In a desktop environment, we can choose modes from the menu of the mode line. You can start the tutorial by typing C-h t or select Help Emacs Tutorial from the menu. The Emacs man page give the man emacs command in the command line will give you a summary of the basic options when calling Emacs from the command line.

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A quite detailed manual can be found in the Emacs info pages Using info needs some training, but using the Emacs interface is quite intuitive and similar to using a web browser. Type the command C-h r or choose Help Emacs Tutorial from the menu and you will open the front page of the emacs manual in a new window. By using the keys SPC and Backspace we can read the documentation page by page.

Using the navigation icons on the toolbar, you can go to the previous or to the next pages, go up one level etc. There are commands that can be given by typing single characters. For example, type d in order to jump to the main info directory. Type g emacs and go to the top page of the Emacs manual. Type g info and read the info manual. Emacs is structured in an intuitive and user friendly way. These make them quite long sometimes, but by using auto completion of their names this does not pose a grave problem. You can customize everything in Emacs. From key bindings to programming your own functions in the Elisp language.

Emacs reads and executes all these commands just before starting a session. Such a. Everything after a ; is a comment. You may also customize Emacs from the menu commands Options Customize Emacs. This is not a systematic exposition and you are expected to learn what is needed in this book by example. So, please, if you have not already done it, get in front of a computer and do what you read. We can put more than one command on each line by separating them with a semicolon. Proliferation of comments is necessary for documenting our code. Good documentation of our code is an integral part of programming.

If we plan to have our code read by others or by us at a later time, we have to make sure to explain in detail what each line is supposed to do. You and your collaborators will save a lot of time in the process of debugging, improving and extending your code. These lines start with a and are interpreted by a separate program, the preprocessor.

This acts like an editor! Within the parentheses placed after the name of the function, we put the arguments that we pass to the function. Notice that it ends with a semicolon. This statement performs an output operation printing a string to the standard output. The sentence Hello World! Splitting them over a number of lines is only a matter of making the code legible. In order to execute the commands in a program, it is necessary to compile it. This is a job done by a program called the compiler that translates the human language programming statements into binary commands that can be loaded to the computer memory for execution.

The compilation command is:. The extension. If the compilation is successful, the program runs with the command:. The dot is the current working directory and. Now, we will try a simple calculation. Given the radius of a circle we will compute its length and area. The expressions 2.

## C++ Programming for the Absolute Beginner

Note the explicit decimal points at the constants 2. If we write 2 or 4 instead, then these are going to be constants of the int type and by using them the wrong way we may obtain surprising results We compile and run the program with the commands:. Now we will try a process that repeats itself for many times. We will store the values of the radii in an array R[10] of the double type. This way, the elements of the array are referred to by an index that takes values from 0 to 9. The way it works is the following: In the round brackets after the keyword for , there exist three statements separated by semicolons.

If it is true , then the statements in the loop are executed. If it is false, the control of the program is transferred after the end of the loop. Now we will write an interactive version of the program. Instead of hard coding the values of the radii, we will interact with the user asking her to give her own values. The program will read the 10 values of the radii from the standard input stdin. A const declares a variable to be a parameter whose value does not change during the execution of the program and, if it is of int type, it can be used to declare the size of an array.

The array elements R[i] are read using the command:. The calculation of the length and the area of the circle is performed by the function. Calling a function, transfers the control of the program to the statements within the body of the function. The above function has arguments R[i], perimeter, area. The argument R[i] is intended to be only an input variable whose value is not going to change during the calculation. The arguments perimeter and area are intended for output.

Upon return of the function to the main program, they store the result of the computation. The user of a function must learn how to use its arguments in order to be able to call it in her program. These must be documented carefully by the programmer of the function. In order to use a function, we need to declare it the same way we do with variables or, as we say, to provide its prototype.

We may provide just enough details that determine the types of its arguments and the value returned. In our program this is done on the line:. The argument list does not need to include the argument names, only their types. Including the names is a matter of style that improves legibility of the code. The argument R is intended to be left unchanged during the function execution. This is why we used the keyword const in its declaration. This is why const is not used for them. The type of the value returned by a function is declared by the keyword before its name.

In our case, this is void which declares that the function does not return a value. The arguments R,L,A must be declared in the function and need not have the same names as the ones that we use when we call it. All arguments are declared to be of type double. This makes possible to change their values from within the function.

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This happens because, in this case, only the value of the variable L of the calling program is copied to a local variable which is used only within the function. The names of variables in a function are only valid within the scope of the function, i. Therefore the variable const int N is valid only within the scope of main.

You may use any name you like, even if it is already used outside the scope of the function. The names of arguments need not be the same as the ones used in the calling program. Only their types have to match. We summarize all of the above in a program trionymo. After a check whether , it computes the discriminant by calling the Discriminant a,b,c. The value returned to the calling program is the value of the expression given as an argument to the return statement.

Plotting data is an indispensable tool for their qualitative, but also quantitative, analysis. Its great advantage is the possibility to use it from the command line, as well as from shell scripts and other programs. Gnuplot is programmable and it is possible to call external programs in order manipulate data and create complicated plots. There exist interactive terminals where the user can transform a plot by using the mouse and keyboard commands. This section is brief and only the features, necessary for the following chapters, are discussed.

You can start a gnuplot session with the gnuplot command:. Type a command an press [Enter]. Type quit in order to quit the program. Plotting a function is extremely easy. Use the command plot and x as the independent variable of the function In order to plot many functions simultaneously, you can write all of them in one line:.

The above command plots the functions , , and. Within the square brackets [:] , we set the limits of the and axes, respectively. The bracket [] sets and the bracket [] sets. You may leave the job of setting such limits to gnuplot, by omitting some, or all of them, from the respective positions in the brackets. For example, typing [1:][:5] changes the lower and upper limits of and and leaves the upper and lower limits unchanged In fact, gnuplot ignores everything after a. After the keyword using , we instruct gnuplot which columns to use as the and coordinates, respectively.

The keywords with points instructs gnuplot to add each pair to the plot with points. We can combine several plots together in one plot:. There are many powerful ways to use the keyword using. Instead of column numbers, we can put mathematical expressions enclosed inside brackets, like using Gnuplot evaluates each expression within the brackets and plots the result.

The second line adds the plot of the function. We can plot the data written to the standard output of any command. Assume that there is a program called area that prints the perimeter and area of a circle to the stdout in the form shown below:. The interesting data is at the second and fourth columns. These can be plotted directly with the gnuplot command:. The data printed by the last program is in two columns and we plot the results using Gnuplot can produce plots in several languages e. The last line chooses the interactive terminal qt to be the output of the next plot.

Use set terminal pdf,postscript eps or svg , respectively. And now a few words for 3-dimensional 3d plotting. The next example uses the command splot in order to make a 3d plot of the function. We close this section with a few words on parametric plots. A parametric plot on the plane 2-dimensions is a curve , where is a parameter. A parametric plot in space 3-dimensions is a surface , where are parameters. The following commands plot the circle and the sphere :. They are much more simple to use than to incorporate them into your program.

This is a very simple shell script. This can be any program in the system, which in our case is the tcsh shell. The following lines are valid commands for the shell, one in each line. If not, we have to give the command:. Some of the versions of the programs that we wrote are asking for input from the stdin , which, normally, you have to type on the terminal. The stdin of the command. The last EOF has to be placed exactly in the beginning of the line.

The power of shell scripting lies in its programming capabilities: Variables, arrays, loops and conditionals can be used in order to create a complicated program. Shell variables can be used as discussed in section 1. The command echo echoes its argument to the stdin. These values can be used by the commands in the loop. It executes the command cat AREA. DAT is true. DAT exists. We close this section by presenting a more complicated and advanced script.

It only serves as a demonstration of the shell scripting capabilities. Read carefully the commands, as well as the comments which follow the mark. Notice how these arguments are manipulated by the script. Then, the script asks for the values of the radii of ten or more circles interactively, so that it will compute their perimeter and area. You will not regret the time investment! When a particle moves on the plane, its position can be given in Cartesian coordinates. The position vector is , where and are the unit vectors on the and axes respectively.

## Bibliography

The velocity vector is where. In this section we study the kinematics of a particle trajectory, therefore we assume that the functions are known. By taking their derivatives, we can compute the velocity and the acceleration of the particle in motion. The continuous functions are approximated by a discrete sequence of their values at the times such that.

We will start the design of our program by forming a generic template to be used in all of the problems of interest. Figure 2.

The program prints these values to the stdout so that the user can check them for correctness and store them in her data. The main calculation is performed in a loop executed while. The equations giving the position of the particle at time are. The data structure is quite simple. The constant angular velocity is stored in the double variable omega. The center of the circle , the radius of the circle and the angle are stored in the double variables x0, y0, R, theta. The current position is calculated and stored in the double variables x, y and the velocity in the double variables vx, vy.

The declarations of the variables are put in the beginning of the program:. The user interface of the program is the interaction of the program with the user and, in our case, it is the part of the program where the user enters the parameters omega, x0, y0, R, t0, tf, dt. The program issues a prompt with the names the variables expected to be read. The variables are read from the stdin by reading from the stream cin and the values entered by the user are printed to the stdout using the stream cout 1 :.

There are a couple of things to explain. Notice that after reading each variable from the standard input stream cin , we call the function getline. By calling getline cin,buf , a whole line is read from the input stream cin into the string buf 2. Then the statement. Since we never use buf , this is a mechanism to discard the rest of the line of input.

The reason for doing so will become clear later. In order to use them you have to include the header. Finally, endl is used to end all the cout statements. Next, the program initializes the state of the computation. This includes checking the validity of the parameters entered by the user, so that the computation will be possible. For example, the program computes the expression 2. We will also demand that and. An if statement will make those checks and if the parameters have illegal values, the exit statement 4 will stop the program execution and print an informative message to the standard error stream cerr 5.

The line myfile. If or the corresponding exit statements are executed which end the program execution. The optional error messages are included after the stop statements which are printed to the stderr. The value of the period is also calculated and printed for reference. The statements between within the scope of the while condition are executed as long as condition has a true value. Notice the use of the functions sin and cos that calculate the sine and cosine of an angle expressed in radians. The header cmath is necessary to be included. Compilation and running can be done using the commands:.

A typical session looks like:. The lines shown above that start with a character are printed by the program and the lines without are the values of the parameters entered interactively by the user. The user types in the parameters and then presses the Enter key in order for the program to read them. Here we have used , , , , and. Each line has the parameters that we want to pass to the program with each call to cout. The rest of the line consists of comments that explain to the user what each number is there for. We want to discard these characters during input and this is the reason for using getline to complete reading the rest of the line.

The program can read the above values of the parameters with the command:. Its contents can be inspected after the execution of the program with the command cat :. We use gnuplot for plotting the data produced by our programs. Therefore we can plot the functions and by using the gnuplot commands:.

Note that. Use the online help system in gnuplot in order to see its usage:. Therefore, the right way to call the function is atan2 y-y0,x-x0. We also plot the lines which graph the constant functions and which mark the limit values of. The velocity components as function of time as well as the trajectory can be plotted with the commands:. We close this section by showing how to do a simple animation of the particle trajectory using gnuplot. We are not going to explain how it works 10 but how to use it in order to make your own animations.

A full animation session can be launched using the commands:. We are now going to apply the steps described in the previous section to other examples of motion on the plane. The motion is periodic with angular frequency and period. The angular velocity is computed from which gives. We note that the acceleration of gravity is hard coded in the program and that the user can only set the length of the pendulum.

A simple session for the study of the above problem is shown below 12 :. Then, the equations describing the trajectory of the particle and its velocity are given by the parametric equations. The structure of the program is similar to the previous ones. The initial time is taken to be.

The program calculates and and prints them to the stdout. The solutions to the equations of motion. Long commands have been continued to the next line as before. We can use them in order to construct the asymptotes of the plotted functions of time. The last example of this section will be that of the anisotropic harmonic oscillator.

The force on the particle is. The solutions of the dynamical equations of motion for , , and are. We have set in the program above. The user must enter the two angular frequencies and and the corresponding times. A typical session for the study of the problem is shown below:. By slightly generalizing the methods described in the previous section, we will study the motion of a particle in three dimensional space.

All we have to do is to add an extra equation for the coordinate and the component of the velocity. The structure of the programs will be exactly the same as before. The particle moves on the plane with constant angular velocity. The equations of motion are derived from the relations. Their solution 15 is.

We take. The full program is listed below:. In order to plot the functions , , , , , we give the following gnuplot commands:. In order to make a three dimensional plot of the trajectory, we should use the gnuplot command splot :. We can change the plot limits with the command:. The program animate3D. All we have to do is to change the value of the file variable in gnuplot. The magnetic force on the particle is and the equations of motion are. We are now ready to write a program that calculates the trajectory given by 2.

The components of the initial velocity are and. The initial position is calculated from the equation. In this section we will study the motion of a particle that is free, except when bouncing elastically on a wall or on certain obstacles. This motion is calculated by approximate algorithms that introduce systematic errors.

These types of errors 16 are also encountered in the study of more complicated dynamics, but the simplicity of the problem will allow us to control them in a systematic and easy to understand way. When it reaches the boundaries and it bounces and its velocity instantly reversed. Its potential energy is.

The force within the box and at the position of the walls. As long as the particle moves within the box, its motion is free and. Therefore, real numbers are represented in some approximation that depends on the amount of memory that is used for their storage. In most implementations 17 , float uses 4 bytes of memory and double 8. Moreover, float represent numbers with magnitude in the, approximate, range while double in.

Note that variables of the integer type int , long , We also used numerical constants of type float. This is indicated by the letter f at the end of their names: 2. Finally we changed the form of the output. That is why we used the statements. For purposes of studying the numerical accuracy of our results, we used 9 digits of output, which is, of course, slightly redundant. The use of setw requires the header iomanip. Compiling, running and plotting the trajectory using gnuplot can be done as follows:.

The reader should have already realized that the above mentioned error can be made to vanish by taking arbitrarily small. Therefore, we naively expect that as long as we have the necessary computer power to take as small as possible and the corresponding time intervals as many as possible, we can achieve any precision that we want.

Well, that is true only up to a point. The remedy to such a problem can only be a change in the algorithm. This is not always possible, but in the case at hand this is easy to do. For example, consider the equation that gives the position of a particle in free motion. Then, all we have to do is to reverse the direction of the motion and reset the initial position and time to be the position and time of the collision.

This can be done by using the loop:. We considered two types of errors: First, the systematic error of determining the instant of the collision of the particle with the wall. This error is reduced by taking a smaller time step. This error is increased with decreasing. Such a situation is found in many interesting problems, therefore it is quite instructive to study it in more detail. When the exact solution of the problem is not known, the systematic errors are controlled by studying the behavior of the solution as a function of. We will take , , , , , so that the particle will collide with the wall every 10 units of time.

We will measure the position of the particle 20 as a function of and study its convergence to a limit 21 as. The analysis requires a lot of repetitive work: Compiling, setting the parameter values, running the program and calculating the value of for many values of. The third number in the above line is the value of the velocity. We repeat for 12 more times until reaches the value The result is.

The 2nd method has much better behavior that the 1st one. How many steps does it really take to reach , when the expected number of those is? Each time you take a measurement, issue the command. The result is interesting:. We conclude that one important source of accumulation of systematic errors is the calculation of time. This will be discussed further in chapter 4. The particle starts moving at time from and our program will calculate its trajectory until time with time step.

The last line repeats the same animation at half speed. The arrow drawn is not the position vector with respect to the origin of the coordinate axes, but the one connecting the initial with the current position of the particle. The next step should be to test the accuracy of your results.

This can be done by generalizing the discussion of the previous section and it is left as an exercise for the reader. We will start with a game of If the ball leaves out of the box through its open side, the player loses. In order to check if the ball is in the hole when it is at position , all we have to do is to check whether. Initially we place the ball at the position at time. The player hits the ball which leaves with initial velocity of magnitude at an angle degrees with the axis.

You should construct the plots of the position and the velocity of the particle. Copy it from the accompanying software to the current directory and give the gnuplot commands:. The next example with be three dimensional. The collisions of the particle with the cylinder are elastic. We take the axis of the cylinder to be the axis and the two bases of the cylinder to be located at and. The collisions of the particle with the bases of the cylinder are easy to program: we follow the same steps as in the case of the simple box.

The projection of the particle moves within a circle of radius and center at the intersection of the axis with the plane. The velocity of the particle before the collision is. The transformation , will be performed in the function reflectVonCircle vx,vy,x,y,xc,yc,R. Upon entry to the function, we provide the initial velocity vx,vy , the collision point x,y , the center of the circle xc,yc and the radius of the circle 24 R.

Upon exit from the function, vx,vy have been replaced with the new values Note that the function atan2 is used for computing the angle theta. This function, when called with two arguments atan2 y,x , returns the angle in radians. The correct quadrant of the circle where lies is chosen. The angle that we want to compute is given by atan2 y-yc,x-xc. Then we apply equations 2. In order to plot the position and the velocity as a function of time, we use the following gnuplot commands:. In order to plot the trajectory, together with the cylinder, we give the commands:.

The command set parametric is necessary if one wants to make a parametric plot of a surface. The cylinder without the bases is given by the parametric equations with ,. The last example will be that of a simple model of a spacetime wormhole. Of course we should note that such macroscopic and stable wormholes are not known to be possible to exist in the framework of general relativity. One needs an exotic type of matter with negative energy density which has never been observed.

We will study a very simple model of the above geometry on the plane with a particle moving freely in it We identify the points on the two circles such that the point 1 of the left circle is the same as the point 1 on the right circle, the point 2 on the left with the point 2 on the right etc.

The two circles are given by the parametric equations , , for the right circle and , , for the left. Crossing the circle is determined by the relation. The angle is calculated from the equation. Similarly we calculate the case of entering from and emerging from. The condition now is:. The angle is given by. For mapping , we calculate the vectors. The emerging velocity is:. Systematic errors are now coming from crossing the two mouths of the wormhole. There are no systematic errors from crossing the boundaries , why? Try to think of ways to control those errors and study them.

The closed trajectories that we are looking for come from the initial conditions. They are unstable, as can be seen by taking. It is easy to compile and run the program. The most celebrated application of the logistic map comes from the study of population growth in biology. The simplest and most naive model is the one that makes the reasonable assumption that the rate of population growth of a population is proportional to the current population:.

The general solution of the above equation is showing an exponential population growth for an decline for. It is obvious that this model is reasonable as long as the population is small enough so that the interaction with its environment adequate food, diseases, predators etc can be neglected. The simplest model that takes into account some of the factors of the interaction with the environment e. The parameter gives the maximum growth rate of the population and controls the ability of the species to maintain a certain population level.

The equation 3. Then and equation 3. The solutions of the above equation are well approximated by so that we have population growth when and decline when. Equation 3. In what follows, the derivative of will be useful:. Since we interpret to be the fraction of the population with respect to its maximum value, we should have for each 1. The function has one global maximum for which is equal to.

Therefore, if , then , which for an appropriate choice of will lead to for some value of. Therefore, the interval of values of which is of interest for our model is. The logistic map 3. Given an initial value , a sequence of values is produced. This will be referred 2 to as the trajectory of. In the following sections we will study the properties of these trajectories as a function of the parameter. The solutions of the logistic map are not known except in special cases. For we have. For , whereas for we have periodic trajectories resulting in rational and non periodic resulting in irrational.

For other values of we have to resort to a numerical computation of the trajectories of the logistic map. It is obvious that if the point is a solution of the equation , then for every. For the function we have two solutions. We will see that for appropriate values of , these solutions are attractors of most of the trajectories.

This means that for a range of values for the initial point , the sequence approaches asymptotically one of these points as. Obviously the measure zero sets of initial values and result in trajectories attracted by and respectively. Then we obtain. In the following discussion, these points will be shown to be bifurcation points. For the function with we have that and. Therefore, if the point is an attractor, whereas the point is irrelevant. When , the point results in , therefore is unstable.

Any initial value near deviates from it. Since for we have that , the point is an attractor. Any initial value approaches. Indeed, when we have that and for the point becomes unstable. Consider the solution of the equation. If is one of its solutions and for some we have that , then and therefore is also a solution. The points , are such that they are real solutions of the equation. By expanding the polynomials 3. The roots of the trinomial in 3. The study of the stability of the solutions of requires the same steps that led to the equation 3.

By noting that 5 , we see that for , and for ,. For the intermediate values the derivatives for. Therefore, these points are stable solutions of and the points bifurcate to , for. These are real solutions of the equation that gives the 4-cycle. For , the points , are a stable 4-cycle which is an attractor of almost all trajectories of the logistic map 6. The points are getting closer to each other as increases so that. As we will see, marks the onset of the non-periodic, chaotic behavior of the trajectories of the logistic map.

Computing the bifurcation points becomes quickly intractable and we have to resort to a numerical computation of their values. Initially we will write a program that computes trajectories of the logistic map for chosen values of and. Its stdout is redirected to the stdin of the command. Similarly, we can study trajectories which are -cycles when crosses the values.

Another way to depict the 2-cycles is by constructing the cobweb plots: We start from the point and we calculate the point , where. This point belongs on the curve. The point is then projected on the diagonal and we obtain the point. We repeat times obtaining the points and on and respectively. If we have a -cycle, we will observe a periodic trajectory going through points which are solutions to the equation. Compiling and running this program is done exactly as in the case of the program in logistic.

We can plot the results using gnuplot. The plot command shown above, runs the program exactly as it is done on the command line. Only the second command "echo trj. The following line adds the plots of the functions , and of the diagonal.

Figures 3. An example of a non periodic trajectory is also shown, which exhibits chaotic behavior which can happen when. By plotting these points as a function of we construct the bifurcation diagram. These can be calculated numerically by using the program bifurcate.

In this program, the user selects the values of that she needs to study and for each one of them the program records the point of the -cycles 8 ,. This is easily done by computing the logistic map several times until we are sure that the trajectories reach the stable state. The parameter NTRANS in the program determines the number of points that we throw away, which should contain all the transient behavior. The program is listed below:.

When goes past , the trajectories become non-periodic and exhibit chaotic behavior. Chaotic behavior will be discussed more extensively in the next section. It is also interesting to note the appearance of a 3-cycle right after! By using the theorem of Sharkovskii, Li and Yorke 9 showed that any one dimensional system has 3-cycles, therefore it will have cycles of any length and chaotic trajectories. The stability of the 3-cycle can be studied from the solutions of in exactly the same way that we did in equations 3.

Figure 3. It is also quite interesting to note that if we magnify a branch withing these windows, we obtain a diagram that is similar to the whole diagram! We say that the bifurcation diagram exhibits self similarity. There are more interesting properties of the bifurcation diagram and we refer the reader to the bibliography for a more complete exposition. We close this section by mentioning that the qualitative properties of the bifurcation diagram are the same for a whole class of functions.

Feigenbaum discovered that if one takes any function that is concave and has a unique global maximum, its bifurcation diagram behaves qualitatively the same way as that of the logistic map. Examples of such functions 10 studied in the literature are , and. The constants and of equations 3. In order to determine the bifurcation points, one has to solve the nonlinear, polynomial, algebraic equations and. For this reason, one has to use an approximate numerical calculation of the roots, and the simple Newton-Raphson method will prove to be a good choice.

In order to understand how it works, we assume that is an analytic function for all the values of used in the computation. Then, by Taylor expanding around we obtain. If we wish to have , we choose. The equation above gives the Newton-Raphson method for one equation of one variable. When , are non zero at the root and is bounded, the convergence of the method is quadratic with the number of iterations. This means that there is a neighborhood of the root such that the distance is.

If the root has multiplicity larger than 1, convergence is slower. In the general case it works well only close enough to a root. We should also keep in mind that there are simple reasons for the method to fail. For example, when for some , the method stops. For functions that tend to as , it is easy to make a bad choice for that does not lead to convergence to a root. Sometimes it is a good idea to combine the Newton-Raphson method with the bisection method.

When the derivative diverges at the root we might get into trouble. For example, the equation with , does not lead to a convergent sequence. For some functions the basin of attraction of a root the values of that will converge to the root can be tiny.

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See problem The parameters and. Given , we solve for which gives the energy. The function and its derivative are. In the program listed above, the user is asked to set the initial point. It is instructive to make the plot of the left and right hand sides of 3. Then we can make appropriate choices of the initial point. Using gnuplot , the plots are made with the commands:.

We conclude that one of the roots of the equation is. The reader can compute more of these roots by following these steps by herself. The method discussed above can be easily generalized to the case of two equations. Suppose that we need to solve simultaneously two algebraic equations and. In order to compute a sequence , , , , , that may converge to a root of the above system of equations, we Taylor expand the two functions around. As an example, consider the equations with ,. In order to guess the region where the real roots of the systems lie, we make a 3-dimensional plot using gnuplot :.

We plot the functions together with the plane. The intersection of the three surfaces determine the roots we are looking for. Compiling and running the program can be done by using the commands:. The computation above leads to the roots , ,. The Newton-Raphson method for many variables becomes hard quite soon: One needs to calculate the functions as well as their derivatives, which is prohibitively expensive for many problems. It is also hard to determine the roots, since the method converges satisfactorily only very close to the roots.

In order to determine the bifurcation points for we will solve the algebraic equations and. At these points, -cycles become unstable and -cycles appear and are stable. This happens when , where. We will look for solutions for. We will solve the algebraic equations:. The above listing shows the points of the 2-cycle and some of the points of the 4-cycle.

It is also possible to compare the calculated value with the expected one. Improving the accuracy of the calculation is left as an exercise for the reader who has to control the systematic errors of the calculations and achieve better accuracy in the computation of.

We have seen that when , the trajectories of the logistic map become non periodic and exhibit chaotic behavior. Chaotic behavior mostly means sensitivity of the evolution of a dynamical system to the choice of initial conditions. Assume that two trajectories have , as initial points and. The exponent is called a Liapunov exponent. A useful equation for the calculation of is. We modify the program logistic. The last line plots the stdout of the command "echo 3.

The gnuplot command set logscale y , puts the y axis in a logarithmic scale. The slopes of these lines are equal to the Liapunov exponent. We estimate the error of our computation from the standard error of the mean. The reader should perform such a computation as an exercise.

Using gnuplot , the relevant commands are:. Now we are going to use equation 3. This equation is approximately correct when a we have already reached the steady state and b in the large limit. This is a point where the system exhibits strong chaotic behavior and convergence is achieved after we compute a large number of steps.

The plot can be constructed with the gnuplot commands:. In order to determine the regions of chaotic behavior we have to study the dependence of the Liapunov exponent on the value of. This is recommended for programs that run for a long time, so that the shell returns the prompt to the user and the program continues to run even after the shell is terminated.

The intervals with correspond to stable -cycles. The intervals where correspond to manifestation of strong chaos. These intervals are separated by points with where the system exhibits weak chaos. This means that neighboring trajectories diverge from each other with a power law instead of an exponential, where is a positive exponent that needs to be determined. The parameter is the one usually used in the literature.

Strong chaos is obtained in the limit. For larger , switching between chaotic and stable periodic trajectories is observed each time changes sign. The critical values of can be computed with relatively high accuracy by restricting the calculation to a small enough neighborhood of the critical point. You can do this using the program listed above by setting the parameters rmin and rmax.

We can also study the chaotic properties of the trajectories of the logistic map by computing the distribution of the values of in the interval. After the transitional period, the distribution for the cycles will have support only at the points of the cycles, whereas for the chaotic regimes it will have support on subintervals of.

The distribution function is independent for most of the initial points of the trajectories. If one obtains a large number of points from many trajectories of the logistic map, it will be practically impossible to understand that these are produced by a deterministic rule. For this reason, chaotic systems can be used for the production of pseudorandom numbers, as we will see in chapter As we will see in chapter 12 , it is given by the equation.

In our case, we can make an approximate calculation of by dividing the interval to subintervals of width. For given we obtain a large number of values of the logistic map and we compute the histogram of their distribution in the intervals. The probability density is obtained from the limit of as becomes large and small large.

Indeed, converges to. The output of your first program. Producing that one-line message took a lot of work! Actually, of the eight lines in the program, only two—the ones that start with printf —do the work that produces the output. To see a much longer program, glance at Appendix B. Although the Draw Poker game there spans several pages, it contains elements common to the shorter program you just saw.

Look through both the programs just discussed and notice any similarities. Be careful when typing C programs into your C compiler. C gets picky, for instance, if you accidentally type a square bracket [ when you should type a brace. I typed the previous program in Word for this book and then copied it to Code::Blocks. When compiling the program, I received a number of errors because my quotes on the printf line were smart quotes created by the word processor to give that cool slanted look , and the compiler did not recognize them. After I deleted the quotes on the line and retyped them in my programming editor, the code compiled just fine.

So if you get errors in programs, make sure the quotes are not the culprit.

### C.hierclass: Designing classes in a hierarchy:

For instance, most of the spacing you see in C programs makes the programs clearer to people, not to C. As you program, add blank lines and indent sections of code that go together to help the appearance of the program and to make it easier for you to find what you are looking for. Use the Tab key to indent instead of typing a bunch of spaces. Most C editors let you adjust the tab spacing the number of spaces that appear when you press Tab. Some C program lines get long, so a tab setting of three provides ample indention without making lines too long.

C requires that you use lowercase letters for all commands and predefined functions. You learn what a function is in the next section. About the only time you use uppercase letters is on a line with define and inside the printed messages you write. See All Related Store Items.