The role of optimism and pessimism in the dynamics of emotional states

Figure 1.  Example of an influence function. We see that for $|\xi_2|=\xi_1$ we have $f(\xi_1)\leq -f(\xi_2)$

Figure 2.  All possible types of the phase space portrait of System (1) illustrating Theorem 3.3. (a): unique globally attractive focus; (b): unique globally attractive node; (c): bifurcation from one stable steady state to bistability; (d): bistability

Figure 3.  An example of the situation when two actors reach the same emotional state independently if they meet each other or not

Figure 4.  An example of the change of emotional states of two friends with the opposite initial states

Figure 5.  Emotional states of two strongly dependent on each other friends with the same parameters describing both of them but with the opposite initial states

Figure 6.  Examples of changes in emotional states of two actors with the opposite attitude to each other. Top: the steady state is a stable focus and emotions oscillate. Bottom: the steady state is a stable node and emotions slowly approach the equilibrium

Figure 7.  En example of two actors for which different initial conditions yield different opposite final emotional states

Figure 8.  Scheme of six possible interactions of the pessimist ($y$) with the other actor ($x$) who could be a pessimist or optimist. (a): $y$ is negatively ($c_{1} < 0$) and $x$ is positively ($c_{2}>0$) oriented; (b): $y$ is positively ($c_{1}>0$) and $x$ is negatively ($c_{2} < 0$) oriented; (c): both $x$ and $y$ are positively oriented ($c_{1}, c_{2}>0$) and $m_{1}m_{2} > c_{1}c_{2}$ indicating the existence of exactly one steady state; (d): both $x$ and $y$ are negatively oriented ($c_{1}, c_{2} < 0$) and $m_{1}m_{2} > c_{1}c_{2}$ indicating the existence of exactly one steady state; (e): both $x$ and $y$ are positively oriented ($c_{1}, c_{2}>0$) and $m_{1}m_{2} < c_{1}c_{2}$ indicating that there might exist more than one steady state; (f): both $x$ and $y$ are negatively oriented ($c_{1}, c_{2}>0$) and $m_{1}m_{2} < c_{1}c_{2}$ indicating that there might exist more than one steady state

Figure 9.  An example of emotional states of two pessimists who have strong influence on each other

Figure 10.  Graph of emotional states of two pessimists who are enemies. Signs of the initial conditions are opposite. Horizontal doted line indicates the uninfluenced equilibriums of both of actors

Figure 11.  Relationships between two optimists: (g) $y$ is positively while $x$ is negatively oriented ($c_1 < 0, c_2>0$); (h) $y$ is negatively while $x$ is positively oriented ($c_1>0, c_2 < 0$); (i) both actors are negatively oriented $(c_1, c_2 < 0)$ and $m_{1}m_{2} > c_{1}c_{2}$ indicating the existence of exactly one steady state; (j) both actors are negatively oriented $(c_1, c_2 < 0)$ and $m_{1}m_{2} < c_{1}c_{2}$ indicating the existence of up to three steady states; (k) both actors are positively oriented $(c_1, c_2>0)$ and $m_{1}m_{2} > c_{1}c_{2}$ indicating the existence of exactly one steady state; (l) both actors are positively oriented $(c_1, c_2>0)$ and $m_{1}m_{2} < c_{1}c_{2}$ indicating the existence of up to three steady states

Figure 12.  Possible changes of the dynamics of emotional states initiated by one of the interacting actors

Figure 13.  Comparison of the dynamics of emotional states of two pessimists when they are friends ($c_{1}, c_{2}>0$) and when they have inconsistent relations ($c_{1}>0$, $c_{2} < 0$)