# Search Patterns Based on Trajectories Extracted from the Response of Second-Order Systems
## 1 Introduction  
Metaheuristic algorithms refer to generic optimization schemes that emulate the operation of different natural or social processes. In metaheuristic approaches, the optimization strategy is performed by a set of search agents. Each agent maintains a possible solution to the optimization problem, and is initially produced by considering a random feasible solution. An objective function determines the quality of the solution of each agent. By using the values of the objective function, at each iteration, the position of the search agents is modified, employing a set of search patterns that regulate their movements within the search space. Such search patterns are abstract models inspired by natural or social processes [1]. These steps are repeated until a stop criterion is reached. Metaheuristic schemes have confirmed their supremacy in diverse real-world applications in circumstances where classical methods cannot be adopted. Essentially, a clear classification of metaheuristic methods does not exist. Despite this, several categories have been proposed that considered different criteria, such as a source of inspiration, type of operators or cooperation among the agents. In relation to inspiration, nature-inspired metaheuristic algorithms are classified into three categories: Evolution-based, swarm-based, and physics-based. Evolution-based approaches correspond to the most consolidate search strategies that use evolution elements as operators to produce search patterns. Consequently, operations, such as reproduction, mutation, recombination, and selection are used to generate search patterns during their operations. The most representative examples of evolution-based techniques, include Evolutionary Strategies(ES) [2; 3; 4], Genetic Algorithms (GA) [5], Differential Evolution (DE) [6] and Self-Adaptative Differential Evolution (JADE) [7]. Swarm-inspired techniques use behavioral schemes extracted from the collaborative interaction of different animals or species of insects to produce a search strategy. Recently, a high number of swarm-based approaches have been published in the literature. Among the most popular swarm-inspired approaches, include the Crow Search Algorithm (CSA) [8], Artificial Bee Colony (ABC) [9], Particle Swarm Optimization (PSO) algorithm [10; 11; 12], Firefly Algorithm (FA) [13; 14], Cuckoo Search (CS) [15], Bat Algorithm (BA) [16], Gray Wolf Optimizer (GWO) [17], Moth-flame optimization algorithm (MFO) [18] to name a few. Metaheuristic algorithms that consider the physics-based scheme use simplified physical models to produce search patterns for their agents. Some examples of the most representative physics-based techniques involve the States of Matter Search (SMS) [19; 20], the Simulated Annealing (SA) algorithm [21; 22; 23], the Gravitational Search Algorithm (GSA) [24], the Water Cycle Algorithm (WCA) [25], the Big Bang-Big Crunch (BB-BC) [26] and Electromagnetism-like Mechanism (EM) [27]. Figure 1 visually exhibits the taxonomy of the metaheuristic classification. Although all these approaches consider very different phenomena as metaphors, the search patterns used to explore the search space are exclusively based on spiral elements or attraction models [10; 11; 12; 13; 14; 15; 16; 17; 28]. Under such conditions, the design of many metaheuristic methods refers to configuring a recycled search pattern that has been demonstrated to be successful in previous approaches for generating new optimization schemes through a marginal modification. On the other hand, the order of a differential equation refers to the highest degree of derivative considered in the model. Therefore, a model whose input-output formulation is a second-order differential equation is known as a second-order system [29]. One of the main elements that make a second-order model important is its ability to present very different behaviors, depending on the configuration of its parameters.