scholarly journals Phase portraits, Lyapunov functions, and projective geometry

2020 ◽  
Author(s):  
Lilija Naiwert ◽  
Karlheinz Spindler
Keyword(s):  
Dynamical System ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Phase Portrait ◽  
Projective Geometry ◽  
Lyapunov Functions ◽  
Phase Portraits ◽  
Classroom Experiences

AbstractWe discuss two problems which grew out of an introductory differential equations class but were solved only later, each after having been put into a different context. First, how do you find a rather complicated Lyapunov function with your bare hands, without using a fully developed theory (while reconstructing the steps leading up to such a theory)? Second, how can you obtain a global picture of the phase-portrait of a dynamical system (thereby invoking ideas from projective geometry)? Since classroom experiences played an important part in the making of this paper, didactical aspects will also be discussed.

Control Vector Lyapunov Functions for Large-Scale Impulsive Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Large Scale ◽  
Lyapunov Theory ◽  
System Stability ◽  
Dynamical System Theory ◽  
Control Vector ◽  
Vector Lyapunov Functions

This chapter extends the notion of control vector Lyapunov functions to impulsive dynamical systems. Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In particular, the use of vector Lyapunov functions in dynamical system theory offers a very flexible framework since each component of the vector Lyapunov function can satisfy less rigid requirements as compared to a single scalar Lyapunov function. Using control vector Lyapunov functions, the chapter develops a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. These results are used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty.

On Stability with Respect to a Part of the Variables for Nonlinear Discrete-Time Systems with a Random Disturbances

2021 ◽  
Vol 22 (1) ◽  
pp. 12-18
Author(s):  
V. I. Vorotnikov ◽  
Yu. G. Martyshenko
Keyword(s):  
Lyapunov Function ◽  
Finite Difference ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
General Problem ◽  
Equilibrium Position ◽  
Lyapunov Functions ◽  
Stochastic Version ◽  
Random Disturbances ◽  
Zero Equilibrium

Nonlinear discrete (finite-difference) system of equations subject to the influence of a random disturbances of the "white" noise type, which is a difference analog of systems of stochastic differential equations in the Ito form, is considered. The increased interest in such systems is associated with the use of digital control systems, financial mathematics, as well as with the numerical solution of systems of stochastic differential equations. Stability problems are among the main problems of qualitative analysis and synthesis of the systems under consideration. In this case, we mainly study the general problem of stability of the zero equilibrium position, within the framework of which stability is analyzed with respect to all variables that determine the state of the system. To solve it, a discrete-stochastic version of the method of Lyapunov functions has been developed. The central point here is the introduction by N. N. Krasovskii, the concept of the averaged finite difference of a Lyapunov function, for the calculation of which it is sufficient to know only the right-hand sides of the system and the probabilistic characteristics of a random process. In this paper, for the class of systems under consideration, a statement of a more general problem of stability of the zero equilibrium position is given: not for all, but for a given part of the variables defining it. For the case of deterministic systems of ordinary differential equations, the formulation of this problem goes back to the classical works of A. M. Lyapunov and V. V. Rumyantsev. To solve the problem posed, a discrete-stochastic version of the method of Lyapunov functions is used with a corresponding specification of the requirements for Lyapunov functions. In order to expand the capabilities of the method used, along with the main Lyapunov function, an additional (vector, generally speaking) auxiliary function is considered for correcting the region in which the main Lyapunov function is constructed.

Control of Large-Scale Dynamical Systems via Vector Lyapunov Functions

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Feedback Control ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Large Scale ◽  
Nonlinear Dynamical System ◽  
Control Vector ◽  
Nonlinear Dynamical ◽  
Vector Lyapunov Functions

This chapter introduces the notion of a control vector Lyapunov function as a generalization of control Lyapunov functions, showing that asymptotic stabilizability of a nonlinear dynamical system is equivalent to the existence of a control vector Lyapunov function. These control vector Lyapunov functions are used to develop a universal decentralized feedback control law for a decentralized nonlinear dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. The chapter also describes the connections between the notion of vector dissipativity and optimality of the proposed decentralized feedback control law. The proposed control framework is then used to construct decentralized controllers for large-scale nonlinear dynamical systems with robustness guarantees against full modeling uncertainty.

scholarly journals THE EFFECT OF NONLINEAR DAMPING TO A DYNAMICAL SYSTEM WITH CENTER PHASE PORTAIT

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Kus Prihantoso Kurniawan ◽  
Husna Arifah
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Phase Portrait ◽  
Nonlinear Damping ◽  
Phase Portraits ◽  
Damped System ◽  
Generalized Hopf Bifurcation

This paper discusses the effect of nonlinear damping to a 2-dimesional system that has center phase portrait. The phase portraits of the damped system are drawn for 3 different values of parameter. These phase portraits stand as the numerical proof of phase portrait change. To prove the change analiticaly, we use the theorem that guarantee the existence of periodic solution. The result shows that nonlinear damping changes the phase portrait topologically. It means that the system undergoes a generalized Hopf bifurcation. Keywords: generalized Hopf bifurcation, center phase portrait, periodic solution

Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5217-5239 ◽  
Author(s):  
Ravi Agarwal ◽  
Snehana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Time Intervals ◽  
Dini Derivative ◽  
Comparison Results ◽  
Strict Stability ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

scholarly journals Stabilization of Stochastic Differential Equations Driven by G-Brownian Motion with Aperiodically Intermittent Control

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 988
Author(s):  
Pengju Duan
Keyword(s):  
Brownian Motion ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Mild Solution ◽  
Intermittent Control

The paper is devoted to studying the exponential stability of a mild solution of stochastic differential equations driven by G-Brownian motion with an aperiodically intermittent control. The aperiodically intermittent control is added into the drift coefficients, when intermittent intervals and coefficients satisfy suitable conditions; by use of the G-Lyapunov function, the p-th exponential stability is obtained. Finally, an example is given to illustrate the availability of the obtained results.

scholarly journals Lyapunov Functions and Lipschitz Stability for Riemann–Liouville Non-Instantaneous Impulsive Fractional Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 730
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Initial Time ◽  
Lipschitz Stability ◽  
Comparison Results ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Definition Of

In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is defined and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufficient conditions for the defined stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

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