Fold-Fold Singularity in a Piecewise Smooth Mathematical Model Describing the Dynamics of a Stockless Market

Abstract

Fold-fold singularities are critical points or singularities in piecewise smooth dynamical systems (PWS) where both the stability and the structure of the system change. These singularities are of great importance in the study of specific dynamics, such as those in markets, as they indicate significant transformations in their evolution, including sudden variability in prices or changes in the behavior of offers and demand. Despite the substantial increase in the use of mathematical and computational tools applied to market dynamics, the current literature does not thoroughly address the study of the existence of fold-fold singularities in piecewise smooth systems within this context. Therefore, due to the importance of markets as economic activities, this paper proves the existence of such a singularity in a mathematical model that describes the dynamics of a stockless market, which is represented by a system of ordinary differential equations defined with piecewise smooth functions.

1. Introduction

The classical theory of dynamical systems, described by smooth differential equations, has developed in both a geometrical and qualitative sense. It has been used to analyze the dynamical behavior of different applied models. Conversely, many practical situations are defined by systems of equations that exhibit discontinuities in certain regions or surfaces. In these scenarios, the theoretical basis for the most important results remains to be fully explored.

Lately, the study of systems of piecewise smooth differential equations, which Filippov has called, in [1], discontinuous righthand sides, has increased considerably due to their applicability in different phenomena presented in science, engineering, economics, medicine, and other fields; see [2,3,4,5,6,7]. These systems are characterized by having surfaces (in the case of ${\mathbb{R}}^3$) or hypersurfaces (in higher-dimensional spaces) that divide the state space, where discontinuities of the functions that define the system of differential equations occur, thus determining different vector fields in the different regions. Depending on how the system is defined, these surfaces, or so-called switching manifolds, may have crossing or sliding zones [8].

Sliding zones can be characterized by stability (attracting) or instability (repulsing or escaping). A vector field can be defined on a switching surface as a convex combination of the two vector fields adjacent to such a switching surface, which is called a Filippov vector field. The solutions are known as sliding segments, whose evolution is subject to the switching surface; see [8]. It should be clarified that this definition, called the Filippov convention, is the one implemented in this work. At this point, we highlight the efforts of some authors, such as [9], who, in their research, have defined this vector field in a more general way, including non-linear terms that generalize the definition of sliding dynamics, leading to what they have called hidden dynamics.

The aim of this paper is to prove the existence of a fold-fold singularity in a mathematical model defined by a system of piecewise smooth differential equations, which describes the dynamics of a stockless market. A fold-fold singularity is a point that satisfies Definitions 4 and 7 in the following sections. We highlight the importance of this study, as bibliographic searches in various databases and academic repositories reveal that there are few papers that combine the theory of piecewise smooth systems, fold-fold singularities, and market models. The lack of literature on this subject creates a broad field of study and an opportunity for future research to explore more deeply the implications of the existence of these singularities in market dynamics and their possible consequences for financial decision-making.

In order to ensure that this document is self-contained and to facilitate its reading, the rest of Section 1 provides a brief description of the theory framing piecewise smooth dynamical systems and some singularities. Here, we reproduce some basic definitions and concepts that can be accessed in [1,8,10,11,12,13,14,15]. In Section 2, we present the basic theory of piecewise smooth systems and tangencies. In Section 3, we show a mathematical model defined through a system of piecewise smooth differential equations that describes the dynamics of a stockless market. The main part of this paper, which is presented in Section 4, aims to prove the existence of a fold-fold singularity in the aforementioned model. To complement the study, Section 5 illustrates some numerical simulations that verify the analytical results obtained, which are of great relevance since the literature is scarce regarding the identification of such singularities or tangencies in everyday dynamics. Finally, Section 6 and Section 7 present the discussion and conclusions, respectively.

2. Review of Piecewise Smooth Systems

In this paper, we analyze, through the lens of the theory of piecewise smooth systems, the singularities presented in a mathematical model describing the dynamics of a stockless market. In particular, we focus on the definitions of fold-fold singularities and classify tangencies.

Definition 1.

In general, a piecewise smooth system (PWS) is defined as a finite set of ordinary differential equations

$$\dot{\mathbf{x}}=f_i\left(\mathbf{x}\right);\mathbf{x}\in S_i\subset {\mathbb{R}}^n$$

(1)

where the vector fields $f_i$ are smooth and defined on the different disjoint open sets $S_i$, $i=1,2,3,\dots ,n$ [15].

The sets $S_i$ are defined in terms of smooth scalar fields $H^{\left(i\right)}\left(\mathbf{x}\right)$, with non-zero gradient $H_{\mathbf{x}}^{\left(i\right)}\left(\mathbf{x}\right)$ on the i-th surface, for $i=1,2,\dots ,n-1$, as follows:

$$\begin{array}{ccc}\hfill S_i& =& \{\mathbf{x}\in {\mathbb{R}}^n:H^{\left(i\right)}\left(\mathbf{x}\right)<0\}\hfill \\ \hfill S_{i+1}& =& \{\mathbf{x}\in {\mathbb{R}}^n:H^{\left(i\right)}\left(\mathbf{x}\right)>0\},\hfill \end{array}$$

which are separated by the ${\Sigma }_i$ surfaces, $i=1,2,\cdots ,n-1$, called switching manifolds or switching boundaries. These are $(n-1)$-dimensional and are defined by the 0 level set of the smooth scalar fields $H^{\left(i\right)}\left(\mathbf{x}\right)$. In particular,

$${\Sigma }_i=\{\mathbf{x}\in {\mathbb{R}}^n:H^{\left(i\right)}\left(\mathbf{x}\right)=0\}.$$

(2)

Definition 2.

An equilibrium of the system $\dot{\mathbf{x}}=f_i\left(\mathbf{x}\right)$ is a point $\mathbf{x}$ where

$$f_i\left(\mathbf{x}\right)=0.$$

In order to analyze the geometry of the solutions on or near the switching surfaces ${\Sigma }_i$, it is necessary to introduce the following notion of derivatives. This allows us, among other aspects, to detect tangencies between solutions and boundaries. The directional derivative of $H^{\left(i\right)}$ with respect to the vector field $f_i$ is defined in terms of the Lie derivative, denoted by ${\mathcal{L}}_{f_i}$, and through the inner product

$${\mathcal{L}}_{f_i}{H}^{\left(i\right)}\left(\mathbf{x}\right)=\langle H_{\mathbf{x}}^{\left(i\right)}\left(\mathbf{x}\right),f_i\left(\mathbf{x}\right)\rangle .$$

(3)

The m-th order Lie derivative, as noted by [13], is denoted by ${\mathcal{L}}_{f_i}^m{H}^{\left(i\right)}\left(\mathbf{x}\right)$. Specifically,

$${\mathcal{L}}_{f_i}^2{H}^{\left(i\right)}\left(\mathbf{x}\right)={\mathcal{L}}_{f_i}\left({\mathcal{L}}_{f_i}{H}^{\left(i\right)}\left(\mathbf{x}\right)\right)=\langle \nabla \langle H_{\mathbf{x}}^{\left(i\right)}\left(\mathbf{x}\right),f_i\left(\mathbf{x}\right)\rangle ,f_i\left(\mathbf{x}\right)\rangle ,$$

where $\nabla (·)$ represents the gradient.

By applying the Lie derivative, we can identify the regions or points on the switching surfaces ${\Sigma }_i$ where crossing or sliding occurs. To do this we, define

$${\sigma }^{\left(i\right)}\left(\mathbf{x}\right)=\left({\mathcal{L}}_{f_i}{H}^{\left(i\right)}\right)\left({\mathcal{L}}_{f_{i+1}}{H}^{\left(i\right)}\right);i=1,2,\dots ,n-1.$$

Now, we have the following.

• We have crossing at the points on ${\Sigma }_i$ where ${\sigma }^{\left(i\right)}\left(\mathbf{x}\right)>0$. This set is referred to as the set of crossing points and is represented by ${\Sigma }_i^c$.
• We have sliding at the points on ${\Sigma }_i$ where ${\sigma }^{\left(i\right)}\left(\mathbf{x}\right)<0$. This set is called the set of sliding points and is denoted by ${\Sigma }_i^s$.

As mentioned earlier, for sliding to occur, the following conditions must be met: ${\sigma }^{\left(i\right)}\left(\mathbf{x}\right)<0$, meaning that the expressions ${\mathcal{L}}_{f_i}{H}^{\left(i\right)}$ and ${\mathcal{L}}_{f_{i+1}}{H}^{\left(i\right)}$ must have opposite signs. Therefore, the following cases are considered.

• ${\mathcal{L}}_{f_{i+1}}{H}^{\left(i\right)}<0<{\mathcal{L}}_{f_i}{H}^{\left(i\right)}$. This implies that the vector fields $f_i$ and $f_{i+1}$ are oriented towards the surface at a point $\mathbf{x}\in {\Sigma }_i$. These points constitute the set of points of stable sliding and are denoted as ${\Sigma }_i^{ss}$.
• ${\mathcal{L}}_{f_{i+1}}{H}^{\left(i\right)}>0>{\mathcal{L}}_{f_i}{H}^{\left(i\right)}$. This implies that the vector fields $f_i$ and $f_{i+1}$ point away from the switching surface at a point $\mathbf{x}\in {\Sigma }_i$; see [13]. These points form the set of unstable sliding or escape points, which is denoted by ${\Sigma }_i^{us}$.

It is worth noting that ${\Sigma }_i^s={\Sigma }_i^{ss}\cup {\Sigma }_i^{us}$.

For $\mathbf{x}\in {\Sigma }_i^s$, it is defined a vector field $g_i\left(\mathbf{x}\right)$, called a Filippov vector field, using the convex Filippov method [15]. This vector field reflects dynamics through

$$g_i\left(\mathbf{x}\right)={\lambda }_i\left(\mathbf{x}\right)f_i\left(\mathbf{x}\right)+(1-{\lambda }_i\left(\mathbf{x}\right))f_{i+1}\left(\mathbf{x}\right),$$

where

$${\lambda }_i\left(\mathbf{x}\right)={\displaystyle \frac{\langle H_{\mathbf{x}}^{\left(i\right)}\left(\mathbf{x}\right),f_{i+1}\left(\mathbf{x}\right)\rangle }{\langle H_{\mathbf{x}}^{\left(i\right)}\left(\mathbf{x}\right),f_{i+1}\left(\mathbf{x}\right)-f_i\left(\mathbf{x}\right)\rangle }}.$$

Thus, in ${\Sigma }_i^s$, we have the Filippov system

$$\dot{\mathbf{x}}=g_i\left(\mathbf{x}\right).$$

As can be found in [15], points $\mathbf{x}$ in ${\Sigma }_i^s$, where

$${\mathcal{L}}_{f_{i+1}-f_i}{H}^{\left(i\right)}\left(\mathbf{x}\right)=\langle H_{\mathbf{x}}^{\left(i\right)}\left(\mathbf{x}\right),f_{i+1}\left(\mathbf{x}\right)-f_i\left(\mathbf{x}\right)\rangle =0;f_{i,i+1}\left(\mathbf{x}\right)\ne 0,$$

are called singular sliding points. At such points, either both vectors $f_i\left(\mathbf{x}\right)$ and $f_{i+1}\left(\mathbf{x}\right)$ are tangent to ${\Sigma }_i$, one of them vanishes while the other is tangent to ${\Sigma }_i$, or they both vanish.

It is important to note that some authors have defined this vector field in a more general manner, including non-linear terms that generalize the definition of sliding dynamics [9,16].

Moreover, any smooth solution $\mathbf{x}\left(t\right)$ completely contained in one of the regions $S_i$, defined for any $t\in I$, with $I\subset \mathbb{R}$, will be referred to as a segment. Segments confined to the switching surfaces and determined by the Filippov vector fields will be called sliding segments; the others will be referred to as non-sliding segments. Furthermore, an orbit is a continuous concatenation of sliding and non-sliding segments. See [13].

Finally, we have the following definition, which can be viewed in [11].

Definition 3.

A pseudo-equilibrium is an equilibrium of the Filippov vector field, which lies on the switching surface where the Filippov vector field has been defined.

2.1. Tangencies

The concept of tangencies is fundamental to the understanding of the dynamics in a switching manifold. They form the boundaries between the crossing, sliding, and escape regions on the ${\Sigma }_i$ surfaces. Generally, these boundaries can occur in two ways: either where the ${\Sigma }_i$ surfaces are non-smooth, which will be called a boundary intersection, or where the ${\Sigma }_i$ surfaces are smooth but one of the adjacent vector fields is tangent, satisfying the tangency condition

$${\mathcal{L}}_{f_i}{H}^{\left(i\right)}=0.$$

The three simplest types of tangencies that can be observed in the smooth regions of the ${\Sigma }_i$ are a fold or quadratic tangency, a cusp or cubic tangency, and a fold-fold or double tangency. The latter is detailed below. According to [11], the fold-fold tangency or double tangency occurs when there is quadratic contact on both sides of the switching surface. More specifically, it is defined as follows.

Definition 4.

A point $\mathbf{x}\in {\Sigma }_i$ is called a tangency of the fold-fold type or a double tangency if it simultaneously satisfies the following conditions:

• ${\mathcal{L}}_{f_i}{H}^{\left(i\right)}={\mathcal{L}}_{f_{i+1}}{H}^{\left(i\right)}=0$,
• ${\mathcal{L}}_{f_i}^2{H}^{\left(i\right)}\ne 0$; ${\mathcal{L}}_{f_{i+1}}^2{H}^{\left(i\right)}\ne 0$;
• the gradient vectors of $H^{\left(i\right)}$, ${\mathcal{L}}_{f_i}{H}^{\left(i\right)}$, and ${\mathcal{L}}_{f_{i+1}}{H}^{\left(i\right)}$ are linearly independent.

This tangency is of the following type:

• Visible-visible, if ${\mathcal{L}}_{f_i}^2{H}^{\left(i\right)}>0>{\mathcal{L}}_{f_{i+1}}^2{H}^{\left(i\right)}$;
• Invisible-invisible, if ${\mathcal{L}}_{f_i}^2{H}^{\left(i\right)}<0<{\mathcal{L}}_{f_{i+1}}^2{H}^{\left(i\right)}$;
• Visible-invisible, if $\left({\mathcal{L}}_{f_i}^2{H}^{\left(i\right)}\right)\left({\mathcal{L}}_{f_{i+1}}^2{H}^{\left(i\right)}\right)>0$.

A tangency is called visible or invisible depending on whether the orbits curve locally away from or towards the surface. This is also true for sliding segments that are tangent to the boundary of the sliding region [11]. Furthermore, it follows that if the fold-fold tangency is invisible-invisible, then it is called a T-singularity of Teixeira or T-singularity; see [17].

2.2. Bifurcations

The next definition that can be found in [13] defines the topological equivalence between piecewise smooth systems, a fundamental concept for the understanding of bifurcations.

Definition 5.

Two piecewise smooth systems are said to be topologically equivalent if there is a homeomorphism that maps the maximal segments of one system into the maximal segments of the other, preserving the direction in time and whether a segment is in the sliding zone, not sliding, or not in contact with the switching manifolds ${\Sigma }_i$.

Based on the above, by [11], a bifurcation is defined.

Definition 6.

A bifurcation occurs when arbitrary small perturbations generate a topologically non-equivalent system. The bifurcation is said to be discontinuity-induced if it affects the phase portrait in more than one region or in the manifolds ${\Sigma }_i$.

The following theorem, which constitutes the principal focus of study in [13], defines a sliding bifurcation and is presented below.

Theorem 1.

A generic uniparametric sliding bifurcation in ${\mathbb{R}}^n$, for any $n\ge 3$, may occur in a fold, cusp, or fold-fold tangency.

It is important to note that the objective of this paper is to prove the existence of a fold-fold singularity, but it will not study bifurcations in the model presented below. Although, according to the aforementioned theorem, which is referred to as Theorem 1.4 in [13], a uniparametric sliding bifurcation occurs at a fold-fold tangency in ${\mathbb{R}}^n$, for any $n\ge 3$.

3. Application to a System Describing Market Dynamics

The following model is analyzed, which describes the dynamics of a stockless market through a system of piecewise smooth differential equations. The state variables considered are the offer, the developing production capacity, and the demand at time t, which are denoted, respectively, by $x\left(t\right)$, $y\left(t\right)$, and $z\left(t\right)$. This model was proposed and analyzed by the authors in [18]. Here, we present a summary of the model, which is defined by the system

$$\left\{\begin{array}{ccc}\dot{x}& =& qy-rx\hfill \\ \dot{y}& =& B(x,z)-qy\hfill \\ \dot{z}& =& k·\left(1-{\displaystyle \frac{B(x,z)+\xi }{I_m}}\right)·z\hfill \end{array}.\right.$$

(4)

In the aforementioned system, the function $B(x,z)$ represents the investment function, which translates into the capacity to produce. This function is piecewise defined as follows:

$$B(x,z)=\left\{\begin{array}{cc}0,\hfill & R(x,z)\le 0\hfill \\ I_m,\hfill & 0<R(x,z)\le \delta \hfill \\ I_h,\hfill & R(x,z)>\delta \hfill \end{array}\right.$$

(5)

where $\delta$ represents a positive threshold at $(0,1)$ that defines the decision to invest. Hereafter, medium and high investment are denoted by $I_m$, $I_h$, respectively. Furthermore, the return on investment, denoted by $R(x,z)$, is defined in [19] as the difference between the profit obtained and the fixed costs of production, divided by the fixed costs of production. Therefore, the return on investment is expressed as follows:

$$R(x,z)={\displaystyle \frac{(P(x,z)-C_v)f_p-C_f}{C_f}},$$

(6)

where $C_v$ represents the costs and $f_p$ the production factor, which is interpreted as the ratio between the actual quantity of items produced during a period of time and the quantity of items that would have been produced if the same period of time had been fully worked. $C_f$ represents the operational costs. The price $P(x,z)$ is defined by

$$P(x,z)={\displaystyle \frac{P_{max}·(1-M(x,z))+2·P_{min}·M(x,z)}{1+M(x,z)}},$$

with $P_{max}$, $P_{min}$ as the maximum and minimum price, respectively, and $M(x,z)$ representing the reserve margin. If we consider $x\ge z>0$, then the expression for $M(x,z)$ is

$$M(x,z)={\displaystyle \frac{x-z}{x+m^2}},$$

(7)

where $m^2$ is a positive constant representing the minimum offer at the beginning of the market. Therefore, it is possible to interpret x as the actual offer and the term $x+m^2$ as the effective offer.

The remaining parameters of the system (4) are the developing production rate, represented by q; the consumption rate, denoted by r; the minimum investment required to start the market, designated by $\xi$; and the demand growth rate, represented by k.

Depending on how the system (4) is defined, it can be presented as follows.

• Let $S_1$ be the set of points in ${\mathbb{R}}^3$ for which null inversion is defined and satisfy

  $$z\le a_{1}x+b_1,$$

  with

  $$a_1=2\left[{\displaystyle \frac{C-P_{min}}{C+P_{max}-2·P_{min}}}\right];b_1=m^2\left[{\displaystyle \frac{C-P_{max}}{C+P_{max}-2·P_{min}}}\right],$$

  (8)

  and

  $$C={\displaystyle \frac{C_f+f_p{C}_v}{f_p}}.$$
• A medium investment is defined in the set of points in ${\mathbb{R}}^3$, denoted $S_2$, with

  $$a_{1}x+b_1<z\le a_{2}x+b_2,$$

  with $a_1$, $b_1$ as defined in (8) and

  $$a_2=2\left[{\displaystyle \frac{C_{\delta }-P_{min}}{C_{\delta }+P_{max}-2·P_{min}}}\right];b_2=m^2\left[{\displaystyle \frac{C_{\delta }-P_{max}}{C_{\delta }+P_{max}-2·P_{min}}}\right].$$

  (9)
  Furthermore,

  $$C_{\delta }={\displaystyle \frac{(\delta +1)C_f+f_p{C}_v}{f_p}}.$$
• The points of ${\mathbb{R}}^3$ that exhibit a high degree of inversion are denoted as $S_3$ and are defined by

  $$z>a_{2}x+b_2.$$

In summary, the state space is divided into three areas denoted by $S_1$, $S_2$, and $S_3$, which are conditioned by null, medium, and high inversion, respectively. These areas are defined by

$$\begin{array}{ccc}\hfill S_1& =& \left\{\mathbf{x}\in {\mathbb{R}}^3:z\le a_{1}x+b_1\right\},\hfill \\ \hfill S_2& =& \left\{\mathbf{x}\in {\mathbb{R}}^3:a_{1}x+b_1<z\le a_{2}x+b_2\right\},\hfill \\ \hfill S_3& =& \left\{\mathbf{x}\in {\mathbb{R}}^3:z>a_{2}x+b_2\right\}.\hfill \end{array}$$

These regions are separated by two surfaces or switching planes:

$$\begin{array}{ccc}\hfill {\Sigma }_1& =& \left\{\mathbf{x}\in {\mathbb{R}}^3:z=a_{1}x+b_1\right\},\hfill \\ \hfill {\Sigma }_2& =& \left\{\mathbf{x}\in {\mathbb{R}}^3:z=a_{2}x+b_2\right\}.\hfill \end{array}$$

In this case, we have the smooth scalar fields $H^{\left(i\right)}:{\mathbb{R}}^3⟶\mathbb{R}$, which are defined by

$$H^{\left(i\right)}(x,y,z)=a_{i}x+b_i-z,i=1,2.$$

As $\mathbf{x}=(x,y,z)$, the gradient of $H^{\left(i\right)}\left(\mathbf{x}\right)$ is

$$H_{\mathbf{x}}^{\left(i\right)}\left(\mathbf{x}\right)=\left(a_i,0,-1\right).$$

The above is evident in Figure 1, where the non-negative state variables in the first octant have been plotted. The blue plane represents the switching surface ${\Sigma }_1$, the yellow plane represents ${\Sigma }_2$, and the green plane is $z=x$.

Thus, for $\mathbf{x}=(x,y,z)\in {\mathbb{R}}^3$, the system (4) is a piecewise smooth system, defined in this case as follows:

$$\dot{\mathbf{x}}=\left\{\begin{array}{cc}{f}_1\left(\mathbf{x}\right);& \mathbf{x}\in S_1\\ f_2\left(\mathbf{x}\right);& \mathbf{x}\in S_2\\ f_3\left(\mathbf{x}\right);& \mathbf{x}\in S_3,\end{array}\right.$$

where the vector fields $f_1$, $f_2$, and $f_3$ differ by the value taken by the function $B(x,z)\in \{0,I_m,I_h\}$ in the system given by Equation (4).

The behavior of non-smooth system orbits in the regions approaching the ${\Sigma }_1$ and ${\Sigma }_2$ switching planes is studied using Filippov’s method, as detailed in [15].

For $\mathbf{x}\in {\Sigma }_1$, ${\Sigma }_2$, the following definitions are provided:

$$\begin{array}{ccc}\hfill {\sigma }^{\left(1\right)}\left(\mathbf{x}\right)& =& \left({\mathcal{L}}_{f_1}{H}^{\left(1\right)}\left(\mathbf{x}\right)\right)\left({\mathcal{L}}_{f_2}{H}^{\left(1\right)}\left(\mathbf{x}\right)\right)\hfill \\ \hfill {\sigma }^{\left(2\right)}\left(\mathbf{x}\right)& =& \left({\mathcal{L}}_{f_2}{H}^{\left(2\right)}\left(\mathbf{x}\right)\right)\left({\mathcal{L}}_{f_3}{H}^{\left(2\right)}\left(\mathbf{x}\right)\right)\hfill \end{array}$$

Recall that, as described in the previous section, there is crossing at the points $\mathbf{x}\in {\Sigma }_i$ where ${\sigma }^{\left(i\right)}\left(\mathbf{x}\right)>0$. Otherwise, there is sliding. Figure 2 illustrates this for each switching surface.

On the set of sliding points of each of the switching planes ${\Sigma }_1^s$ and ${\Sigma }_2^s$, the following Filippov vector fields are, respectively, defined:

$$g_1\left(\mathbf{x}\right)=\dot{\mathbf{x}}=\left\{\begin{array}{c}qy-rx\\ I_m(1-{\lambda }_1\left(\mathbf{x}\right))-qy\\ a_1(qy-rx)\end{array}\right.;g_2\left(\mathbf{x}\right)=\dot{\mathbf{x}}=\left\{\begin{array}{c}qy-rx\\ {\lambda }_2\left(\mathbf{x}\right)(I_m-I_h)+I_h-qy\\ a_2(qy-rx)\end{array}\right.$$

with

$$\begin{array}{ccc}\hfill {\lambda }_1\left(\mathbf{x}\right)& =& {\displaystyle \frac{a_1(qy-rx)}{k(a_{1}x+b_1)}}+{\displaystyle \frac{\xi }{I_m}}\hfill \\ & & \\ \hfill {\lambda }_2\left(\mathbf{x}\right)& =& {\displaystyle \frac{a_2{I}_m(qy-rx)}{k(a_{2}x+b_2)(I_h-I_m)}}+{\displaystyle \frac{\xi }{I_h-I_m}}+1.\hfill \end{array}$$

The possible pseudo-equibria for $g_1\left(\mathbf{x}\right)$ and $g_2\left(\mathbf{x}\right)$ are, respectively,

$$\begin{array}{ccc}\hfill {\tilde{\mathbf{x}}}_1& =& (x,y,z)=\left({\displaystyle \frac{I_m-\xi }{r}},{\displaystyle \frac{I_m-\xi }{q}},a_1\left({\displaystyle \frac{I_m-\xi }{r}}\right)+b_1\right)\hfill \\ & & \\ \hfill {\tilde{\mathbf{x}}}_2& =& (x,y,z)=\left({\displaystyle \frac{I_m-\xi }{r}},{\displaystyle \frac{I_m-\xi }{q}},a_2\left({\displaystyle \frac{I_m-\xi }{r}}\right)+b_2\right).\hfill \end{array}$$

It can be proven that ${\tilde{\mathbf{x}}}_1\in {\Sigma }_1^s$ but that ${\tilde{\mathbf{x}}}_2\notin {\Sigma }_2^s$. Therefore, we will henceforth only analyze the dynamics on the switching plane ${\Sigma }_1$. The stability of the pseudo-equilibrium ${\tilde{\mathbf{x}}}_1$ was analyzed in [18].

In addition to pseudo-equilibria, in sliding zones, we can also find other points that play an important role in the analysis of smooth systems by sections. One of these points is defined below.

Definition 7.

The singular sliding points satisfy, for $\mathbf{x}\in {\Sigma }_1^s$,

$${\mathcal{L}}_{f_2-f_1}{H}^{\left(1\right)}\left(\mathbf{x}\right)=\langle H_{\mathbf{x}}^{\left(1\right)}\left(\mathbf{x}\right),f_2\left(\mathbf{x}\right)-f_1\left(\mathbf{x}\right)\rangle =0;f_{1,2}\left(\mathbf{x}\right)\ne 0.$$

(10)

In our case, the singular sliding point, for $\mathbf{x}\in {\Sigma }_1^s$, is

$${\widehat{\mathbf{x}}}_1=(x,y,z)=\left(-{\displaystyle \frac{b_1}{a_1}},-{\displaystyle \frac{b_{1}r}{a_{1}q}},0\right).$$

(11)

Geometrically, this point is at the intersection of the stable sliding zone designated as ${\Sigma }_1^{ss}$, and the unstable sliding zone, denoted by ${\Sigma }_1^{us}$, and lies on the $xy$-plane. The following section will demonstrate that this point is a fold-fold tangency.

It should be noted that other authors have described the dynamics of a market through piecewise smooth systems, but these systems, due to the nature of the functions that define them, do not present sliding regions, nor do they study the characteristics of the tangencies [3,20]. However, in the work presented here, as has just been mentioned and analyzed, there are sliding regions. We now proceed to study and classify the tangencies or singularities.

4. Test of the Double Tangency Conditions

In this section, we will demonstrate that the singular sliding point defined in (11) satisfies the conditions to be classified as a fold-fold tangency. Recall that three conditions must be satisfied: ${\mathcal{L}}_{f_1}{H}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)={\mathcal{L}}_{f_2}{H}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)=0$, ${\mathcal{L}}_{f_1}^2{H}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)\ne 0$, and the gradient vectors of $H^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)$, ${\mathcal{L}}_{f_1}{H}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)$ and ${\mathcal{L}}_{f_2}{H}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)$ are linearly independent.

Proposition 1.

In the dynamical system (4), the singular sliding point ${\widehat{\mathbf{x}}}_1$ given in (11) is a fold-fold tangency.

Proof. 

We shall prove that the three conditions mentioned above are satisfied.

• Let us prove that ${\mathcal{L}}_{f_1}{H}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)={\mathcal{L}}_{f_2}{H}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)=0$. It is henceforth considered $\mathbf{x}=(x,y,z)$.

  $$\begin{array}{ccc}\hfill {\mathcal{L}}_{f_1}{H}^{\left(1\right)}\left(\mathbf{x}\right)& =& \langle H_{\mathbf{x}}^{\left(1\right)}\left(\mathbf{x}\right),f_1\left(\mathbf{x}\right)\rangle \hfill \\ & =& (a_1,0,-1)·\left(qy-rx,-qy,kz\left(1-{\displaystyle \frac{\xi }{I_m}}\right)\right)\hfill \\ & =& a_1(qy-rx)-kz\left(1-{\displaystyle \frac{\xi }{I_m}}\right).\hfill \end{array}$$
  Replacing ${\widehat{\mathbf{x}}}_1$ in the above expression, it can be verified that ${\mathcal{L}}_{f_1}{H}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)=0$.
  Similarly,

  $$\begin{array}{ccc}\hfill {\mathcal{L}}_{f_2}{H}^{\left(1\right)}\left(\mathbf{x}\right)& =& \langle H_{\mathbf{x}}^{\left(1\right)}\left(\mathbf{x}\right),f_2\left(\mathbf{x}\right)\rangle \hfill \\ & =& (a_1,0,-1)·\left(qy-rx,I_m-qy,kz\left(1-{\displaystyle \frac{I_m+\xi }{I_m}}\right)\right)\hfill \\ & =& a_1(qy-rx)-kz\left(1-{\displaystyle \frac{\xi }{I_m}}\right).\hfill \end{array}$$
  Replacing ${\widehat{\mathbf{x}}}_1$ in the above expression, it can be verified that ${\mathcal{L}}_{f_2}{H}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)=0$.
  For the time being, the singular sliding point ${\widehat{\mathbf{x}}}_1=(x,y,z)=\left(-{\displaystyle \frac{b_1}{a_1}},-{\displaystyle \frac{b_{1}r}{a_{1}q}},0\right)$ satisfies the first condition, and, as mentioned above, this point connects the stable sliding zone with the unstable one, as can be seen in Figure 3.
• Let us now prove that ${\mathcal{L}}_{f_1}^2{H}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)=\langle \nabla \langle H_{\mathbf{x}}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right),f_1\left({\widehat{\mathbf{x}}}_1\right)\rangle ,f_1\left(\widehat{\mathbf{x}}\right)\rangle \ne 0$

  $$\begin{array}{ccc}\hfill {\mathcal{L}}_{f_1}^2{H}^{\left(1\right)}\left(\mathbf{x}\right)& =& \left(-a_{1}r,a_{1}q,-k\left(1-{\displaystyle \frac{\xi }{I_m}}\right)\right)·\left(qy-rx,-qy,kz\left(1-{\displaystyle \frac{\xi }{I_m}}\right)\right)\hfill \\ & =& -a_{1}r(qy-rx)-a_1{q}^{2}y-k^{2}z{\left(1-{\displaystyle \frac{\xi }{I_m}}\right)}^2\hfill \end{array}$$
  Replacing ${\widehat{\mathbf{x}}}_1$ in the last expression,

  $$-a_{1}rq\left(-{\displaystyle \frac{b_{1}r}{a_{1}q}}\right)+a_{1}r\left(-{\displaystyle \frac{b_1}{a_1}}\right)-a_1{q}^2\left(-{\displaystyle \frac{b_{1}r}{a_{1}q}}\right)=b_{1}rq\ne 0$$
  This is fulfilled, since $b_1,r,q\ne 0$. Moreover, as $b_1<0$, then ${\mathcal{L}}_{f_1}^2{H}^{\left(1\right)}\left(\mathbf{x}\right)<0$.
  Similarly for ${\mathcal{L}}_{f_2}^2{H}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_{\mathbf{1}}\right)=\langle \nabla \langle H_{\mathbf{x}}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_{\mathbf{1}}\right),f_2\left({\widehat{\mathbf{x}}}_{\mathbf{1}}\right)\rangle ,f_2\left({\widehat{\mathbf{x}}}_{\mathbf{1}}\right)\rangle \ne 0$, we see

  $$\begin{array}{ccc}\hfill {\mathcal{L}}_{f_2}^2{H}^{\left(1\right)}\left(\mathbf{x}\right)& =& \left(-a_{1}r,a_{1}q,-k\left(1-{\displaystyle \frac{I_m+\xi }{I_m}}\right)\right)·\left(qy-rx,I_m-qy,kz\left(1-{\displaystyle \frac{I_m+\xi }{I_m}}\right)\right)\hfill \\ & =& -a_{1}r(qy-rx)+a_{1}q(I_m-qy)-k^{2}z{\left(1-{\displaystyle \frac{I_m+\xi }{I_m}}\right)}^2\hfill \end{array}$$
  In the same way, replacing ${\widehat{\mathbf{x}}}_1$ in the latter expression, we obtain

  $$-a_{1}rq\left(-{\displaystyle \frac{b_{1}r}{a_{1}q}}\right)+a_1{r}^2\left(-{\displaystyle \frac{b_1}{a_1}}\right)+a_{1}q{I}_m-a_1{q}^2\left(-{\displaystyle \frac{b_{1}r}{a_{1}q}}\right)=a_1{I}_{m}q+b_{1}rq\ne 0,$$

  provided that $-{\displaystyle \frac{b_1}{a_1}}\ne {\displaystyle \frac{I_m}{r}}$. Moreover, ${\mathcal{L}}_{f_2}^2{H}^{\left(1\right)}\left(\mathbf{x}\right)>0$ for $-{\displaystyle \frac{b_1}{a_1}}>{\displaystyle \frac{I_m}{r}}$.
  Under the above assumptions, the tangency is fold-fold invisible-invisible; therefore, it is a Teixeira singularity or Teixeira T-singularity [17].
• Let us now check that the gradient vectors of the $H^{\left(1\right)}$, ${\mathcal{L}}_{f_1}{H}^{\left(1\right)}$, and ${\mathcal{L}}_{f_2}{H}^{\left(1\right)}$ are linearly independent. Recall that

  $$\begin{array}{ccc}\hfill H_{\mathbf{x}}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)& =& (a_1,0,-1)\hfill \\ \hfill \nabla \left({\mathcal{L}}_{f_1}{H}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)\right)& =& \left(-a_{1}r,a_{1}q,-k\left(1-{\displaystyle \frac{\xi }{I_m}}\right)\right)\hfill \\ \hfill \nabla \left({\mathcal{L}}_{f_2}{H}^{\left(1\right)}\left({\widehat{\mathbf{x}}}_1\right)\right)& =& \left(-a_{1}r,a_{1}q,-k\left(1-{\displaystyle \frac{I_m+\xi }{I_m}}\right)\right)\hfill \end{array}$$
  To check their linear independence, we calculate the determinants of these vectors.

  $$\left|\begin{array}{ccc}{a}_1& -a_{1}r& -a_{1}r\\ 0& a_{1}q& a_{1}q\\ -1& -k\left(1-{\displaystyle \frac{\xi }{I_m}}\right)& -k\left(1-{\displaystyle \frac{I_m+\xi }{I_m}}\right)\end{array}\right|=a_1^{2}qk\ne 0.$$
  Therefore, the vectors are linearly independent.

Thus, the singular sliding point ${\widehat{\mathbf{x}}}_1=(x,y,z)=\left(-{\displaystyle \frac{b_1}{a_1}},-{\displaystyle \frac{b_{1}r}{a_{1}q}},0\right)$ is a fold-fold tangency; in particular, it is a T-singularity. □

As mentioned above, Theorem 1.4 is stated in [13], which indicates that, in particular, at a fold-fold tangency, a uniparametric sliding bifurcation occurs in ${\mathbb{R}}^n$, $n\ge 3$.

5. Numerical Simulations

This section presents some numerical scenarios that illustrate the behavior of trajectories near the fold-fold singularity. In order to achieve this, both positive and negative initial conditions are considered, with the latter being taken above and below the singularity. Although these scenarios are not applicable in practice, they are included here to complement the work mathematically. It should first be noted that the green plane in the graphs illustrates the $z=0$ plane, the regions shown in cyan illustrate the sliding region in the switching plane ${\Sigma }_1$, and the point on the sliding region represents the pseudo-equilibrium. In addition, curves of different colors represent trajectories that start from the points that indicate the initial conditions; to start the numerical computations, specific parameter values were assigned, as detailed in

Table 1. Model parameter values.

Symbol	Value/Unit	Name
$M^2$	1000 U.A.	Minimum quantity of articles to start the market
$\xi$	U.A.	Minimum investment quantity to start the market
$\delta$	0.5	Investment threshold
r	1/35 $d^{-1}$	Consumption rate
q	1/5 $d^{-1}$	Production rate in development
$C_f$	60 M.U.	Fixed production costs
$C_v$	150 M.U.	Variable costs
$f_p$	0.7	Production factor
k	0.03 $d^{-1}$	Demand growth rate
$P_{max}$	350 U.M.	Maximum price
$P_{min}$	180 U.M.	Minimum price
$I_m$	2500 U.A.	Medium investment
$I_h$	4200 U.A.	High investment

U.A.: units of articles, M.U.: monetary units.

. The value of $\xi$ will be defined in each numerical scenario.

Figure 4 illustrates that, by assuming negative initial conditions as starting points above the pseudo-equilibrium that is in the escape zone (${\widehat{\mathbf{x}}}_1\in {\Sigma }_1^{us}$), the trajectories for the offer, developing production capacity, and demand transit the spiral-shaped crossover zone and move around the unstable sliding zone of the switching surface ${\Sigma }_1$ and upwards until they reach the fold-fold singularity. Moreover, in Figure 5, negative initial conditions are also considered, but, in this case, they are situated below the pseudo-equilibrium. Here, the trajectories for the offer and developing production capacity tend to zero, whereas that for the demand decreases without bounds.

The objective of Figure 4 and Figure 5 is to illustrate the dynamics of the state variables in the neighborhood of the double tangency, taking negative initial values for the variables. Moreover, depending on the parameter values, it is possible for the pseudo-equilibrium ${\widehat{\mathbf{x}}}_1$ to move from the stable sliding zone to the unstable sliding zone through the fold-fold tangency for some values of $\xi$. In the model, this represents the minimum investment required to start the market. In particular, if $\xi <I_m+{\displaystyle \frac{b_{1}r}{a_1}}$ then ${\widehat{\mathbf{x}}}_1\in {\Sigma }_1^{ss}$. Otherwise, ${\widehat{\mathbf{x}}}_1\in {\Sigma }_1^{us}$. We clarify that, according to the parameter values assigned in Table 1 for the numerical simulations, $I_m+{\displaystyle \frac{b_{1}r}{a_1}}\approx 2470.7$.

Now, it is worth considering the scenario depicted in Figure 6. This scenario also assumes negative initial conditions and the pseudo-equilibrium in the stable sliding zone. As mentioned before, this can be in the stable or unstable sliding zone depending on the values of $\xi$. Under these conditions, the trajectories involve the escape zone through the crossing zone to a point where the offer and developing production capacity tend to zero and the demand decreases without bounds. It is similar to the situation shown in Figure 5.

In Figure 7, we take positive initial conditions and the pseudo-equilibrium in the stable sliding zone ${\Sigma }_1^{ss}$. The orbits transit through the crossing zone above the plane $z=0$ until they collide with the stable sliding zone, after which the trajectories slide, seeking the pseudo-equilibrium, which is locally asymptotically stable, as mentioned before.

Finally, in the scenario shown in Figure 8, positive initial conditions and the pseudo-equilibrium in the unstable sliding zone are taken. The orbits cross the switching surface ${\Sigma }_1$ above the plane $z=0$ until they reach the sliding zone and then slide to the fold-fold singularity.

It is noteworthy that, in this paper, based on the work and research carried out by other authors, we propose to make an additional contribution to the practical applications of the theory framing piecewise smooth systems and related topics of singularities. For example, in [12], this theory is analyzed and applied to a canonical switched feedback control model, and the classification of fold-fold singularities and their dynamics is demonstrated. Moreover, in [13], it is shown that, in general, the behavior of a uniparametric sliding bifurcation is as follows: a sliding orbit starting from the stable sliding zone can leave such a set of points either when the orbit reaches the border points of the stable sliding zone or when it passes through the fold-fold and passes into the unstable sliding region; here, the trajectory leaves the set of points by crossing the switching surface in spiral motions or when it reaches the unstable sliding boundary. The latter motion is also described in [14] and is called a non-smooth diabolo. In contrast to the above, in our work, and according to the dynamics of the model analyzed, as the $z=0$ plane is invariant, the sliding orbits reach the double tangency and stop, preventing the trajectories from passing into the unstable sliding sector. We highlight that, although the pseudoqeulibrium is in this escape zone, the trajectories stop when sliding to the fold-fold tangency. The fact that the $z=0$ plane is invariant makes the application studied in this work a non-generic case.

6. Discussion

In this paper, we have detected and proved the presence of a fold-fold singularity in a mathematical model described by a system of piecewise smooth ordinary differential equations, which describes the dynamics of a stockless market.

This identification of the fold-fold singularity in our system suggests complex and potentially chaotic behavior in the dynamics of a stockless market. In contrast to previous studies on the subject, the model studied offers new insights into how discontinuities and abrupt transitions can influence market stability.

The results in this paper may have significant implications for the design and regulation of stockless markets, highlighting the need for strategies that consider possible singularities and their effects. This study contributes to the field of discontinuous dynamical systems and their application in economics, since there is little literature that combines the mathematical modeling of a market, sliding in the switching surfaces, and this type of singularity at the same time.

One of the main simplifications is the definition of the vector field in the sliding zone using the Filippov convention, which is described as a convex combination of the vector fields adjacent to the switching surface, rather than considering non-linear terms as other authors do, generalizing the Filippov definition of the vector field. This simplification, while useful for the initial analysis, may not fully capture all dynamics of the system.

Future research on this issue should explore the bifurcations in this system, using both analytical and numerical analysis to better understand how the dynamics vary under different conditions. In addition, considering non-linear terms in the vector field defined in sliding zones could provide a more complete picture of the system’s behavior.

In conclusion, this study provides a new understanding of the dynamics of stockless markets, highlighting the importance of fold-fold singularities. Although we have made significant progress, much remains to be explored, especially in the area of bifurcations and their impact on market stability.

7. Conclusions

In this paper, a brief overview of the basic concepts of piecewise smooth systems, tangencies, and the types of tangencies, and a brief description of the conditions required for bifurcations to occur in such systems, is given.

The above topics are applied to a mathematical model describing the dynamics of a stockless market, defined through a system of differential equations with piecewise smooth functions, capturing external factors such as the investment decision. This is made possible by a piecewise smooth function, which divides the state space into three regions depending on the returns obtained. Unlike other models found in the literature, the one analyzed in this paper presents sliding in the regions of the switching surfaces. Therefore, it was possible to analyze tangencies and singularities.

According to the definitions and main results, it is proven that the analyzed model has a fold-fold tangency, which could also be classified as a T-singularity or Teixeira singularity. Such a point is the boundary between the stable sliding region and the escape zone and lies on the $xy$-plane. The trajectories around this point indicate that, when taking negative initial conditions, below the singularity and above the pseudo-equilibrium, they pass through the crossover zone, enclosing, in a spiral shape, the escaping region, ascending until reaching the singularity. Furthermore, by taking the positive initial conditions and above the singularity, the trajectories travel through the crossover zone until they collide with the stable sliding zone. Subsequently, the trajectories slide in search of the pseudo-equilibrium, when it is in the stable sliding zone, which is locally asymptotically stable. When the pseudo-equilibrium is in the escape zone, the orbits slide through the set of sliding points, descending to the singularity, reflecting that the $z=0$ plane is an invariant plane.