Chaos Control 1.6.2
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Chaos Control was created to help you manage your goals and desired outcomes in both your business and personal life.
'Chaos Control Freaks' is the first episode of the anime series, Sonic X. It first aired on 6 April 2003 and 23 August 2003 in Japan and the United States, respectively. The episode starts out in a forest at night time, as it shows Big the Cat and his pet frog named Froggy sleeping. Just then, an alarm is heard going off at a nearby tower switching on its lights. Sonic the Hedgehog runs. Chaos Control (カオスコントロール Kaosu Kontorōru) is a recurring technique in the Sonic the Hedgehog series. It is a Chaos Power that allows the user to warp time and space with the mystical Chaos Emeralds. While first introduced as a way to teleport over large distances, Chaos Control has since been evolved into a overall term for any supernatural reality manipulation conducted. Chaos Control includes many editing and task-creation features, and you can also sync them to the cloud so you can access them from any device. In addition, you can set alerts for specific lists, forcing you to get things done if you don't want to keep receiving alarms all the time. Technical information.
People don’t usually achieve impressive results simply by being good at task management. It’s the ability to set legitimate goals that makes the difference. Just write down your desired outcomes to make them real. This simple technique helps you to prioritize your goals before acting on them.
Chaos Control is a task manager based on the best ideas of GTD (Getting Things Done) methodology created by David Allen. Whether you are running a business, launching an app, working on a project or simply planning your holiday trip, Chaos Control is a perfect tool to manage your goals, juggle your priorities, and organize your tasks to get things done. And the best part is, you can handle both heavyweight project planning and simple daily routine like shopping list management in one flexible app. Also, Chaos Control is available across all major mobile and desktop platforms with seamless sync.
HERE IS HOW IT WORKS:
- MANAGE YOUR PROJECTS
Project is a goal combined with a set of tasks you need to complete in order to achieve it. Create as many projects as you like to write down all the desired outcomes you have - ORGANIZE YOUR GOALS
Create unlimited number of projects and group them by category using Folders - USE GTD CONTEXTS
Organize tasks from different projects using flexible context lists. If you are familiar with GTD you would just love this feature - PLAN YOUR DAY
Set due dates for tasks and make plans for any particular day - USE CHAOS BOX
Put all the incoming tasks, notes and ideas into Chaos Box in order to process them later. It works similar to GTD inbox, but you can use it as a simple to-do list - SYNC YOUR DATA
Chaos Control works on both desktop and mobile devices. Setup an account and sync your projects across all of your devices
This app is designed with creative people in mind. Designers, writers, developers, startup founders, entrepreneurs of all kinds and pretty much anyone with ideas and desire to make them happen. We combined the power of GTD with the convenient interface to help you with:
- personal goal setting
- task management
- time management
- planning your business and personal activities
- building your routine
- handling simple to do lists, checklists and shopping lists
- catching your ideas and thoughts to process them later
KEY FEATURES
- Seamless cloud sync across all major mobile and desktop platforms
- GTD-inspired Projects and Contexts supplemented with Folders, sub-folders and sub-contexts
- Recurring tasks (daily, weekly, monthly and chosen days of the week)
- Chaos Box – Inbox for your unstructured tasks, notes, memos, ideas and thoughts. Great tool for staying on track inspired by GTD ideas
- Notes for tasks, projects, folders and contexts
- Fast and smart search
What’s New:
Version 1.6.2
- Various minor improvements and performance optimization.
Screenshots
![Chaos control 1 6 2006 bmw Chaos control 1 6 2006 bmw](https://static.wikia.nocookie.net/sonic/images/3/3b/Title045.jpg/revision/latest?cb=20111101113118)
International Journal of Modern Nonlinear Theory and Application
Vol.1 No.3(2012), Article ID:23085,3 pages DOI:10.4236/ijmnta.2012.13011
Vol.1 No.3(2012), Article ID:23085,3 pages DOI:10.4236/ijmnta.2012.13011
Chaos Control in a Discrete Ecological System
Department of Mathematics and Finance-Economics, Sichuan University of Arts and Science, Dazhou, China
Email: [email protected]
Received August 23, 2012; revised September 7, 2012; accepted September 14, 2012
Keywords: Chaos Control; Discrete Ecological System; Numerical Simulation
ABSTRACT
In research [1], the authors investigate the dynamic behaviors of a discrete ecological system. The period-double bifurcations and chaos are found in the system. But no strategy is proposed to control the chaos. It is well known that chaos control is the first step of utilizing chaos. In this paper, a controller is designed to stabilize the chaotic orbits and enable them to be an ideal target one. After that, numerical simulations are presented to show the correctness of theoretical analysis.
1. Introduction
Population dynamics in ecology are generally governed by discrete and continuous systems. In recent years, the study of discrete ecological systems has attracted extensive attentions [1-6]. This is because that some natural populations have non-overlapping generations, thus discrete models are more realistic than continuous ones to study these species. Another reason is that people always study population changes by one year (mouth, week or day). Such investigations are often required discrete models. Especially, using discrete models is more efficient for numerical simulations. Recently, Zhang and Li [1] studied the following discrete ecological model:
(1)
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where xn, yn denote the two ecological species’ densities respectively in generation n; δ is the integral step size. The more meaning of system (1) can refer to the reference [1,2]. It is shown that the system (1) generates period-double bifurcations and chaos. But the authors did not investigate the chaos control of the system.
It is well known that chaos control is the first step of utilizing chaos. The possibility of chaos control in biological systems has been stimulated by recent advances in the study of heart and brain tissue dynamics. Recently, some authors have investigated that such a method can be applied to population dynamics and even play a nontrivial evolutionary role in ecology [7-9]. In this paper, we design a proper controller to control the chaos of system (1).
2. Chaos Control
In this section, chaotic orbits to an unstable fixed point are stabilized by utilizing some control techniques. Firstly, we introduce the following lemma which is useful to establish our results Lemma 1[1]. If a > b, then system (1) has an unique positive fixed point at, where,.
Consider the following map which is the feedback is applied to system (1)
(2)
where Xn = (xn, yn)T, μn is control variable and satisfies,. Evidently, map (2) degenerates to original system (1) only if μn = 0. We select the feedback variable μn in the range (–ε, ε), so that the orbit holds in the neighborhood of fixed point E as long as the control arises. The ergodic nature of the chaotic dynamics guarantees that the mode trajectory in the neighborhood of the wishful orbit. In the neighborhood of E, map (2) can be approximated by the following form:
(3)
where A is the Jacobian matrix at E and B is a column vector, and they are given by:
,
.
Let X* = (x*, y*)T and suppose that μn is a linear function of Xn, which is expressed as μn = PT(Xn – X*),. Substitute the result into (3), we get
According to the study [10], the fixed point E will be stable if the matrix (A – BPT) is asymptotically stable, that is to say, all its eigenvalues are less than 1 in modulus. Now, we make use of “pole placement technique” [11] to determine the specific values in (A – BPT). If system (1) is chaotic, we obtain
.
Then we choose ,
as the desired eigenvalues of the matrix (A – BPT). The controllability matrix Swinsian 1 12 0 – music manager and player.
has two rank. Thus the solution to the pole placement problem is obtained as
where Q = CW, p1 and p2 are the coefficients of characteristic polynomial of the matrix A, and ,;
q1 and q2 are the coefficients of characteristic polynomial of the matrix (A – BPT),
and , q2 = 0.
After calculations, we get
Chaos Control 1 6 2006 Full
(4)
Furthermore, the controller has the following form:
where,.
However, the above considerations only are fit for a local small neighbor of E. In view of the global situationwe can specify μn by making μn = 0 if
is too large. This is because the range of μn is restrained by and. Thus, we limit the number value
.
Therefore, in practice we take μn as
(5)
According to the above analysis, we get the following result.
Theorem 1. If then the control variable can stabilize chaotic trajectory of system (1) to the fixed point E, where PT is given by Equation (4).
3. Numerical Simulations
In the section, we use density-time diagrams and phase portraits to confirm the above theoretical analysis.
Let a = 2.21, b = 1.02, δ = 0.9666. At the condition, has the value 7.70732. According to Lemma 1, system (1) has and only has a positive fixed point E(x*, y*) = (2.16667, 0.53846). We adopt
.
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When ε is given the value 0.03 and 0.09, Theorem 1 is satisfied. Density-time diagram of ecological specie xn is given by Figure 1(a), which is characterized by switches between apparently regular and chaotic behaviors. Actually, it is intermittency, which is a basic characteristic of chaos. At the same parameters, phase portrait is illustrated by Figure 1(b), which is a chaotic attractor. Figure 2 is the chaos control diagrams corresponding toFigure 1. With the same parameters of Figure 1, system (1) is chaotic if n < 800 when ε = 0.03 (Figures 2(a) and (b)) according to the control strategy. Actually, Figure 2(a) is supertransient, which is used to denote an unusually long convergence to an attractor. Figure 2(b) is phase portrait corresponding to Figure 2(a). When ε increases to 0.09, supertransient disappears and the system
(a)(b)
Figure 1. (a) Density-time diagram of xn; (b) phase portrait of xn and yn. Where the parameters given by a = 2.21, b = 1.02, δ = 0.9666.
(a)(b)(c)(d)
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Figure 2. (a) Density-time controlled diagram of xn with ε = 0.03; (b) controlled phase portrait with ε = 0.03; (c) Density-time controlled diagram of xn with ε = 0.09; (d) controlled phase portrait with ε = 0.09. The other parameters given by a = 2.21, b = 1.02, δ = 0.9666.
stabilizes to the fixed point (2.16667, 0.53846), which is simulated by Figures 2(c) and 2(d).
4. Conclusion
In this paper, we design a proper controller to control the chaos of system (1) which was firstly studied by Zhang and Li [6]. From the theoretical analysis, we concluded that the control variable can stabilize chaotic trajectory of system (1) to the fixed point E(x*, y*) under the condition of
where PT is given by Equation (4). Then simulations are presented to show the correctness of theoretical analysis. Figure 1 demonstrates system (1) is chaotic with parameters a = 2.21, b = 1.02, δ = 0.9666. Figure 2 indicates system (1) processes from supertransient to the fixed point when the control variable applied to the system.
5. Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 30970305), the Sichuan Provincial Natural Science Foundation (No. 10ZB136), the Sichuan Provincial Old Revolutionary Base Areas Foundation (No. SLQ2010C-17).
![Chaos Chaos](https://ygoprodeck.com/wp-content/uploads/2017/07/jerry_wang_tomato_warrior.png)
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NOTES
*Corresponding author.