On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities
Abstract
:1. Introduction
- The recurrence relations are obtained for calculating the coefficients of the expansion of solutions in a power series for any dynamical system with quadratic nonlinearities in a general form;
- The convergence of the power series is studied. A simple formula for calculations is derived (in comparison with that of [21] obtained in the known literature) for calculating the length of the integration step in a general form;
- The criteria for checking the accuracy of the approximate solution are obtained. The control of the accuracy and configuration of the approximate solution of a dynamical system uses forward and backward time, which makes the numerical method reliable (degrees of piecewise polynomials, the value of the maximum integration step, etc.);
- The FGBFI method allows to construct high-precision approximations to non-extendable solutions of a system of autonomous differential equations with a quadratic right-hand side, like, for instance, the system of the following form:In this case, the numerical solution computed with FGBFI will never cross the asymptote and will approach it arbitrarily closely.
2. Bohr’s Almost Periodic Functions
3. Dissipative System of the Fourth Order
4. The FGBFI Method
- Set the number of bits under the mantissa of a real number and the precision for an estimate of the common term of the power series. Note that defines the machine epsilon . Let us choose so that the precision of representation of the real number is with a margin, i.e.,
- ;
- Set for the system (2) and value the number that determines the direction in time: is gone forward in time, is gone backward in time;
- SetT as the length of the time interval on which the numerical integration will be performed;
- ;
- Calculate the integration step by Formula (5);
- If, then,Else;
- ;
- Let;
- Calculate the approximate value by summing the terms of the series (3) to such a value i, where the following inequality holds:
- Print
- If, then we got out the compact . Then, write ;
- If, then;
- If, then terminate the algorithm;
- ;
- Go to step 6.
- The accuracy of approximation at is a frequently used criterion in applications of numerical methods for solving differential equations. When the inequality (6) is true, it is necessary to increase the degrees of all polynomials , obtaining the next approximation, and compare the distance between the obtained approximate solutions on the interval with the value . If , then we increase the powers of ; otherwise, we have to use the obtained solution;
- The radius of the neighborhood of the initial point, to which the approximate solution should return in backward time, is another criterion. In other words, we need to select the accuracy of so that the following inequality holds:
- The comparison of configurations of approximate solutions in forward and backward time is necessary to determine the numbers (7), describing approximate solutions. Next, check the following:
5. Analysis of the Poincaré Recurrences for a Fourth-Order System
6. Calculation of the Lyapunov Exponents
- Divide a given time interval (usually the value T is large) into short intervals by the following length:
- Let be a point of the researched solution, e.g., (10);
- ;
- ;
- Let be the column vectors of initial perturbations from n-components (the real numbers), (p is a number of the perturbation vector);
- Assign to each component of the vector random number in the range ;
- , (the initial values of the sums at calculating the Lyapunov exponents);
- Orthogonalize and normalize to unity the system of vectors , ,..., ;
- ;
- Forp from 1 to n with step 1:Beginning of cycleLetAssign to vectors and the matching vector blocks , i.e.,Note that for all values of p the vector , is the same.End of cycle
- Forp from 1 to n with step 1:Beginning of cycleEnd of cycle
- If, then Go to step 9;
- , ;
- Print, .
7. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
References
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Time Moment | Value |
---|---|
0.3655 | 0.0423764 |
0.7310 | 0.0157840 |
1.0965 | 0.0323213 |
1.4620 | 0.0196461 |
1.8275 | 0.0173865 |
2.1930 | 0.0209083 |
2.5585 | 0.0135900 |
2.9240 | 0.0206764 |
3.2895 | 0.0194286 |
3.6550 | 0.0197196 |
4.0205 | 0.0237665 |
4.3860 | 0.0213908 |
4.7515 | 0.0263917 |
5.1170 | 0.0259931 |
5.4825 | 0.0288078 |
5.8480 | 0.0311426 |
6.2135 | 0.0320319 |
6.5790 | 0.0356643 |
6.9445 | 0.0362604 |
7.3101 | 0.0340862 |
7.6756 | 0.0325096 |
8.0411 | 0.0305410 |
8.4066 | 0.0280697 |
8.7721 | 0.0268155 |
9.1376 | 0.0243143 |
9.5031 | 0.0231314 |
9.8686 | 0.0211890 |
Time Moment | Value |
---|---|
0.3655 | 0.00133861 |
0.7310 | 0.00267728 |
1.0965 | 0.00401586 |
1.4620 | 0.00535439 |
1.8275 | 0.00669292 |
2.1930 | 0.00803134 |
2.5585 | 0.00936979 |
2.9240 | 0.01070810 |
3.2895 | 0.01204650 |
3.6550 | 0.01338470 |
4.0205 | 0.01472300 |
4.3860 | 0.01606120 |
4.7515 | 0.01739930 |
5.1171 | 0.01774010 |
5.4826 | 0.01640090 |
5.8481 | 0.01506170 |
6.2136 | 0.01372250 |
6.5791 | 0.01238330 |
6.9446 | 0.01104430 |
7.3101 | 0.00970522 |
7.6756 | 0.00836622 |
8.0411 | 0.00702727 |
8.4066 | 0.00568835 |
8.7721 | 0.00434949 |
9.1376 | 0.00301067 |
9.5031 | 0.00167189 |
9.8686 | 0.00033316 |
Time Moment | Value |
---|---|
0.365504 | 1.105410 |
0.731007 | 1.096510 |
1.096510 | 0.348186 |
1.462010 | 0.752419 |
1.827520 | 1.799740 |
2.193020 | 0.697982 |
2.558530 | 0.397935 |
2.924030 | 1.498770 |
3.289530 | 1.051780 |
3.655040 | 0.048264 |
4.020540 | 1.146250 |
4.386040 | 1.403530 |
4.751550 | 0.304245 |
5.117050 | 0.793828 |
5.482550 | 1.754720 |
5.848060 | 0.656571 |
6.213560 | 0.442487 |
6.579070 | 1.541010 |
6.944570 | 1.008280 |
7.310070 | 0.091034 |
7.675580 | 1.189060 |
8.041080 | 1.359960 |
8.406580 | 0.261577 |
8.772090 | 0.837369 |
9.137590 | 1.711900 |
9.503100 | 0.613282 |
9.868600 | 0.485547 |
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Pchelintsev, A.N. On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities. Mathematics 2021, 9, 2057. https://doi.org/10.3390/math9172057
Pchelintsev AN. On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities. Mathematics. 2021; 9(17):2057. https://doi.org/10.3390/math9172057
Chicago/Turabian StylePchelintsev, Alexander N. 2021. "On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities" Mathematics 9, no. 17: 2057. https://doi.org/10.3390/math9172057
APA StylePchelintsev, A. N. (2021). On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities. Mathematics, 9(17), 2057. https://doi.org/10.3390/math9172057