Consider a nonlinear ODE \[\begin{array}{l} \dot{x}=Ax+F(x,y), \\ \dot{y}=By+G(x,y). \end{array}\] where x is the stable direction and y is the center-unstable direction. Under suitable assumptions, the ODE possesses a foliation in the phase space, where leaves on it can be characterized by the exponential growth/decay rates of the differences between solutions that start on them. In particular, the invariant stable and inertial (center-unstable) manifolds are special cases of leaves in the foliation. Moreover, each leaf is a graph of a certain Lipschitz function. Thus, one can restrict the flow of the original ODE to its stable or inertial manifold. In the case of the inertial manifold this restriction is known as the inertial form, which is an ODE in the variable y alone which shares the long time behavior of the original ODE.

FOLI8PAK can

• compute the function whose graph is inertial manifold, stable manifold and stable foliation.
• compute trajectories restricted on the inertial and stable manifold.
• compute "tracking" solutions for any given initial condition.
• generate AUTO-07p compatible files for the inertial form.

As illustration, see the following two figures:

Figure 1: (Left frame) Flows in the stable manifold of a test problem (see the reference) with different initial conditions $x_0$, i.e., solutions to $$x′ = Ax+F(x,\Phi_0(x)), x(0)=x_0.$$ (Right frame) Flows in the inertial manifold for theKuramoto-Sivashinsky equation (KSE), i.e., solutions to $$y′ = By + G(\Psi_0(y), y), y(0) = y_0.$$
Figure 2: Bifurcation diagrams for KSE. The plots are created by AUTO-07p and the Equations-File for AUTO-07p is generated by FOLI8PAK. This figure shows that 3-mode inertial form (see the left frame) has the same long time behaviors of 12-mode approximation (see the right frame).