The Role of Global Dynamics in Analysis, Control and Safe Design

The analysis of the uncoupled system, represented by a single mechanical equation with a constant membrane temperature, demonstrates similar buckling behavior. To understand the discrepancy observed, we must conduct a thorough global analysis of the coupled system. By increasing the initial membrane temperature, we effectively reduce the duration of the thermal transient. This approach allows us to examine varying cross-sections for different initial membrane temperature values, facilitating a transition from the initial single modes to the fully buckled state at the end of the transient.

Additionally, this analysis is illustrated in the membrane temperature-velocity plane, where vertical lines represent the mechanical cross-sections. It is crucial to emphasize that only a comprehensive global analysis can reveal the differences in the varying scenarios of the multi-field coupled system, which accounts for transient effects.

Now, let us address a broader analytical challenge concerning the robustness and overall stability of the system's response in the presence of global bifurcations. This involves transitioning from detached manifolds associated with a specific saddle point to manifolds that are tangent to strong manifold intersections. This transition may occur, for example, when increasing the excitation amplitude. In global dynamic terms, this represents a shift from uneroded basins to the onset of erosion, eventually leading to full erosion and the phenomenon of escape.

Such phenomena are observed across various physical systems, including both macro and micro-nano mechanics. It is essential that the basins are sufficiently large and compact to ensure robustness against finite variations in initial conditions. We must also assess whether the evolution of the basin with changing parameters remains smooth.