#### Qualitative Analysis for the Bilinear Influence Function

We consider system (

4) with influence function of bilinear type given as:

These functions define different systems, depending on the quadrant.

The equilibrium points for each case,

$({H}^{*},{W}^{*})$, can be written as:

where the values of

m and

M are taken, according to the quadrant under consideration, to be

${m}_{-}$ or

${m}_{+}$ and

${M}_{-}$ or

${M}_{+}$.

If the value of $W\left(or\text{}H\right)$ is positive, it means the wife (or husband) is happy. If $W\left(or\text{}H\right)$ is negative, it means the wife (or husband) is unhappy. Therefore, if the equilibrium point is in the first quadrant, then both wife and husband are happy. In the third quadrant, both are unhappy. In the second quadrant, the husband is unhappy, and the wife is happy; and, in the fourth quadrant, the wife is unhappy, and the husband is happy.

As usual, the stability of the equilibria is analyzed by linearizing the system around the equilibrium points. The corresponding Jacobians of the linearized system are:

Therefore,

$trace={r}_{H}+{r}_{W}=p$, which is negative for the given

r’s,

$\mathrm{determinant}={r}_{W}{r}_{H}-mM=q$; then, the discriminant is

$\phantom{\rule{0.166667em}{0ex}}\Delta ={p}^{2}-4q={({r}_{H}-{r}_{W})}^{2}+4mM\phantom{\rule{0.166667em}{0ex}}$, and the eigenvalues are

Therefore, the critical point of the linearized system will be:

- (1)
- (2)
It will be stable if $\phantom{\rule{0.166667em}{0ex}}p<0\phantom{\rule{0.166667em}{0ex}}$ or unstable if $\phantom{\rule{0.166667em}{0ex}}p>0\phantom{\rule{0.166667em}{0ex}}$.

- (3)
A spiral if:

it will be stable if

$p<0$ and unstable if

$p>0$.

- (4)
- (5)
A proper or improper node if:

It is known (Hartman-Grobman Theorem) that the fixed points of the non-linear system and of its linearization will be of the same kind except when the eigenvalues have a real part equal to zero.

For the problem under investigation, we have $p<0$, and, for the validating model, $m>0,M>0\phantom{\rule{0.166667em}{0ex}}$ always; therefore, $\Delta \ge 0\phantom{\rule{0.166667em}{0ex}}$ and, using the above results, we have proved the following:

**Theorem** **1.** The equilibrium points, or steady states of system (4) for the bilinear influence function, are given by for the appropriate values of m and M on the corresponding quadrant. These points are:

- (i)
Saddle points (and therefore unstable) if and only if $\phantom{\rule{0.166667em}{0ex}}{r}_{H}{r}_{W}<mM\phantom{\rule{0.166667em}{0ex}}$, and $\phantom{\rule{0.166667em}{0ex}}{({r}_{H}-{r}_{W})}^{2}+4mM\ge 0\phantom{\rule{0.166667em}{0ex}}$ or

- (ii)
Stable nodes if and only if $\phantom{\rule{0.166667em}{0ex}}{r}_{H}{r}_{W}>mM\phantom{\rule{0.166667em}{0ex}}$ and $\phantom{\rule{0.166667em}{0ex}}{({r}_{H}-{r}_{W})}^{2}+4mM\ge 0\phantom{\rule{0.166667em}{0ex}}$.

It is interesting to note that, while the stability of the equilibrium points (or steady state solutions) depends on the emotional inertia (${r}_{H}$ and ${r}_{W}$) and the influence function (m and M), their existence depends also on the values of the individual constants ${a}_{H}$ and ${a}_{W}$, which are proportional to the uninfluenced equilibrium points. For the influence function of the bilinear type, it was discovered that, changing only the slopes m and M, the equilibrium points will move around the quadrants, and their stability will also change. This means that, by changing only the influence functions, the satisfaction of the couple can approach a constant value favorable to both if $({H}^{*},{W}^{*})$ is in quadrant (I), favorable only to the wife if $({H}^{*},{W}^{*})$ is in quadrant (II), unfavorable to both if $({H}^{*},{W}^{*})$ is in quadrant (III), and favorable only to the husband if $({H}^{*},{W}^{*})$ is in quadrant (IV). These steady state solutions can be stable or unstable. If ${r}_{H}{r}_{W}>mM$, the satisfaction of the couple will not change with time; otherwise, it will and can get very far from the values $({H}^{*},{W}^{*})$. If we are given fixed values for the r’s and the a’s, varying the values of the slopes m and M, we can control the position and stability of the equilibrium states. The same is true if the r’s and the slopes are given; then, varying the a’s, stability and position are determined.

In terms of therapeutic interventions, this describes the situation when, without changing each person (the a’s and the r’s), the satisfaction of the couple can be changed by working jointly with both members to modify the values of the interaction (bilinear influence function), or, when working with husband and wife separately, their uninfluenced equilibrium points (the a’s) are changed. On the other hand, the stability of the equilibria can only be altered by changing the relationship between the inertia, ${r}_{W}$ and ${r}_{H}$, and the slopes $m,M$.

To illustrate the above, we take the values for the variables of the model given in Table 10.1 from Gottman for the validating couple, which are: Inertia for the husband =

${\tilde{r}}_{H}=0.37$, and inertia for the wife =

${\tilde{r}}_{W}=0.14$; therefore,

${r}_{H}={\tilde{r}}_{H}-1=-0.63,\phantom{\rule{0.277778em}{0ex}}{r}_{W}={\tilde{r}}_{W}-1=0.86.$ We chose influence functions with

${m}_{-}=0.31,\phantom{\rule{0.277778em}{0ex}}{m}_{+}=0.21,\phantom{\rule{0.277778em}{0ex}}$${M}_{-}=0.28,$${M}_{+}=0.15$. Then, the Jacobians have the form:

where the values of the slopes

$m,M$ depend on the quadrant. For the first quadrant:

$m={m}_{+}=0.21,\phantom{\rule{0.166667em}{0ex}}M={M}_{+}=0.15$. The equilibrium points are

and

then, from Theorem 1, the equilibrium points, if they exist, are stable nodes.

Conditions for the existence of equilibrium points in the first quadrant are

${H}^{*}>0\phantom{\rule{0.166667em}{0ex}}$ and

${W}^{*}>0$, which mean:

Similar analyses can be carried out for the other quadrants, which will end up in inequalities between the a’s. In this particular example, for all quadrants, ${r}_{H}{r}_{W}>mM$ and $\Delta >0$; therefore, Theorem 1 implies that all the equilibrium points are stable nodes.

Note that, for general validating personality types, ${r}_{H}$ and ${r}_{W}$ are both negative, so one of the eigenvalues is negative, but the second one may be positive for large enough values of the radicand. This is consistent with Theorem 1.

For the case of influence functions given by piece-wise constants (

11) and (

12), the number and values of the equilibrium points depend on the values of the constants, but all equilibrium points have the same local stability.

**Theorem** **2.** For validating personalities, all the equilibrium points for influence functions given by piece-wise constants (11) and (12) are stable. The proof is straightforward since the influence functions only add a constant to each equation, and the partial derivatives with respect to H and W are zero, except at the jumps.

Therefore, the Jacobian matrix is

with eigenvalues

${\lambda}_{1}={r}_{W},{\lambda}_{2}={r}_{H}$, and both are negative for validating personalities.

For the other types of influence functions studied, due to the complexity, we will only give the equilibrium points and their stability for fixed values of the parameters. This will be done in the next section.