and implementation of a ZAD and sliding mode control (ZAD-SMC) for a multiphase step-down converter.  
The ZAD-SMC operates at a fixed switching frequency and allows the implementation of an interleaving technique that provides current-ripple cancellation at the converter output. In addition, the control with ZAD has also been used to implement a buck converter in a field programmable gate array (FPGA) [10]; applying ZAD with a pulse on the side fulfills the requirement of the fixed switching frequency. Additionally, bifurcation analyses were performed to study the transition from periodicity to chaos [11] in buck and boost DC-DC converters operated with the ZAD control technique. Other results show robustness with low output error and a fixed switching frequency [12; 13].  
Additionally, the converter displays different behaviors when the parameters vary and their operating conditions are modified [14]. In previous research, some authors have analyzed the bifurcations to find the optimal range of parameter design in power converters, thus establishing the conditions under which the system exhibits changes in stability or periodicity (periodic, quasi-periodic, or chaotic orbits) [13; 15; 16]. Furthermore, the existence of saturated periodic orbits in the boost converter leads to the presence of the period addition phenomenon, which has been partially studied in [17] and reported in one-dimensional discrete dynamical systems. For example, an effective and efficient solution using an interleaved boost converter was proposed in [2], which was not limited to only mitigate or alleviate partial shading effects but can also to extract the maximum power available from the whole partially shaded photovoltaic system with 100 % mismatch loss index (MML).  
Although the literature review shows that authors have dealt with the stability issues in power converters, they have not presented a detailed analysis of the existence of \(nT\)-periodic orbits and determined equations that define the problem. Therefore, this paper illustrates the conditions for the existence of \(nT\)-periodic orbits. The stability of the orbits is studied based on eigenvalues of the Jacobian matrix of the Poincare application. Contributions in this paper are defined as follows:  
* An exact formula is presented to provide sufficient conditions for the existence of \(nT\)-periodic orbits with and without saturation in the duty cycle.
* The application of Poincare is generalized and an equation is presented that allows calculating the Jacobian matrix to determine the stability of the \(nT\)-periodic orbits. A wide range of the \(\gamma\) parameter, \(1T\)- and \(2T\)-periodic orbits in the boost converter, and Floquet exponents are determined to analyze the stability limit of the \(1T\)-periodic orbit.
* A biparametric diagram that generates \(nT\)-periodic orbits is created to perform the analysis from \(1T\)- to \(7T\)-periodic orbits.
* The variational equation that admits the perturbation of the \(1T\)-periodic orbits is calculated to determine the system stability.  
The paper is divided into three more sections. Section 3 presents the materials and methods, where the description of the boost converter and the mathematical model used in this research are presented. In addition, this section defines the duty cycle formulation and the discretization of the system. Moreover, the Poincare map application is included to analyze the periodic orbits. Finally, the stability analysis is explained in this section to find the \(nT\)-periodic orbits. Section 4 illustrates the results of these methods applied to find the different periodic orbits and analyze the findings. Finally, the conclusions are included in the last section of the paper.