Sine map bifurcation

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Support Answers MathWorks. Search MathWorks. MathWorks Answers Support. Open Mobile Search. Trial software. You are now following this question You will see updates in your activity feed. You may receive emails, depending on your notification preferences. How to output all X values of bifurcation diagram for logistic map?

Sinead on 10 Aug Vote 0. Accepted Answer: Fangjun Jiang. I'm using the code below in matlab to produce a bifurcation diagram for the logistic map. I want to get every value of X for each a value but the array editor only shows one value of x for each a. Any help appreciated. Accepted Answer. Fangjun Jiang on 10 Aug Vote 1.

Cancel Copy to Clipboard. If you want to keep a history and observe it, you can save it to a different array, X for example. The code below saves the x data in each iteration to a row in X. Thank you.A technique to generate periodic or nonperiodic oscillations systematically in first-order, continuous-time systems via a nonlinear function of the state, delayed by a certain timeis proposed.

This technique consists in choosing a nonlinear function of the delayed state with some passivity properties, tuning a gain to ensure that all the equilibrium points of the closed-loop system be unstable, and then imposing conditions on the closed-loop system to be semipassive.

We include several typical examples to illustrate the effectiveness of the proposed technique, with which we can generate a great variety of chaotic attractors. We also include a physical example built with a simple electronic circuit that, after applying the proposed technique, displays a similar behavior to the logistic map.

Control and synchronization of chaos have become an intense field of research since several years ago. Some publications pointed out the impact this field may have in many areas of science and technology [ 1 — 8 ]. Chaos anticontrol, that is, producing chaos in a controlled way in a nonchaotic system, has also attracted attention due to its potential usefulness for some important problems of mechanical, electronic, telecommunications, optical, chemical, or biological systems, among others [ 9 — 16 ].

Some techniques to generate chaos in discrete-time systems was proposed in [ 17 — 19 ]. In those papers the aim is to design a feedback nonlinear controller for a linear system such that the closed-loop system displays a chaotic behavior in the sense of Li and Yorke [ 20 ]. The continuous-time counterpart seems to be, however, a more difficult problem. Introducing a time-delayed state in the feedback control law to modify the behavior of an oscillatory system has been proposed since some time ago [ 21 ].

Traditionally, controllers are designed such that the negative effects of time-delay terms on the system performance are attenuated, even suppressed. On the contrary, a time-delay controller makes use of delays to attain some control objectives, for example, stabilizing unstable periodic orbits embedded in a chaotic attractor [ 2452223 ]. One of the first papers in this line is due to Pyragas [ 21 ]. The method proposed therein uses a control signal proportional to the difference between the measured system output and the same output signal delayed by a certain time.

As a result, chaos is suppressed and the system oscillates periodically, with a period close to the introduced delayrendering a control signal with small amplitude.

Time delay has also been proposed to generate oscillations, even in first-order systems, which is the aim of the present paper. In [ 24 — 26 ] some conditions are given such that a continuous system can reproduce period-doubling bifurcations displaying by a map, using a singular perturbation technique for delay systems.

To find conditions such that a continuous-time system with an input depending on a delayed state displays a chaotic behavior is, however, a more difficult task, and one must rely on numerical or physical experiments to support the investigation.

Patterns in the sine map bifurcation diagram

A very known numerical example is given in [ 27 ], where it is described how a first-order continuous-time system, controlled with a piecewise linear function of the delayed state, displays chaotic behavior. Some other results following the same idea were presented in [ 28 — 31 ] and in [ 3233 ], where fuzzy techniques have been employed. Similarly, a model to generate any number of scrolls from a first-order time-delay system is proposed in [ 34 ].

A recent and detailed study of this problem was presented in [ 35 ], using Fourier series to analyze the generation of oscillations via time-delay states. The results mentioned were got from approximate methods or oriented to particular systems and supported on numerical or physical experiments.

All these results have contributed to establishing that chaotifying a regular system with a simple function of the delayed state is possible. However, developing formal tools or systematic procedures to produce chaotic behavior is a more involved problem. In this paper, another technique to produce periodic or nonperiodic, even chaotic oscillations in first-order, continuous-time systems via a nonlinear function of the delayed state is proposed.

Given a first-order passive system, this technique consists in synthesizing a feedback signal given by a nonlinear function of the delayed state, then tuning a gain to ensure that all the equilibrium points of the closed-loop system be unstable, and imposing conditions on the closed-loop system to be semipassive. The main property of this procedure is that it can induce a dynamical behavior to the closed-loop system, similar to the behavior displayed by the map defined by the nonlinear function of the delayed state.

If this map shows chaotic behavior, then the closed-loop system can also display this dynamics by tuning a parameter. At present time this tuning must be performed via numerical simulation; however, under some conditions established in this paper, it can be ensured that the reproduction of a dynamic behavior is similar to the map.

We include several examples to illustrate the effectiveness of the proposed technique, with which a variety of chaotic attractors can be generated. Let us consider a first-order, continuous-time system given by whereis the state, is the input, and is smooth.

Suppose the input is set to where is a function such that the delay differential equation has a solution. Note that, without loss of generality, we can assume a unit time delay. A different delay can be used, but it can be transformed to unity by a time scaling. Then this system is related to the discrete system in such a way that at time instants. Therefore, the solutions of the previous systems 4 and 6 coincide at. This means that any dynamical behavior displayed by the discrete-time system 6 will be displayed also by the continuous-time, pure-shift system 4.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Home Questions Tags Users Unanswered.

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sine map bifurcation

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It only takes a minute to sign up. So, in Matlab I did. Is this correct? If not, what did I do wrong here? Or, just help me find the bifurcation values here.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Matlab code for logistic map bifurcation Ask Question. Asked 6 years, 3 months ago. Active 6 years, 3 months ago. Viewed 3k times. Active Oldest Votes. Similarly, the second bifurcation point occurs in the change from period-2 to period-4 points.

Hope that helps! Daryl Daryl 5, 3 3 gold badges 21 21 silver badges 37 37 bronze badges. But, how do I find such a value for b? This will eventually give you a bracket for the value, at which point bisecting could be used. Sign up or log in Sign up using Google.See what's new with book lending at the Internet Archive. Uploaded by jakej on September 21, Search icon An illustration of a magnifying glass. User icon An illustration of a person's head and chest. Sign up Log in.

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Images Donate icon An illustration of a heart shape Donate Ellipses icon An illustration of text ellipses. Patterns in the sine map bifurcation diagram Item Preview.

EMBED for wordpress. Want more? Advanced embedding details, examples, and help! Numerical computations of bifurcation maps for one dimensional maps show patterns regular jumps in point density in the zones of chaotic behaviour.

In this work, empiric formulas are given for these patterns for an entire class of maps.

What are Logistic Maps (and what they tell us about free will)

Addeddate External-identifier urn:arXiv There are no reviews yet. Be the first one to write a review. Additional Collections.Using Python to visualize chaos, fractals, and self-similarity to better understand the limits of knowledge and prediction.

Chaos theory is a branch of mathematics that deals with nonlinear dynamical systems. A system is just a set of interacting components that form a larger whole. Lastly, dynamical means the system changes over time based on its current state.

How does that happen? The logistic function uses a differential equation that treats time as continuous. The logistic map instead uses a nonlinear difference equation to look at discrete time steps. If the growth rate is set too low, the population will die out and go extinct.

Here are the values we get:. The columns represent growth rates and the rows represent generations. The model always starts with a population level of 0. In the column for growth rate 2. So the growth rate of 2. Here you can easily see how the population changes over time, given different growth rates.

The blue line represents a growth rate of 0. The population dies out.

Bifurcations of the sine map.

The cyan line represents a growth rate of 2. The growth rates of 3. While the yellow line for 3. An attractor is the value, or set of values, that the system settles toward over time.

Oscillations in First-Order, Continuous-Time Systems via Time-Delay Feedback

When the growth rate parameter is set to 0. In other words, the population value is drawn toward 0 over time as the model iterates. When the growth rate parameter is set to 3. This attractor is called a limit cycle. It never hits the same point twice and its structure has a fractal form, meaning the same patterns exist at every scale no matter how much you zoom into it. When we produced the line chart above, we had only 7 growth rates. Thus, each vertical slice depicts the population values that the logistic map settles toward for that parameter value.

For growth rates less than 1. For growth rates between 1. But for some growth rates, such as 3. It never settles into a fixed point or a limit cycle. So, why is this called a bifurcation diagram?

At the vertical slice above growth rate 3. At growth rate 3. In other words, at that growth rate, applying the logistic equation to one of these values yields the other.A python library for creating cobweb plots, orbit diagrams, and calculating Lyapunov exponents.

sine map bifurcation

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If nothing happens, download the GitHub extension for Visual Studio and try again. A python library for creating cobweb plots, bifurcation diagrams, and calculating the Lyapunov exponents of one-dimensional maps.

sine map bifurcation

When calculating Lyapunov exponents you can either pass the map's first-order derivative, or the plotting function can calculate the derivative using scipy. Unfortunately, scipy's derivate function can be very slow and may raise errors depending on the map's shape.

The package can generate whole plots, or you can feed it axes to populate. For example, the following will show two bifurcation diagrams fror the Gauss Iterated Map side-by-side Here are four cobweb plots for the Sine map showing long-run behavior under different parameters. Skip to content. MIT License. Dismiss Join GitHub today GitHub is home to over 50 million developers working together to host and review code, manage projects, and build software together.

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View code. Bifurcation diagram with Lyapunov exponents To create a bifurcation diagram paired with Lyapunov exponents for the sine map About A python library for creating cobweb plots, orbit diagrams, and calculating Lyapunov exponents.

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