Asymptotic analysis is a key tool to study nonlinear difference equations which arise in the mathematical modelling of real-world phenomena. It is not expected that explicit solutions can be found for the solutions of nonlinear difference equations; however, some nonlinear equations can be transformed into equivalent linear equations by a change of dependent variable. In this work, we transform a discrete logistic equation, which is a nonlinear difference equation, into a linear equation and we determine its explicit solution. This result able us to study the behavior of this solution and check the results known in stability theory. 2 | P a g e 1. Introduction Many types of problems are naturally described by recurrence relations said difference equations [2, 3], which usually …show more content…
To determine the stability of a fixed point is due to the fact that we may not be able to find the solution in a closed form even for the deceptively simple-looking equation (1.1). Definition 3 (basin of attraction) [1] Let ̅ be a fixed point of map f. Then the basin of attraction (or the stable set) ( ̅) of ̅ is defined as ̅ { ̅ } ̅ consists of all points that are forward asymptotic to ̅. 4 | P a g e 1. The Problem and Objectives of the study 2.1. The Discrete Logistic Equation [2] Let be the size of a population at time and μ is the rate of growth of the population from one generation to another, the discrete logistic equation is the mathematical model in the form ( ) This equation is the simplest nonlinear first-order difference equation. In spite of its simplicity, this equation exhibits complicated dynamics. If we know the initial population given by then we find its solution, by simple iteration we it is the orbit of In this work we are interested in the special case where the rate . We consider the mathematical model To find the fixed points of (2.1) we solve ̅ ̅ ̅ ̅, therefore ̅ ̅ , or ̅ ̅ Thus, we have two fixed points: ̅ ̅ ̅ ̅ ̅ ̅
In other words T(n) can be expressed as sum of T(n-1) and two operations using the following recurrence relation:
In, The Population Bomb by, Paul R Ehrlich, he explains the problem of population increase, and how there are people everywhere! The feeling of feeling over populated. He talks about how if there are more people then there is more food that needs to be produced then ate. He explains on the rich people becoming wealthier and the poor are going to be even poorer and there is going to be a starvation. Population is doubling every year and how our energy is turning into
...cal techniques which are numerically stable and can be used to both linear and nonlinear fractional differential equation [28]have been developed in the literature. For finding the numerical approximation more accurate and for reducing the computational cost here we choose the Caputo version and predictor-corrector algorithm for fractional differential equation [29],.According to the fractional predictor-corrector algorithm, we find that hyper chaos exist in new four-dimensional fractional order system. In the numerical results the system parameters are chosen as a = 10, b = 28, c = 8/3 and d = 1. Numeric results shows that when 0.9 1, the fractional- order system (2) always exhibits hyper chaotic behaviors. The two Lyapunov exponents τ_1=0.9,τ_2=0.08 are obtained when = 0.9 is chosen. If < 0.9 then in system (2) there is no hyper chaos exist. As shown
= 3 ´ E(C-H) + 1 ´ E(C-O) + 1 ´ E(O-H) + 1.5 ´ E(O=O)
“Only make moves when ya heart's in it, and live the phrase Sky's The Limit”. (Christopher Wallace “Sky’s The Limit”). In this lyric, Christopher Wallace explained how one should focus on personal goals, even though obstacles and social issues are present. Hip hop evolved during the 1970’s as a liberation movement in the form of a diverse culture. (Emmett G Price lll). Hip hop is used to express one’s feelings about personal and social issues. It has been known to be both positive and negative, however, over a period of time, Hip Hop has shed light upon the social issues within the country. Many Hip Hop artist such as Christopher Wallace, Jermaine Cole, Kendrick Duckworth, Dewayne Carter Jr., Lauryn Hill, Onika Maraj, and many other artist
method can be produced and a graph of the function can be made. From the graph,
ADVANTAGES & LIMITATIONS Advantages Ratio analysis is an important and age-old technique of financial analysis. The following are some of the advantages of ratio analysis. 1. What is the difference between a. and a. Simplifies financial statements It simplifies the comprehension of financial statements. Ratios tell the whole story of changes in the financial condition of the business.
Newton-Raphson method is of use when it comes to approximating the root or roots of an equation.
Many years ago humans discovered that with the use of mathematical calculations many things can be calculated in the world and even the universe. Mathematics consists of many different operations. The most important that is used by mathematicians, scientists and engineers is the derivative. Derivatives can help make calculations of anything with respect to another event or thing. Derivatives are mostly common when used with respect to time. This is a very important tool in this revolutionary world. With derivatives we can calculate the rate of change of anything with respect to time. This way we can have a sort of knowledge of upcoming events, and the different behaviors events can present. For example the population growth can be estimated applying derivatives. Not only population growth, but for example when dealing with plagues there can be certain control. An other example can be with diseases, taking all this events together a conclusion can be made.
in exponential form. For instance, in a base 2 system, 4 can be written as 2
Chapter one presents the Introduction. It also contains the statement of the problem and its signi...
-In order to solve this differential equation you look at it till a solution occurs to you.
There have been many great mathematicians in the world, though many are not well known. People have been studying math for ages, the oldest mathematical object dated all the way back to around 35,000 BC. There are still mathematicians today, studying math and figuring out ways to improve the mathematical world. Some of the most well-known mathematicians include Isaac Newton, Albert Einstein, and Aristotle. These mathematicians (and many more) have influenced the mathematical world and mathematics would not be where it is today without them. There were many great individuals who contributed greatly in mathematics but there was one family with eight great mathematicians who were very influential in mathematics. This was the Bernoulli family. The Bernoulli family contributed a lot to mathematics, medicine, physics, and other areas. Even though they were great mathematicians, there was also hatred and jealousy between many of them. These men did not want their brothers or sons outdoing them in mathematics. Most Bernoulli fathers told their sons not to study mathematics even if they wanted. They were told to study medicine, business, or law, instead, though most of them found a way to study mathematics. The mathematicians in this family include Jacob, Johann, Daniel, Nicolaus I, Nicolaus II, Johann II, Johann III, and Jacob II Bernoulli.
Human population growth was relatively slow for most of human history. Within the past 500 years, however, the advances made in the industrial, transportation, economic, medical, and agricultural revolutions have helped foster an exponential, "J-shaped" rise in human population (Southwick, Figure 15.1, p. 160). The statistics associated with this type of growth are particularly striking: "Human beings took more than 3 million years to reach a population of 1 billion people...The second billion came in only 130 years, the third billion in 30 years, the fourth billion in 15 years, the fifth billion in 12 years..." (Southwick, p. 159). As human population has grown, there has been simultaneous growth within the industrial sector. Both of these increases have greatly contributed to environmental problems, such as natural resource depletion, ecosystem destruction, and global climate change. Also linked with the increasing human population are many social problems, such as poverty and disease. These issues need to be addressed by policy makers in the near future in order to ensure the survival and sustainability of human life.
Here we have assumed that input v(t) is equal to zero. This assumption is valid as we are only concentrating on the internal stability of the system.