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Two methods of stability analysis of systems described by dynamical equations are being considered. They are based on an analysis of eigenvalues spectrum for the evolutionary matrix or the spectral equation and they allow determining the conditions of stability and instability, as well as the possibility of chaotic behavior of systems in case of a stability loss. The methods are illustrated for nonlinear Lorenz and Rossler model problems.

The work is dedicated to the methods of practical stability analysis for systems described by nonlinear autonomous equations. The analysis of such systems is of a particular interest due to the dynamical chaos phenomena, which can be observed in cases of stability loss [

The more common methods of system stability investigation are the spectral methods, which consist of dynamics spectrum analysis for small perturbations. The problem is defined in the following way.

Let’s assume that the system state is defined by a combination of macroscopic characteristics

The linearization of these equations leads to a system of equations describing the dynamics of small perturbations

The condition of solvability of the system (2) which is its spectral equation―(SE)

defines the eigenvalues spectrum―

In the Fourier-Laplace transform for the perturbations in case of stationary initial states and also in cases when the initial dependences are weak in comparison with the high-speed and high-gradient perturbations (method of local dispersion relation (LDR)), the spectral equation acquires polynomial form

In the classical posing of the stability analysis problem, it comes down to the analysis of spectral equation roots.

The indication of an instability is the presence of a SE root with positive real part Re

Due to the fact that the exact solution of Equation (4) with complex coefficients for

In studies [

1) NSE (neutrality, separation, exclusion) method is based on the spectral Equation (4) and it is implemented according to the following outline

(5.1), (5.2), (5.3), (5.4)

(5.1) Neutrality―the condition of the real part of SE roots being equal to zero;

(5.2) The separation of the spectral equation into two if the neutrality condition is fulfilled;

(5.3) The exclusion of frequency or one of the parameters from the equations and obtaining of a neutral surface―

(5.4) Indication of stability and instability areas in relation to the neutral surface.

The NSE outline is fully realized for the polynomial SE. The general neutrality conditions (3.3) in this case are given by

Specifically for a third order system the conditions (6) are

2) The

which consists of the evolutionary matrix

The neutrality criterion and the equation for the critical frequencies in this method have the following form

The commutation of

Assuming that Equation (1), which describe the system that is being analyzed in terms of stability, represent a combination of nonlinear autonomic equations

The perturbation dynamics of system (10) in this case are described by Equation (11)

If all the time derivatives in (11) are negative, the perturbations attenuate and the system is Lyapunov stable. If there is at least one positive derivative, the solution curves scatter; the system is not stable. The correlation of derivative signs allows to determine the possibility of chaotic behavior and the formation of complex localized structures―strange attractors [

In these cases, the spectral equation method (4) and NSE method (5) for the stability analysis in their classical forms are not applicable.

Due to the fact that the

Specifically, the criterion (12) being equal to zero corresponds to the presence of zero-order derivatives (eigen- values), the criterion sign change-corresponds to a sign change of time derivatives in dynamical Equation (11). As a result, the multiplication factor analysis in (12) represents an analysis of evolutionary matrix eigenvalues spectrum for a nonlinear system and therefore, an analysis of time derivatives signs in Equation (11).

Such generalization can technically be conducted for the NSE (5) and spectral Equation (4) methods, but in that case

So now we will use the generalized NSE and

The Lorenz problem is of a particular interest because nonlinear equations of Lorenz model result from the dynamics equations of a whole range of physical systems: The convection inside a fluid layer heated from underneath, a single-mode laser, water-wheel and other. Besides that, it demonstrates the formation of chaotic dyna- mics (

The Lorenz model equations have the following form

In the phase space of variables

If

i.e. the Lorenz system is dissipative.

The system (13) has two stationary solutions-stationary states

The linearization of system (13) in relation to a solution

The spectral equation and its coefficients in a stationary case are

(It should be noted that the dissipation condition coincides in absolute value with the first coefficient of the spectral equation and is equal to the sum of eigenvalues of the evolutionary matrix. It may be shown that there is a common result.)

The NSE method for SE (17) produces two critical-neutral modes

1)

2)

The mode (18) occurs for the first stationary state and corresponds to its instability when

The mode (19) occurs for the second stationary state and for the classical values of Lorenz parameters the critical values of frequency and the parameter

When

The

As could be expected, the criterion (22) shows an instability of the first stationary state (15) when

The dynamical mode is correspondingly

1) A stable point;

2) A boundary cycle;

3) An attractor (of chaotic dynamics).

This way, in the Lorenz system with

It should be noted that the criterion (23) includes the first critical mode

A nonlinear problem which has an evidentially expressed field of chaotic behavior with an attractor presented in

The Rössler model equations have the following form

From (26) it follows that the Rössler system is dissipative only in a limited field (

The system (25) has two stationary solutions-stationary states

which are possible under the condition

The linearization of Equation (25) in relation to the solution

The spectral equation of the system (25) and its coefficients for the stationary states are correspondingly

The NSE criterion for SE (30) produces two critical-neutral modes (18, 19), which in this case take the following form

The analysis of conditions (31) combined with stationary conditions (27) shows that when

The first critical mode occurs only for the second stationary state

The second critical mode occurs for the first stationary state, which is also unstable when

This way both stationary states are unstable in different ways.

The L-criterion (12) regarding arbitrary solutions

For stationary solutions (27) the criterion (35) is rearranged into

As one would expect, the

which indicates a chaotic behavior of the system with a phase portrait of the type illustrated in

In conclusion, the use of modified NSE methods and the