Periodic Behaviour Concept - Coursework Example

PERIODIC BEHAVIOUR By Student’s name Course code and name Professor’s name University name City, State Date of submission Introduction Periodic phenomena is used to illustrate behaviour that is exhibited over time and again. These include climatic patterns, sinusoidal changes in daily temperature recordings, daylight hour’s variations, the moon’s phases, animal population growth and tidal variations in harbours (Martin, et al., 2012). Most of the systems that exist in nature are periodical meaning that they go back to the initial state from which they originated. The concept of time itself is based on the cyclic nature of the so called periodic ‘common sense’ of the universe. Periodic behaviour differs from one field to the other depending on the descriptors that have been identified for the sake of this study. Periodicity of a given phenomenon is determined mathematically by use of periodic functions which are also known to repeat themselves over and over in horizontal intervals. For this sake, a period is defined as the length of the repetition which is also referred to as a cycle. When repeated over time, these periods form waves which are used to determine various characteristics of a phenomenon such as time and cumulative value of a given quantity. The use of waves has made the periodicity of various natures easy to decipher especially when it comes to mathematical modelling of various problems that require solutions to be determined by this approach. Such modelling techniques that are covered in this investigative article include sine, cosine and tangent functions. In order to understand this concept better naturally occurring processes are used as the patterns are demystified. Population Periodicity This section highlights some of the naturally occurring problems associated to periodicity and that can be modelled mathematically in order to solve them using near-periodical solutions. To begin with, the issue of population has been largely studied by Callahan et al. (1990) as the lead example for the periodicity essence for which they describe predictability of animal population in various national parks. In a habitat where animals prey over the other species, the authors have described two models derived by May and Lotka-Volterra in order to provide evidence of periodical fluctuations of both predators and preys. Data taken by Lotka-Volterra over time was modelled for the lynx as a predator and its main prey the snowshoe hare for a period of up to 40 years. A complex periodic pattern was obtained with one basic bulge and three smaller ones that were discovered to occur over a period of 40 years. These data sets were then used to come up with mathematical functions that were to become the base of their argument when it came to matters regarding population fluctuations (Callahan, et al., 1990). Below is a graphical presentation of this problem that which is preceded by mathematical approach as solutions were sought for future predictions. Figure 1: Population fluctuation for lynx pelts. Lotka-Volterra generated the predator prey equations using the first order differential equations to solve this ecological interaction. The populations of the two species i.e. the predators and the preys shall change in the order of the periodical equations 1 and 2 below. (1) (2) Where = Number of prey species within a given ecological zone. = Number of predators within the same zone. = The growth rate of the prey and predator population. Represent the interaction parameters of the two species. Assuming the above constants of interaction are represented by values established as shown in table 1 below then the problem can be modelled in figure 2 to demonstrate the periodic fluctuations of populations in a prey predator relationship. symbol value α 0.1 β 0.01 γ 0.001 δ 0.05 Table 1: Values of the constants presented by prey predator equations. Sharov (1996) used the same constant values to solve the problem through an analytical method that captured the differential equations by Euler’s method of mathematical calculus. The beauty in his solution lied in the mode of Microsoft Excel programming used in coming up with his final draft document that can be used in calculating for up to nth solution for the problem. While the periodic graph indicates the change in densities of the two populations, a model for asymptotic stability was also generated to indicate the convergence attractor in order to determine the initial conditions in the anticlockwise direction. This is shown in figure 2 as an oval shape graph which also provides an overview of predator population density change although a modification in constants affects them. Figure 2: Mathematical model indicating a periodic solution devised by Sharov (1996). Periodicity in Climate The second issue worth discussion on how periodic behaviour as a natural phenomenon has been demystified mathematically is the climatic pattern. Periodicity in climatic patterns emerged as an issue of mathematical concern back in 1941 when a Serbian geophysicist Milutin Milankovitch came up with a theory on how the earth’s orbit affected climate. The periodic fluctuations that were earlier observed when the earth orbits around the sun were put into use to carry out projections on climate changes (Callahan, et al., 1990). This hypothesis has been tested by geologist who can attest to the possibility of using periodic functions to model for temperature among other climatic characteristics. Further, it has been revealed though thick sedimentation at the lake bottoms that long term projections regarding climate changes can be handled with ease. For example it has been noted that during the dry part of the year, siltation is minimal thus the sedimentation noted during this time is thinner if the soil profile was to be observed. Measuring the annual layer build-up has helped geologists to come up with periodic functions that are reliable in all aspects (Knežević, 2010). For the sake of mathematical modelling, the sine and cosine functions shall be used to indicate the fluctuations that can be measured for temperature. The equation shall take the form of. In this equation, A, B, C and D are constants, is the temperature measured in ˚C while is the month of the year under consideration. In order to create a data model that is responsible for the prediction of weather conditions for the future, weather technologists can model for the coming months given that they have actual averages for the past few years. Considering the averages given for the Wellington Airport for the years ranging from 1971 to 2000 Mackintosh (2001) was able to come up with a periodical graphs that could accurately predict the preceding weather conditions. Month January February March April May June July August September °C 17.09 17.64 15.67 15.19 13.32 11.04 10.72 10.31 12.04 Month October November December °C 13.69 13.07 17.77 Table 2: Average temperatures for Wellington Airport for the years 1971 to 2000. In such a scenario, the data in table 2 above was able to be modelled using periodic functions. For long term averages equation 3 below has been widely used to model for the periodic behavior to be used for prediction purposes. (3) Alternatively equation 4 below can be used in place of 3. (4) Where 1 can be replaced for the months’ values rated from 1 – 12. A = Amplitude . B = . C = units to be translated to the right. D = Figure 3: Monthly mean temperatures for Wellington for the years ranging from 1971 to 2000. Figure 4: A periodic graph obtained for the temperatures spreading through the months of year 2000. Modelling the information contained in the Wellington Airport climate graph in figure 4 gives the temperatures function 5 below. The cosine curve in figure 3 and 4 are generated by entering the function in the excel sheet. Depending on the periodical nature of the curve, the constant changes to 3.75 thus the function changes to, (5) (6) The averaging system used by the theorist assumed that the impact of events that occur on a one off are avoided by all means necessary. The data sheet that was used in table 2 was dependent on random climate data that was obtained during the preceding years. Climate can therefore be predicted very accurately using the averaged values and the periodic functions that can altogether be said to possess this behaviour. Sun Spot Cycle Another important phenomenon that has been determined by use of periodicity or periodic behaviour is the sun spot cycle. Sunspots are helpful in determining the patterns that the meteorological phenomena such as rainfall and carbon dioxide take. The sunspots are said to fluctuate in a span of 11 years thereby reaching a peak. This takes the exact form of the lynx in the population example used above. According to data evidence documented by Sharov (1996), most of the natural periodic phenomena take the 11 year pattern. It is however stated that getting firm evidence of analytical data to come up with peak values has been deemed as difficult. This is because some of the data critical to this analysis may be drowned by some effects that may take a diverse nature. Further, the detection of fluctuations in periodic data due to noise poses difficulties although with computer evolution such software as Matlab and Minitab have been utilised in order to ensure an accurate prediction. Figure 5: Average fluctuations in sunspots over a period of 200 years (De Meyer, 1981). The Venus–Earth–Jupiter (VEJ) frequency that has been used to come up with workable models that can be used to give accurate predictions uses an average of years on a periodic graph also known as the Jose cycle. It has been noted by Salvador (2013) that while some individual sunspot cycles affect the wave frequency, the basic cycles have changed over time resulting to the perturbations such as the Jovian frequency and the VEJ. These principles have been used as the basis of argument for coming up with periodic functions that can be used to model smoothened curves. The working frequency for these models lie within a range of 1253 hertz responsible for the orbital realignment for earth, Venus and Jupiter. This model has been used over and over again with the latest development being the ability by mathematicians to subject it against the Gaussian shape as a way of correcting the phases. The construction of this model does not however state the formation of sunspots which are very important in this study. The periodicity of the sunspot data is transformed by multiplying the sunspot number (SN) from the last occurrence with the polarity plus or minus one for each and every phase consideration (Salvador, 2013). (7) The periodic equation that has been derived for the sake of sunspot modelling refers to and as the phasing parameters and the nonlinear least optimization squares. The polarity of the sunspots was adjusted in terms of the observations that were recorded in the previous years. Therefore; The modulated frequencies include,, and which have been found to change at the 178.8 and 1253. These can also be calculated by using the periodic equations in the order shown in the relationship below. (8) The subsequent modulated frequencies are then imposed on a frequency of 178.8 although this can only be calibrated over a 300 year monthly sunspot data which obviously covers a 20% of the main cycle. In other words to process a more accurate periodic behaviour graph, a longer period is required i.e. Figure 6: Periodic fluctuations in sunspot numbers (Salvador, 2013). The monthly sunspot data responsible for the replication of the periodic cycle is very important since the polarity is sourced here. The monthly periods which are also a reflection of the sunspot numbers are described by geophysicists as Solanki data based on Solanki e al. (2004). A strong correlation between the data interestingly reflected between the years 1749 to 2013 reproduced absolute values for the upcoming years. A reconstructed graph is shown above for the purpose of forecasting the significant sunspot cycles that are required for various purposes. Periodicity of the Earth’s Orbit The last illustration worth for the periodicity topic is the orbiting of the earth around the sun. While the earth returns to its original position after every single year, there has been need to investigate on how this can be forecast using the periodic behaviour. The periodicity behind it is obvious as the cycles such as the seasons are purely dependent on the position of the earth around the sun. A quick research in the cycles shows that more subtle periodicities occur while the earth is in motion. Due to the precession of the earth, in order for a repetition to occur, 23,000 years are required in as much as there may be fluctuations in obliquity and eccentric cycles (Sharov, 1996). The average angle of the earth can be used to estimate the position of the earth as 0.985653˚ represents one calendar day. The shape of the ellipse however lowers the average depending on the position of the earth. For the purpose of estimation, the constant which denotes the measure of elliptic dimensions in comparison to the circle is used. The periodic equations below have been widely utilised in order to give a prediction of what the future is likely to contain. (9) Although this formula is only an estimation of the accurate angle that should be used to give the exact position of the earth, graphic demonstration is also key to explaining this. To change the angle in degree to time, the number of minutes per day which is 1440 is divided with the angle of rotation which is 361˚. This gives an approximate 3.98892 minutes for each degree that the earth rotated. If may be we were to estimate the variation for which our watches would read on 10th January then the only way this can be calculated using this method is as shown below. The equation of time takes the form of -1 minute and 2.5 seconds. This can also be modelled in a periodical curve for the first three months to minimize the errors that may result from the whole process. Conclusion The periodicity phenomenon is very important for the estimation of various values within given cadres. Ranging from population, earth’s orbit, sunspot across the skies and climate/ weather, this method of prediction has been very handy in solving a lot of issues that are beyond human control. Climate for example is easily predictable through values that can be modelled to give an overview of what is expected. Some of the phenomenon modelled through periodicity are accurate as there are no disturbances (noise) in the curves and therefore depict the mathematical importance of this process. List of References The Azimuth Project, 2012. Milankovitch cycle. [Online] Available at: http://www.azimuthproject.org/azimuth/show/Milankovitch+cycle [Accessed 17 August 2014]. Callahan, J. et al., 1990. Periodicity. In: Calculus in Context. Massachusetts: National Science Foundation , pp. 419-460. De Meyer, F., 1981. Mathematical modelling of the sunspot cycle. Solar Physics, Volume 70, pp. pp. 259-272. Knežević, Z., 2010. Milutin Milankovic and astronomical theory of Climate changes. Astronomical Observatory, Belgrade, Volume 2, pp. pp. 17-20. Mackintosh, L., 2001. Using trigonometric functions to model climate. [Online] Available at: http://www.niwa.co.nz/education-and-training/schools/resources/climate/modelling [Accessed 17 August 2014]. Martin, D. et al., 2012. Mathematics for the International Student: Mathematics HL (Core). Ed. 3 ed. Adelaide: Haese Mathematics. Salvador, R. J., 2013. A mathematical model of the sunspot cycle for the past 1000 yr. Pattern Recognition in Physics, Volume 1, p. pp. 117–122. Sharov, A., 1996. 10.2 Lotka-Volterra Model. [Online] Available at: https://home.comcast.net/~sharov/PopEcol/lec10/lotka.html [Accessed 15 August 2014]. Wolfram Mathworld, 2014. Lotka-Volterra Equations. [Online] Available at: http://mathworld.wolfram.com/Lotka-VolterraEquations.html [Accessed 15 August 2014]. Read More

A complex periodic pattern was obtained with one basic bulge and three smaller ones that were discovered to occur over a period of 40 years. These data sets were then used to come up with mathematical functions that were to become the base of their argument when it came to matters regarding population fluctuations (Callahan, et al., 1990). Below is a graphical presentation of this problem that which is preceded by mathematical approach as solutions were sought for future predictions. Figure 1: Population fluctuation for lynx pelts.

Lotka-Volterra generated the predator prey equations using the first order differential equations to solve this ecological interaction. The populations of the two species i.e. the predators and the preys shall change in the order of the periodical equations 1 and 2 below. (1) (2) Where = Number of prey species within a given ecological zone. = Number of predators within the same zone. = The growth rate of the prey and predator population. Represent the interaction parameters of the two species.

Assuming the above constants of interaction are represented by values established as shown in table 1 below then the problem can be modelled in figure 2 to demonstrate the periodic fluctuations of populations in a prey predator relationship. symbol value α 0.1 β 0.01 γ 0.001 δ 0.05 Table 1: Values of the constants presented by prey predator equations. Sharov (1996) used the same constant values to solve the problem through an analytical method that captured the differential equations by Euler’s method of mathematical calculus.

The beauty in his solution lied in the mode of Microsoft Excel programming used in coming up with his final draft document that can be used in calculating for up to nth solution for the problem. While the periodic graph indicates the change in densities of the two populations, a model for asymptotic stability was also generated to indicate the convergence attractor in order to determine the initial conditions in the anticlockwise direction. This is shown in figure 2 as an oval shape graph which also provides an overview of predator population density change although a modification in constants affects them.

Figure 2: Mathematical model indicating a periodic solution devised by Sharov (1996). Periodicity in Climate The second issue worth discussion on how periodic behaviour as a natural phenomenon has been demystified mathematically is the climatic pattern. Periodicity in climatic patterns emerged as an issue of mathematical concern back in 1941 when a Serbian geophysicist Milutin Milankovitch came up with a theory on how the earth’s orbit affected climate. The periodic fluctuations that were earlier observed when the earth orbits around the sun were put into use to carry out projections on climate changes (Callahan, et al., 1990). This hypothesis has been tested by geologist who can attest to the possibility of using periodic functions to model for temperature among other climatic characteristics.

Further, it has been revealed though thick sedimentation at the lake bottoms that long term projections regarding climate changes can be handled with ease. For example it has been noted that during the dry part of the year, siltation is minimal thus the sedimentation noted during this time is thinner if the soil profile was to be observed. Measuring the annual layer build-up has helped geologists to come up with periodic functions that are reliable in all aspects (Knežević, 2010). For the sake of mathematical modelling, the sine and cosine functions shall be used to indicate the fluctuations that can be measured for temperature.

The equation shall take the form of. In this equation, A, B, C and D are constants, is the temperature measured in ˚C while is the month of the year under consideration. In order to create a data model that is responsible for the prediction of weather conditions for the future, weather technologists can model for the coming months given that they have actual averages for the past few years.

Read More