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Liénard Equation
In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a second order differential equation, named after the French physicist Alfred-Marie Liénard. During the development of radio and vacuum tube technology, Liénard equations were intensely studied as they can be used to model oscillating circuits. Under certain additional assumptions Liénard's theorem guarantees the uniqueness and existence of a limit cycle for such a system. A Liénard system with piecewise-linear functions can also contain homoclinic orbits. Definition Let ''f'' and ''g'' be two continuously differentiable functions on R, with ''g'' an odd function and ''f'' an even function. Then the second order ordinary differential equation of the form :+f(x)+g(x)=0 is called the Liénard equation. Liénard system The equation can be transformed into an equivalent two-dimensional system of ordinary differential equations. We define :F(x) := \int_0^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biryukov Equation
In the study of dynamical systems, the Biryukov equation (or Biryukov oscillator), named after Vadim Biryukov (1946), is a non-linear second-order differential equation used to model damped oscillators. The equation is given by \frac+f(y)\frac+y=0, \qquad\qquad (1) where is a piecewise constant function which is positive, except for small as \begin & f(y) = \begin -F, & , y, \le Y_0; \\ ptF, & , y, >Y_0. \end \\ pt& F = \text > 0, \quad Y_0 = \text > 0. \end Eq. (1) is a special case of the Lienard equation; it describes the auto-oscillations. Solution (1) at a separate time intervals when f(y) is constant is given by y_k(t) = A_\exp(s_t) + A_\exp(s_t) \qquad\qquad (2) where denotes the exponential function. Here s_k = \begin \displaystyle \frac\mp\sqrt, & , y, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autonomous Differential Equation
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems. Many laws in physics, where the independent variable is usually assumed to be time, are expressed as autonomous systems because it is assumed the laws of nature which hold now are identical to those for any point in the past or future. Definition An autonomous system is a system of ordinary differential equations of the form \fracx(t)=f(x(t)) where takes values in -dimensional Euclidean space; is often interpreted as time. It is distinguished from systems of differential equations of the form \fracx(t)=g(x(t),t) in which the law governing the evolution of the system does not depend solely on the system's current state but also the parameter , again often interpreted as time; such systems are by definition not autonomous. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stability Theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance. In dynamical systems, an orbit is called ''Lyapunov stable'' if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well-studied problem involvi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Van Der Pol Oscillator
In dynamical system, dynamics, the Van der Pol oscillator is a Conservative force, non-conservative oscillator with nonlinearity, non-linear Damping ratio, damping. It evolves in time according to the second-order differential equation: :-\mu(1-x^2)+x= 0, where ''x'' is the position coordinate system, coordinate—which is a function (mathematics), function of the time ''t'', and ''μ'' is a scalar (mathematics), scalar parameter indicating the nonlinearity and the strength of the damping. History The Van der Pol oscillator was originally proposed by the Dutch electrical engineering, electrical engineer and physicist Balthasar van der Pol while he was working at Philips. Van der Pol found stable oscillations, which he subsequently called relaxation oscillator, relaxation-oscillations and are now known as a type of limit cycle in electrical circuits employing vacuum tubes. When these circuits are driven near the limit cycle, they become entrainment (physics), entrained, i.e. the d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel Equation Of The Second Kind
Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd who offered his firstborn flock up to God as an offering. God accepted his offering but not his brother's. Cain then killed Abel out of jealousy. According to Genesis, this was the first murder in the history of mankind. Genesis narrative Interpretations Jewish and Christian interpretations According to the narrative in Genesis, Abel ( ''Hébel'', in pausa ''Hā́ḇel''; grc-x-biblical, Ἅβελ ''Hábel''; ar, هابيل, ''Hābēl'') is Eve's second son. His name in Hebrew is composed of the same three consonants as a root meaning "breath". Julius Wellhausen has proposed that the name is independent of the root. Eberhard Schrader had previously put forward the Akkadian (Old Assyrian dialect) ''ablu'' ("son") as a more lik ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Order Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autonomous Differential Equation
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems. Many laws in physics, where the independent variable is usually assumed to be time, are expressed as autonomous systems because it is assumed the laws of nature which hold now are identical to those for any point in the past or future. Definition An autonomous system is a system of ordinary differential equations of the form \fracx(t)=f(x(t)) where takes values in -dimensional Euclidean space; is often interpreted as time. It is distinguished from systems of differential equations of the form \fracx(t)=g(x(t),t) in which the law governing the evolution of the system does not depend solely on the system's current state but also the parameter , again often interpreted as time; such systems are by definition not autonomous. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
