## 4 Chaos in the Dissipative Quantum Model of Brain  
We recall that the set of \(SU(1,1)\) coherent states forms a Kahlerian manifold \(\mathbf{H}\), which has a symplectic structure. It may thus represent the phase space for the classical dynamics generated by the \(SU(1,1)\) group action [56]. The \(SU(1,1)\) generalized coherent states, i.e., the memory states \(|0(t)\rangle_{\mathcal{N}}\) considered above, are thus recognized to be \"points\" in \(\mathbf{H}\), and transitions among these points induced by the group action are therefore classical trajectories [56] in \(\mathbf{H}\) with the \(\mathcal{N}\)-set specifying the initial condition at \(t_{0}=0\). In the dissipative model, such a conclusion plays a crucial role; indeed, it confirms the already observed transition from the quantum dynamics to the classical level. The trajectories in the memory space can be shown to satisfy the requirements characterizing the chaotic behavior in non-linear dynamics [57], namely:  
(a) the trajectories are bounded, and each trajectory does not intersect itself (they are not periodic). 2. trajectories specified by different initial conditions do not cross each other. 3. different initial conditions lead to diverging trajectories. The boundedness of the trajectories follows from the finiteness of the norm of the state vectors at any time \(t\), \({\cal N}\langle 0(t)|0(t)\rangle_{\cal N}=1\). As time evolves, the trajectory of given \({\cal N}\) never crosses itself; the \"points\" \(|0(t)\rangle_{\cal N}\) and \(|0(t^{\prime})\rangle_{\cal N}\) for any \(t\) and \(t^{\prime}\), with \(t\
eq t^{\prime}\), never coincide, as shown by Equations (9). The requirement (a) is thus satisfied. One can show [55; 58; 59] that in the infinite volume limit:  
\[{}_{\cal N}\langle 0(t)|0(t^{\prime})\rangle_{\cal N^{\prime}} \underset{V\to\infty}{\longrightarrow}0\quad\forall\,t\quad,\ \forall\,{\cal N}\
eq{\cal N}^{\prime} \tag{10}\] \[{}_{\cal N}\langle 0(t)|0(t^{\prime})\rangle_{\cal N^{\prime}} \underset{V\to\infty}{\longrightarrow}0\quad\forall\,t,t^{\prime}\quad,\ \forall\,{\cal N}\
eq{\cal N}^{\prime}. \tag{11}\]  
Equation (11) holds also for \(t=t^{\prime}\) for any \({\cal N}\
eq{\cal N}^{\prime}\). Equations (10) and (11) imply that trajectories specified by different initial conditions, \({\cal N}\
eq{\cal N}^{\prime}\), never cross each other. Requirement (b) is thus satisfied, which implies that at any time \(t\), there is no interference, i.e., no confusion, among different memories. However, finite volume effects may allow non-zero inner products between memory states with different codes. In such cases, at the crossing points between two different trajectories, one might switch from one to another trajectory, which can be experienced as an \"association\" of memories. Let us now consider the time evolution of the difference between the codes \({\cal N}\
eq{\cal N}^{\prime}\) (\(\theta\
eq\theta^{\prime}\)), specifying the initial time conditions, at \(t_{0}=0\), of two trajectories. This gives us the time evolution of the \"distance\" between the trajectories. For a very small difference \(\delta\theta_{\kappa}\), for any \(\kappa\), in the initial conditions of the two memory states, we obtain:  
\[\Delta{\cal N}_{A_{\kappa}}(t)\equiv{\cal N^{\prime}}_{A_{\kappa}}(\theta^{ \prime},t)-{\cal N}_{A_{\kappa}}(\theta,t)\approx\sinh\bigl{(}2(\Gamma_{ \kappa}t-\theta_{\kappa})\bigr{)}\delta\theta_{\kappa}\, \tag{12}\]  
with \(\delta\theta_{\kappa}\equiv\theta_{\kappa}-\theta^{\prime}_{\kappa}\), assumed in full generality to be positive, and:  
\[\frac{\partial}{\partial t}\Delta{\cal N}_{A_{\kappa}}(t)=2\Gamma_{\kappa} \cosh\bigl{(}2(\Gamma_{\kappa}t-\theta_{\kappa})\bigr{)}\delta\theta_{\kappa}. \tag{13}\]  
The modulus of the difference \(\Delta{\cal N}_{A_{\kappa}}(t)\) and its time derivative thus diverge as \(\exp{(2\Gamma_{\kappa}t)}\), for all \(\kappa\)'s, for large \(t\), suggesting that \(2\Gamma_{\kappa}\), for each \(\kappa\), acts like the Lyapunov exponent in chaos theory [57].