

If $a_2\ge b_1$, then (13) has no positive root.

(ii)

If $a_2<b_1$, then (13) has a unique positive root given by

$${\varpi }_0={\displaystyle \frac{\sqrt{2}}{2}}{\left(-{(\widetilde{{\mu }_1}-{\displaystyle \frac{{\alpha }_2}{{\mu }_2}})}^2-{\widetilde{{\mu }_3}}^2+\sqrt{{[{(\widetilde{{\mu }_1}-{\displaystyle \frac{{\alpha }_2}{{\mu }_2}})}^2+{\widetilde{{\mu }_3}}^2]}^2-4(a_2^2-b_1^2)}\right)}^{{\displaystyle \frac{1}{2}}}.$$

(14)

According to Lemmas 3.5 and 3.7, we obtain the following theorem.

Theorem 3.8.

Assume that ${\mathcal{T}}_0<1$ and ($H_1$) holds. For ${\tau }_2=0$, we have

(i)

If $a_2\ge b_1$, then the equilibrium $E_0$ is locally asymptotically stable for all ${\tau }_1\ge 0$.

(ii)

If $a_2<b_1$, then system (1) undergoes a Hopf bifurcation at $E_0$ when ${\tau }_1={\tau }_{1,j}$, $j\in \mathrm{I}\mathrm{N}$. Moreover, the equilibrium $E_0$ is locally asymptotically stable for ${\tau }_1\in [0,{\tau }_{1,0})$ and unstable for ${\tau }_1>{\tau }_{1,0}$, where

$${\tau }_{1,n}={\displaystyle \frac{1}{{\varpi }_0}}arccos\left({\displaystyle \frac{{\alpha }_2\widetilde{{\mu }_3}+{\mu }_2({\varpi }_0^2-\widetilde{{\mu }_1}\widetilde{{\mu }_3})}{{\alpha }_1(\rho {\mu }_2-{\alpha }_2{\alpha }_3)}}\right)+{\displaystyle \frac{2n\pi }{{\varpi }_0}},n\in \mathrm{I}\mathrm{N}.$$

(15)

Proof. 

The first result $\left(i\right)$ is evident and follows directly from Lemma 3.5 and part $\left(i\right)$ of Lemma 3.7.

On the other hand, it follows from $\left(ii\right)$ of Lemma 3.7 that (13) has a unique positive root ${\varpi }_0$. From (12), we have

$${\tau }_{1,n}={\displaystyle \frac{1}{{\varpi }_0}}arccos\left({\displaystyle \frac{{\alpha }_2\widetilde{{\mu }_3}+{\mu }_2({\varpi }_0^2-\widetilde{{\mu }_1}\widetilde{{\mu }_3})}{{\alpha }_1(\rho {\mu }_2-{\alpha }_2{\alpha }_3)}}\right)+{\displaystyle \frac{2n\pi }{{\varpi }_0}},$$

for $n\in \mathrm{I}\mathrm{N}=\{0,1,2,\cdots \}$. Hence, $\pm i{\varpi }_0$ is a pair of purely imaginary roots of (11) with ${\tau }_1={\tau }_{1,n}$. So, let $\lambda \left({\tau }_1\right)=\nu \left({\tau }_1\right)+i\varpi \left({\tau }_1\right)$ be the root of (11) with $\nu \left({\tau }_{1,n}\right)=0$ and $\varpi \left({\tau }_{1,n}\right)={\varpi }_0$. Differentiating both sides of (11) with respect to ${\tau }_1$ yields

$$\left(2\lambda +a_1-b_1{\tau }_1{e}^{-\lambda {\tau }_1}\right){\displaystyle \frac{d\lambda }{d{\tau }_1}}=b_1\lambda e^{-\lambda {\tau }_1},$$

which implies that

$${\left({\displaystyle \frac{d\lambda }{d{\tau }_1}}\right)}^{-1}={\displaystyle \frac{(2\lambda +a_1)e^{\lambda {\tau }_2}}{b_1\lambda }}-{\displaystyle \frac{{\tau }_1}{\lambda }}={\displaystyle \frac{2\lambda +a_1}{-{\lambda }^3-a_1{\lambda }^2-a_2\lambda }}-{\displaystyle \frac{{\tau }_1}{\lambda }}.$$

Hence,

$$\begin{array}{ccc}\hfill Re{\left({\displaystyle \frac{d\lambda }{d{\tau }_1}}\right)}^{-1}{|}_{\lambda =i{\varpi }_0}& =& Re\left({\displaystyle \frac{a_1+2i{\varpi }_0}{a_1{\varpi }_0^2+i{\varpi }_0({\varpi }_0^2-a_2)}}\right)\hfill \\ & =& {\displaystyle \frac{2{\varpi }_0^2+a_1^2-2a_2}{a_1^2{\varpi }_0^2+{({\varpi }_0^2-a_2)}^2}}\hfill \\ & =& {\displaystyle \frac{2{\varpi }_0^2+{(\widetilde{{\mu }_1}-\frac{{\alpha }_2}{{\mu }_2})}^2+\widetilde{{\mu }_3}}{a_1^2{\varpi }_0^2+{({\varpi }_0^2-a_2)}^2}}>0.\hfill \end{array}$$

Thus, the transversality condition is satisfied. It follows from the Hopf bifurcation theorem [26] that system (1) undergoes a Hopf bifurcation at $E_0$ when ${\tau }_1={\tau }_{1,n}$, $n\in \mathrm{I}\mathrm{N}$. Moreover, by Lemma 3.5, the equilibrium $E_0$ is locally asymptotically stable when ${\tau }_1=0$. Consequently, all roots of the characteristic equation (11) have negative real parts at ${\tau }_1=0$. Since ${\tau }_{1,0}$ is the smallest value among the critical delays ${\tau }_{1,j}$ for which the roots of (11) become purely imaginary, it follows that $E_0$ remains locally asymptotically stable for $0\le {\tau }_1<{\tau }_{1,0}$ and becomes unstable for ${\tau }_1>{\tau }_{1,0}$. This completes the proof. ■

Remark 3.9.

Theorem 3.8 extends the results presented in Theorems 1 and 2 of [23].

The case ${\tau }_1\ne 0$ and ${\tau }_2\ne 0$. In this case, we consider (7) with ${\tau }_2>0$ and ${\tau }_1$ in the stable regions. Regards ${\tau }_2$ as a parameter of bifurcation.

Since ${\tau }_1$ is considered in the stable regions, it suffices to investigate the stability analysis of the first characteristic factor of (7) given as follows:

$$\lambda +{\mu }_2-\kappa +\kappa e^{-\lambda {\tau }_2}=0.$$

(16)

Let $i\psi$ ($\psi >0$) be a root of (16). Then

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mu }_2-\kappa =-\kappa cos\left(\psi {\tau }_2\right),\hfill \\ \psi =\kappa sin\left(\psi {\tau }_2\right),\hfill \end{array}\right.\end{array}$$

(17)

which implies that

$${\psi }^2={\mu }_2(2\kappa -{\mu }_2).$$

(18)

• If $\kappa \le {\displaystyle \frac{{\mu }_2}{2}}$, then (18) has no positive root.
• If $\kappa >{\displaystyle \frac{{\mu }_2}{2}}$, then (18) has a unique positive root given by

  $${\psi }_0=\sqrt{{\mu }_2(2\kappa -{\mu }_2)}.$$

  (19)

By (17), we get

$${\tau }_{2,n}={\displaystyle \frac{1}{{\psi }_0}}arccos\left({\displaystyle \frac{\kappa -{\mu }_2}{\kappa }}\right)+{\displaystyle \frac{2n\pi }{{\psi }_0}},$$

(20)

for $n\in \mathrm{I}\mathrm{N}$. So, $\pm i{\psi }_0$ is a pair of purely imaginary roots of (16) with ${\tau }_2={\tau }_{2,n}$. In addition, it follows from (16) that

$$\left(1-\kappa {\tau }_2{e}^{-\lambda {\tau }_2}\right){\displaystyle \frac{d\lambda }{d{\tau }_2}}=\kappa \lambda e^{-\lambda {\tau }_2}.$$

Then

$${\left({\displaystyle \frac{d\lambda }{d{\tau }_2}}\right)}^{-1}={\displaystyle \frac{e^{\lambda {\tau }_2}}{\kappa \lambda }}-{\displaystyle \frac{{\tau }_2}{\lambda }}=-{\displaystyle \frac{1}{\lambda (\lambda +{\mu }_2-\kappa )}}-{\displaystyle \frac{{\tau }_2}{\lambda }}.$$

Thus,

$$\begin{array}{ccc}\hfill Re{\left({\displaystyle \frac{d\lambda }{d{\tau }_2}}\right)}^{-1}{|}_{\lambda =i{\psi }_0}& =& Re\left({\displaystyle \frac{{\psi }_0+i({\mu }_2-\kappa )}{{\psi }_0[{({\mu }_2-\kappa )}^2+{\psi }_0^2]}}\right)\hfill \\ & =& {\displaystyle \frac{1}{{({\mu }_2-\kappa )}^2+{\psi }_0^2}}>0,\hfill \end{array}$$

which shows that the transversality condition holds. Hence, system (1) undergoes a Hopf bifurcation at the equilibrium point $E_0$ when ${\tau }_2={\tau }_{2,n}$, $n\in \mathrm{I}\mathrm{N}$.

Based on the preceding analysis, we facilely obtain the following results.

Theorem 3.10.

Let ${\tau }_1$ in the stable regions. For ${\tau }_2$, we have

(i)

If $\kappa \le {\displaystyle \frac{{\mu }_2}{2}}$, then the equilibrium $E_0$ is locally asymptotically stable for all ${\tau }_2\ge 0$.

(ii)

If $\kappa >{\displaystyle \frac{{\mu }_2}{2}}$, then system (1) undergoes a Hopf bifurcation at $E_0$ when ${\tau }_2={\tau }_{2,n}$, $n\in \mathrm{I}\mathrm{N}$. Additionally, the equilibrium $E_0$ is locally asymptotically stable when ${\tau }_2<{\tau }_{2,0}$ and becomes unstable when ${\tau }_2>{\tau }_{2,0}$.

Remark 3.11.

Theorem 3.10 provides a more general result that includes Theorem 1 of [22] and Theorem 2 of [27] as special cases.

4. Sensitivity Analysis and Numerical Analysis

This section deals with sensitivity analysis and numerical simulations.

Sensitivity analysis helps to study how variations in model parameters influence the dynamics of a financial system. Since the value of the threshold parameter ${\mathcal{T}}_0$ determines the stability of the trivial equilibrium, we compute the sensitivity index of ${\mathcal{T}}_0$ with respect to each parameter p as follows:

$${{\rm Y}}_{{\mathcal{T}}_0}^p={\displaystyle \frac{p}{{\mathcal{T}}_0}}{\displaystyle \frac{\partial {\mathcal{T}}_0}{\partial p}}.$$

(21)

Based on the explicit formula of ${\mathcal{T}}_0$ established in (5), we have

$$\begin{array}{ccc}\hfill {{\rm Y}}_{{\mathcal{T}}_0}^{{\alpha }_1}& =& {\displaystyle \frac{{\alpha }_1\widetilde{{\mu }_3}({\alpha }_3\widetilde{{\mu }_1}-\rho )}{(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})}},\hfill \\ \hfill {{\rm Y}}_{{\mathcal{T}}_0}^{{\alpha }_2}& =& 1,\hfill \\ \hfill {{\rm Y}}_{{\mathcal{T}}_0}^{{\alpha }_3}& =& {\displaystyle \frac{{\alpha }_1{\alpha }_3}{{\alpha }_1{\alpha }_3+\widetilde{{\mu }_3}}},\hfill \\ \hfill {{\rm Y}}_{{\mathcal{T}}_0}^{{\mu }_1}& =& {\displaystyle \frac{-{\mu }_1\widetilde{{\mu }_3}}{\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho }},\hfill \\ \hfill {{\rm Y}}_{{\mathcal{T}}_0}^{{\mu }_2}& =& -1,\hfill \\ \hfill {{\rm Y}}_{{\mathcal{T}}_0}^{{\mu }_3}& =& {\displaystyle \frac{{\alpha }_1{\mu }_3(\rho -{\alpha }_3\widetilde{{\mu }_1})}{(\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho )({\alpha }_1{\alpha }_3+\widetilde{{\mu }_3})}},\hfill \\ \hfill {{\rm Y}}_{{\mathcal{T}}_0}^{\beta }& =& 0,\hfill \\ \hfill {{\rm Y}}_{{\mathcal{T}}_0}^{\gamma }& =& {\displaystyle \frac{\gamma [\widetilde{{\mu }_3}({\alpha }_1{\alpha }_3+1)+{\alpha }_1({\alpha }_3\widetilde{{\mu }_1}-\rho )]}{\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho }},\hfill \\ \hfill {{\rm Y}}_{{\mathcal{T}}_0}^{\rho }& =& {\displaystyle \frac{-{\alpha }_1\rho }{\widetilde{{\mu }_1}\widetilde{{\mu }_3}+{\alpha }_1\rho }}.\hfill \end{array}$$

Figure 1 presents the sensitivity indices associated with ${\mathcal{T}}_0$. It is known that positive sensitivity indices indicate that an increase in the corresponding parameter leads to an increase in ${\mathcal{T}}_0$, whereas negative indices indicate that increasing the parameter reduces ${\mathcal{T}}_0$. As shown in Table 1, the parameters ${\alpha }_2$, ${\alpha }_3$, and ${\mu }_3$ exert the greatest influence on the dynamics of the financial system. Specifically, increasing any of these parameters results in an increase in the threshold quantity ${\mathcal{T}}_0$. Conversely, increasing the values of ${\alpha }_1$, ${\mu }_1$, ${\mu }_2$, $\rho$ and $\gamma$ leads to a decrease in ${\mathcal{T}}_0$.

To illustrate our analytical results, we use the values of the parameters presented in Table 1 with $\beta =1$ and $\kappa =0.8$. By a simple computation, the threshold parameter is equal to ${\mathcal{T}}_0=0.3065<1$. It follows from Corollary 3.3 that model (1) has a unique equilibrium $E_0\left(0,1.1111,0\right)$. Further, we have ${\tilde{\mu }}_1+{\tilde{\mu }}_3-{\displaystyle \frac{{\alpha }_2}{{\mu }_2}}=1.2689>0$. Hence, the hypothesis ($H_1$) is satisfied. According to Lemma 3.5, the equilibrium $E_0$ is locally asymptotically stable when ${\tau }_1={\tau }_2=0$. We now investigate the dynamical behavior of model (1) for various values of the two time delays ${\tau }_1$ and ${\tau }_2$. The presence of these delays can substantially influence the stability of the equilibrium and may generate complex dynamics, including oscillatory solutions and bifurcation phenomena. By varying ${\tau }_1$ and ${\tau }_2$, we analyze how the delays affect the qualitative behavior of the system and determine the critical conditions under which stability changes occur.

For ${\tau }_2=0$, all the assumptions of Theorem 3.8 (ii) are fulfilled. Therefore, system (1) undergoes a Hopf bifurcation at the equilibrium $E_0\left(0,1.1111,0\right)$ when the first delay reaches the critical value ${\tau }_1={\tau }_{1,0}=1.5663$. Moreover, the equilibrium $E_0$ is locally asymptotically stable for ${\tau }_1\in [0,{\tau }_{1,0})$, while it becomes unstable for ${\tau }_1>{\tau }_{1,0}$. These dynamical behaviors are illustrated in Figure 2, Figure 3 and Figure 4.

Next, fixing ${\tau }_1$ in the stable region $\mathcal{G}=[0,1.5663)$ and choosing ${\tau }_1=1.2\in \mathcal{G}$, we obtain the second critical delay value ${\tau }_2={\tau }_{2,0}=2.1369$. Furthermore, since $\kappa =0.8>{\displaystyle \frac{{\mu }_2}{2}}=0.45$, Theorem 3.10 (ii) implies that system (1) undergoes a Hopf bifurcation at the equilibrium $E_0$ when ${\tau }_2={\tau }_{2,0}$. In addition, the equilibrium $E_0$ remains locally asymptotically stable for ${\tau }_2<{\tau }_{2,0}$ and loses its stability for ${\tau }_2>{\tau }_{2,0}$. These results are illustrated in Figure 5, Figure 6 and Figure 7.