If x⁰(z) is continuous in z over some interval Z, except possibly for a step discontinuity at z = z_θ, and if, for some time t₁ ≥ 0, x(x⁰(z); t₁) is continuous in z except for a step discontinuity at z = z_θ, then a discontinuous threshold phenomenon (DTP) will be said to exist in the phase space.

This definition is tailored to depict a threshold phenomenon characterized by a bounded latency (t₁). The discontinuity in the system’s state at time t₁, as a function of z, can be induced either (a) by a discontinuity in the initial condition at z = z₀, without imposing any special conditions on f(x), or (b) with the initial condition being continuous in z, in which case constraints must be placed on f(x). It may be preferable for both x⁰(z) of Definition I and f(x) of equation (3) to have as components some of the elementary, differentiable functions of calculus. There are several justifications for such a preference. Firstly, economic processes involving numerous agents are commonly described using such functions. Secondly, empirical studies often reveal significant disparities, suggesting that specifying functions with excessive precision may be unwarranted, hence simpler functions should be favored. Lastly, explaining a discontinuous process by employing discontinuous functions can be viewed as an ad hoc assumption, sidestepping the underlying issue. However, according to the Cauchy-Lipschitz theorem for the existence of solutions of differential equations ( Lefschetz, 2016 ; Nemytskii, 2015 ), if f(x) is differentiable and all partial derivatives are uniformly bounded within a certain region, then the solution x(x⁰; t) remains continuous in (x⁰; t) for all t. Consequently, if x⁰(z) is also continuous in z, a Discontinuous Threshold Phenomenon (DTP) becomes untenable.

It is worth noting that to have a Discontinuous Threshold Phenomenon (DTP) with x⁰(z) continuous, it is not mandatory for f(x) to be discontinuous; it only needs to violate a Lipschitz condition at certain points. For instance, a DTP does manifest in the following system:

$\text{d}x/\text{d}t=-x^{1/3}$

$\text{d}y/\text{d}t=y^{1/3}$

If one prefers to utilize differentiable functions, it becomes imperative to establish a revised definition of threshold phenomenon. Definition I can be reworked by relinquishing the requirement of a maximum latency, i.e., finite t₁. A point within a phase space where all $\text{d}{x}_n/\text{d}t=0$ is considered a degenerate trajectory and is termed a singular point.

Figure 4 illustrates a phase plane with a saddle point (one type of singular point) situated at the origin of coordinates, as generated by equations of the following form:

$\text{d}{x}_1/\text{d}t=p_{11}{x}_1+p_{12}{x}_2+q_1\left(x_1,x_2\right)$,

$\text{d}{x}_2/\text{d}t=p_{21}{x}_1+p_{22}{x}_2+q_2\left(x_1,x_2\right)$ (4)

in this case the p’s are constants, and the characteristic equation is

$\left|\begin{array}{cc}{p}_{11}-\lambda & p_{12}\\ p_{21}& p_{22}-\lambda \end{array}\right|$ (5)

when λ has one positive and one negative root, and q₁ and q₂ are power series in x₁ and x₂, starting with terms of degree two or higher. If x⁰(z) is a continuous function of z and describes a line segment labeled “t = 0” in Figure 4 as z varies over the interval Z, the trajectory with x⁰(z) as its initial point undergoes a discontinuous change in shape at a trajectory called the separatrix, where z takes on the value z_θ. This discontinuity differs from that in Definition I. In fact, according to the Cauchy-Lipschitz Theorem, x(x⁰(z); t₁) is continuous in z for every fixed t₁, and as z varies, x(x⁰(z); t₁) continuously traverses a line such as the one labeled “t = t₁” in Figure 4 . Let us arbitrarily define the latency as the time taken for x to move from its initial point along an “all” trajectory to a line such as the one labeled “criterion of shock exposure” (one could think of it as a criterion of excitation as well) in Figure 4 . For a model of the demand for an addictive good, for instance, this line might correspond to the condition that D be halfway between the point of no exposure (and possibly no knowledge about) to an addictive substance and the peak value of the potential impact under unrestricted exposure. With no maximum latency, as z approaches z_θ, the state point remains in the vicinity of the saddle point for an increasingly extended period, where the phase velocity vector dx/dt is very small. Consequently, the latency can become arbitrarily large in this manner. This characteristic of the saddle point can be illustrated by plotting latency against z. In Figure 5 , typical curves of this nature are depicted for a DTP

and a saddle-point threshold phenomenon (STP). If the latter curve were to deviate from the separatrix only for z very close to z_θ, such a model could be deemed a good approximate representation of the demand for an addictive good. Both the STP and the actual demand for an addictive good may exhibit similar increases in latency near the threshold, except that the latency of the STP approaches infinity as z approaches z_θ, while that of the actual demand remains finite.

In Figure 4 , the trajectories in the vicinity of the saddle point can be categorized into three classes based on their behavior:

1) Two trajectories are stable, converging towards the saddle point as time progresses.

2) Two other trajectories are unstable, converging towards the saddle point as time decreases.

3) All remaining trajectories are hyperbolic, initially approaching and subsequently departing from the saddle point as time progresses.

Moreover, the hyperbolic trajectories can be further subdivided into four subclasses based on the directions in which they approach and depart from the saddle point. If two trajectories are selected from any two of these distinct classes or subclasses, it is impossible to transform one into the other by traversing through a continuum of intermediate trajectories. This topological property is responsible for the threshold characteristics of the saddle point.

As z traverses through Z and the initial point x⁰(z) moves along the line “t = 0”, it undergoes a discontinuous transition from one subclass of hyperbolic trajectories to another subclass, exhibiting qualitatively different behaviors as t increases. Both of these subclasses occupy contiguous 2-dimensional regions of the phase plane and are separated by a single stable trajectory, the separatrix.

The saddle-point threshold phenomenon can be extended to phase spaces of any finite number of dimensions. The subsequent definition is pertinent to systems with functions f(x) that are analytic at a singular point, i.e., they can be expanded in a Taylor series around that point. A more comprehensive treatment is available in Kolmogorov and Piskunov (2019) .

A Singular-Point Threshold Phenomenon (STP)1 is deemed to exist in an N-dimensional phase space (N ≥ 1) if there exists an isolated singular point characterized by having one positive root and all other roots (if any) with negative real parts. Moreover, if x⁰(z) is a continuous function that intersects with, and is not tangent to, the (N – 1)-dimensional surface (the separatrix) comprising stable trajectories when z = z_θ.

In two dimensions, the condition on the characteristic roots defines a saddle point. In three dimensions the properties of the singular point can be visualized if we let the trajectories be described by the solutions:

$x_n=a_n{\text{e}}^{\lambda nt}$ ( $n=1,2,3$) (6)

of the differential equations

$\text{d}{x}_n/\text{d}t={\lambda }_n{x}_n$. (7)

The a_n’s are constants of integration and are the coordinates of the initial point. The origin of coordinates is a singular point.

If λ₁ < λ₂ < 0 < λ₃, then the plane x₃ = 0 (the separatrix) encompasses the stable solutions and locally divides the space into two regions, in each of which the trajectories are hyperbolic. However, their behavior for increasing t is qualitatively distinct. Trajectories for which a₃ < 0 approach the negative x₃-axis, extending towards negative infinity along that axis, and may signify the “none” response. Conversely, trajectories for which a₃ > 0 approach the positive x₃-axis, extending towards positive infinity, may represent the “all” response. The separatrix segregates the “all” trajectories from the “none” trajectories.

Conversely, if λ₁ < 0 < λ₂ < λ₃, the hyperbolic trajectories fall into two subclasses that do not differ qualitatively in their behavior for increasing t. Instead, they approach the plane x₁ = 0, pointing in all possible directions within that plane. In this latter scenario, there is no threshold phenomenon.

In general, for a Singular-Point Threshold Phenomenon (STP) to manifest in any N-dimensional phase space, the singular point must possess the property that an (N − 1)-dimensional surface consisting of stable trajectories (the separatrix) serves as a local boundary between two N-dimensional regions. Both of these regions comprise hyperbolic trajectories, which, for sufficiently large t, diverge from the singular point in two opposite directions. If all the characteristic roots have non-zero real parts, then the aforementioned conditions are met if the roots adhere to the specifications outlined in Definition II ( Kolmogorov and Piskunov, 2019 ; Lefschetz, 2016 ). However, cases in which some of the roots have zero real parts are more intricate and necessitate separate examination.

A second approach to revising Definition I involves retaining the concept of a maximum latency while relinquishing the discontinuity between the “all” and “none” trajectories. Figure 6 illustrates an example in a phase plane. For all t₁, x(x⁰(z); t₁) remains continuous in z, but for certain values of t₁, it exhibits rapid variation when z is close to z_θ. Figure 7 depicts how D = F(t₁; z) might appear when plotted against z. The discontinuous curve observed in Figure 2 has been replaced by a continuous curve with a sharp ascent near z = z_θ.

These properties may be described mathematically as follows.

If x⁰(z) is continuous in z over some interval Z, and if there exist a positive time t₁ and two values z₁ and z₂ of z such that the ratio $\left|x\left[x^0\left(z_2\right);t_1\right]-x\left[x^0\left(z_1\right);t_1\right]\right|/\left|z_2-z_1\right|$ is sufficiently large, then a quasi-threshold phenomenon (QTP) will be said to exist in the phase space.

This definition is deliberately imprecise, as the phrase “sufficiently large” is open to subjective interpretation. Consequently, a QTP might even fall into the category of what, in practical terms, is not a threshold phenomenon at all. Nevertheless, one could regard and assess it as such based on statistical considerations.

For comparison, one could argue that in the QTP, the (N − 1)-dimensional separatrix of the STP has been substituted with a “thin” N-dimensional vicinity surrounding an (N − 1)-dimensional surface, as depicted by the cross-hatched region in Figure 6 . The width of this vicinity, or the extent of its smallest dimension, dictates the sharpness of the QTP. Moreover, the single threshold value z_θ of stimulus (shock) may be replaced by the closed interval [z₁, z₂].

The three types of threshold phenomena mentioned above do not encompass all possibilities. It is conceivable to establish a threshold phenomenon with certain properties akin to the saddle-point type by employing a line or surface comprising singular points instead of an isolated one. Additionally, a limit cycle (a periodic closed trajectory), where some trajectories are stable while others are unstable, can be substituted for the saddle point. In this scenario as well, hyperbolic trajectories can be identified with any specified latency, regardless of how extensive it may be. Most of these trajectories may entail the state point oscillating within the vicinity of the limit cycle.