{'fraction_non_alphanumeric': 0.0285662133795747, 'fraction_numerical': 0.00538468558912111, 'mean_word_length': 4.822529224229543, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Mechanism for Stabilization of Dynamics in Nonlinear Systems with Different Time Scales

There are many natural, physical, and biological systems that exhibit multiple time scales. For example, the dynamics of a population of ticks can be described in continuous time during their individual life cycle yet discrete time is used to describe the generation of offspring. These characteristics cause the population levels to be reset periodically. A similar phenomenon can be observed in a sociological college drinking model in which the population is reset by the incoming class each year, as described in the 2006 work of Camacho et al. With the latter as our motivation we analytically and numerically investigate the mechanism by which solutions in certain systems with this resetting conditions stabilize. We further utilize the sociological college drinking model as an analogue to analyze certain one-dimensional and two-dimensional nonlinear systems, as we attempt to generalize our results to higher dimensions.


Introduction
In many dynamical systems, especially systems found in the biological and natural sciences, it is common to find multiple time scales embedded in the same system all of which might be continuous or include a combination of both discrete and continuous time. Dynamical systems with multiple time scales are also called singularly perturbed systems and slow-fast systems. An excellent example of a slow-fast type is the forced Van der Pol equation [5], and is the first system in which the phenomenon of chaos was first observed. However for our purposes and interests, we restrict only to discrete-continuous time scaled systems. Examples of these arise mostly in ecology. To illustrate, recall the famous system of equations known as the Lotka-Volterra model [ 1,3 ]. In this model, x(t) and y(t) denote, the prey and predator populations, respectively, at any time t. Now let us imagine the following prey-predator relationship: The predators are sharks, and the prey are small fish. If there were no more small fish, one might expect that the population of sharks would decline in quantity, according to the differential equation When small fish are present in the habitat or environment, however, it seems more reasonable that the number of encounters between these two species per unit time is jointly proportional to their populations x and y, or in other words, proportional to the product xy, by the law of mass action [4]. The addition of this last rate yields an equation for the shark population, and so in a similar fashion, the dynamics of the prey population dx dt is derived, giving us the nonlinear system of differential equations known as the prey-predator model (2), where a, b, c and d are all positive Although the explicit solution of system (2) is not known, it can be analyzed quantitatively and qualitatively [3], that is we can establish some information about the behavior of its solutions, by seeking a relation for x and y ignoring temporal space ( a better explanation of this can be found in [1] ). Moreover, we can look at this model in a slightly different way by making the growth rate of the prey time dependent, in other words, not constant. If we change the prey population slightly then the population dynamics still evolve continuously in time, but what differs now is the birthrate of the prey, denoted by some given a(t) which evolves discretely in time. This is due to normal ecological changes in temperature or the environment, and as a result the population behaves periodically according to this birthrate. In addition, we also find these type of phenomena in fishing models with harvesting such as the one found in [1] given by the generalized equation Here H(t) is a non-constant rate of members ( fish ) harvested per unit time, and as in the Lotka-Volterra model, the population of fish evolves continuously, yet due to the harvesting rate H(t), the population resets periodically, and this periodic reseting is dependent, more realistically, on harvesting seasons. . This resetting force leads to stabilization but at a different level than the equilibria from the continuous system absent from resetting forces.
From the resetting equation, t k+1 and t − k represent the beginning of year k + 1 and the end of year k, respectively, and the negative superscript indicates that t k+1 is dependent on the previous initial time t k . It can also be noted that N (0) is the total initial population, and c j = is a percentage of the total population which enters each class. Moreover, in the modified model discussed by Camacho et al. [6], numerical tools were the only option available to analyze the nonlinearity of the system, hence numerical simulations have shown that stabilization of the solutions occurred even when the continuous system without resetting did not stabilize ( i.e. solutions of the system with resetting present did not attain the stable equilibrium belonging to the continuous system absent of resetting ). It was also noticed that for certain initial conditions the system ( equations (5) Secondly we have considered both, a logistic model and a Gompertz (1825) model (taken from [1]) For these models we have artificially created a resetting condition given by where γ is a resetting parameter, and 0 < γ < 1. On the other hand, in the two-dimensional case we have considered a Logistic Coupled system [3] and a Gompertz-type System [1] for which we have chosen the resetting condition (A more detailed explanation of the mechanisms chosen is given in the proceeding parts of this prospectus summary).

Summary
Our illustrations have shown that the resetting mechanisms we investigate, certainly stabilize nonlinear system dynamics. As a hypothesis we propose the generalization, that any nonlinear system can be analytically investigated by following simply the theory of dynamical systems along with an advanced calculus approach as follows. It has been established that for a nonlinear autonomous system of ordinary differential equationsẋ which has a given initial condition, the fundamental Existence-Uniqueness Theorem [4] states that as long as there exists an open subset S of R n , which contains x 0 along with the requirement f ∈ C 1 (S), then there must be some a > 0 such that the initial value problem (IVP) stated has locally an explicit analytical, unique solution x(t) on some interval [−a, a]. In addition to the uniqueness of the solution of the system, resetting implies intersection of periodic orbits in the state space of the continuous dynamical system, with certain lower dimensional subspace called Poincaré Sections, transverse to the flow of the system. Initial data are thus periodic orbits with initial conditions on the Poincaré section thereby formulating a Poincaré map This iteration process or map allows us to further, invoke the contraction mapping theorem, and thus we can claim that the dynamics of the system are guaranteed to stabilize, when a generated sequence ( Poincaré Sections ) of initial data converges to a fixed point of the corresponding discrete mapping. But in general, for n-dimensional systems, it is very difficult to find a domain of initial data, where the discrete map might converge analytically. Investigation of the "size"of this region is a typical inverse problem which will be investigated numerically in the following sections.
Furthermore, there are many possibilities and applications to this work. Higher dimensional systems for example, could be analyzed with monotone systems theory, by means of explaining the behavior of high dimensional systems by that of one dimensional flow. But in general we want to find how the interplay of initial conditions and resetting definitions affect the behavior of solutions in the long run. Ultimately we want to find the set of initial conditions that leads to stabilization of dynamics by means of a convergent discrete mapping or Poincaré Mapping. Our future considerations include the investigation of any type of model in which resetting occurs and ideally we want to find applications to ecological and sociological models. It is also of potential interest to consider models in which the periodic administration of a drug, such as chemotherapy for instance, is given to cancer patients.  8) ) without resetting and the the non-smooth curve represent the solution of the system with resetting. Note that the stabilization of the continuous system with resetting occurs at a level different from the equilibria of the continuous system without resetting. That is, when resetting is introduced the system tends to stabilize to an equilibria of its own.  (11) ), numerical simulations show the stabilization expected, which match those observed by Camacho et al. Here again, the smooth curves correspond to the continuous system without resetting. At some time, the resetting rule comes into effect, and the solutions become nonsmooth. Notice in the first case for a resetting parameter γ = 0.67 < 1 and initial condition x 0 = 0.5 ( 0 < x 0 < 0.9, where x * stable = 0.9, and x * unst = 0), the dynamics of the system with resetting stabilize between the equilibrium of the continuous system, yet they follow an equilibrium level of its own. In the second case for initial conditions above stable equilibrium, we see the same phenomena ocurring. It is very interesting to note that through this resetting mechanism ( or rule ) we have chosen for the simple one-dimensional model, stabilization is not the level of the system without resetting. equilibria (x * , y * ) = (1, 0), and (0, y * ) are a saddle point and a set of degenerate isolated unstable equilibria, respectively. Every trajectory is unstable, as the behavior of the system without resetting exhibits a saddle node, unless the initial conditions fall on the stable manifold (1, 0).

Figure 4: Two-Dimensional Logistic Coupled System
Simulations belonging to the twodimensional system ( equations 14-15 ) again show, with the chosen resetting rule once again the system stabilizes, as expected for certain initial conditions and resetting parameter γ. The solution y(t) in the continuous system without resetting ( represented by the pink dashed curve ), is tending to infinity, while with resetting ( represented by the red solid curve) y(t) tends to some nonzero equilibrium ( very close to zero ) but in an oscillatory manner with very small amplitude. The solution x(t) ( represented by the blue solid curve ) in the system with resetting also stabilizes to a different level than the continuous solution x(t) in the system without resetting. Notice the amplitude of the oscillations of the solutions of x(t) surpass those of y(t) ( the red curve ) and this is due to the choice in size of the parameters. It is quite surprising that although the continuous system has no stable solution the introduction of a periodic resetting still causes the system to stabilize.