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are possible. To this end equations for equilibria can be determined from the dynami-
cal model equations for the belief and the preparation state level, which can be ex-
pressed as differential equations as follows (with b(t) the level of the belief, s(t) of the
stimulus, f(t) of the feeling, and p(t) of the preparation for the body state at time t ).
db(t)/dt =
γ 1 (
β 1 (1-(1-s(t))(1-f(t))) + (1-
β 1 )s(t)f(t) - b(t))
dp(t)/dt =
γ 2 (
β 2 (1-(1-b(t))(1-f(t))) + (1-
β 2 )b(t)f(t) - p(t))
To obtain equations for equilibria, constant values for all variables are assumed (also
the ones that are used as inputs such as the stimuli). Then in all of the equations the
reference to time t can be left out, and in addition the derivatives db(t)/dt and dp(t)/dt
can be replaced by 0 . Assuming
γ 1 and
γ 2 nonzero, this leads to the following
equations.
β 1 (1-(1-s)(1-f)) + (1-
β 1 )sf - b = 0
β 2 (1-(1-b)(1-f)) + (1-
β 2 )bf - p = 0
As for an equilibrium it also holds that f = p , this results in the following two equa-
tions in b , f , and s :
(1)
β
1 (1-(1-s)(1-f)) + (1-
β
1 )sf - b = 0
(2)
β 2 (1-(1-b)(1-f)) + (1-
β 2 )bf - f = 0
For the general case (1) can directly be used to express b in f, s and
β
1 . Using this, in
(2) b can be replaced by this expression in f, s and
β 1 , which transforms (2) into a
quadratic equation in f with coefficients in terms of s and the parameters
β 2 .
Solving this quadratic equation algebraically provides a complex expression for f in
terms of s ,
β 1 and
β 1 and
β 2 can be found. As these expressions become rather complex, only an overview for a
number of special cases is shown in Table 1 (for 9 combinations of values 0 , 0.5 and 1
for both
β 1 and
β 2 . Using this, by (1) also an expression for b in terms of s ,
β 2 ). For these cases the equations (1) and (2) can be substantially
simplified as shown in the second column (for equation (1)) and second row (for
equation (2)). The shaded cases are instable (not attracting), so they only occur when
these values are taken as initial values.
As can be seen in this table, for persons that are pessimistic for believing (
β 1 and
β
1 = 0 )
and have a negative profile in generating emotional responses (
β 2 = 0 ), reach a stable
equilibrium for which both the belief and the feeling have level 0 . The opposite case
occurs when a person is optimistic for believing (
β
1 = 1 ) and has a positive profile in
generating emotional responses (
β 2 = 1 ). Such a person reaches a stable equilibrium
for which both the belief and the feeling have level 1. For cases where one of these
β 1
and
β 2 is 0 and the other one is 1 , a stable equilibrium is reached where the belief gets
the same level as the stimulus: b = s . When a person is in the middle between opti-
mistic and pessimistic for believing (
β 1 = 0.5 ), for the case of a negative profile in
generating emotional responses the stable belief reached gets half of the level of the
stimulus, whereas for the case of a positive profile in generating emotional responses
the stable belief reached gets 0.5 above half of the level of the stimulus (which is the
0.65 shown in the second trace in Figure 4). This clearly shows the effect of the feel-
ing on the belief. The case where both
β 1 = 0.5 and
β 2 = 0.5 is illustrated in the first
trace in Figure 4: b = f = s .
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