The success of nations in the path towards economic development hinges heavily on the emergence and dynamism of a modern sector capable of simultaneously absorbing an increasing share of the labour force while reducing the technological gap with the world's frontier. Failure to do so would eventually lead the economy to low- or middle- income traps, in which only a small fraction of the population would benefit from the gains of economic growth and technological progress. Building on previous contributions from Post-Keynesian, Neo-Schumpeterian and Latin-American Structuralism literature, this paper sets up a theoretical model of catching-up among nations aimed at formalizing this idea by exploring the dynamic interactions between structural change and technological upgrading in the process of economic development. The focus of the model is on a "representative" nation of the South that is characterized by having: i) a dual structure (i.e., a large share of labour force working on low-productive-traditional activities that coexists with a small fraction of workers employed in modern activities); ii) a high degree of technological backwardness in the modern activities; and iii) a binding restriction on the external accounts. Under these circumstances, the dynamic behaviour of two key variables will determine the success or failure of this economy over time: the share of labour in the modern sector and the relative stock of technological knowledge of the modern sector compared to that of the world technological leader. Depending on initial conditions and underlying parameters, the southern economy would be attracted towards four different equilibrium points, each of them entailing extremely different implications in terms of long-run development. After analysing the dynamic properties of the model, simple simulations are implemented in order to illustrate a number of structural trajectories that might shed new light on the complex forces acting behind the success or failure of economic development