Sage-Tips

What are nonlinear dynamics?

Nonlinear dynamics is the branch of physics that studies systems governed by equations more complex than the linear, aX+b form. Nonlinear systems, such as the weather or neurons, often appear chaotic, unpredictable or counterintuitive, and yet their behaviour is not random.

What is nonlinear dynamics used for?

Nonlinear dynamics models can be used to study spatially extended systems such as acoustic waves, electrical transmission problems, plasma waves, and so forth. These problems have been modeled by using a linear chain of discrete oscillators with nearest neighbor coupling as shown in Figure 19.

What is linear and non-linear system?

Linear means something related to a line. All the linear equations are used to construct a line. A non-linear equation is such which does not form a straight line. It looks like a curve in a graph and has a variable slope value.

What is meant by Soliton?

A soliton is a solitary wave that behaves like a “particle”, in that it satisfies the following conditions (Scott, 2005): It must maintain its shape when it moves at constant speed. When a soliton interacts with another soliton, it emerges from the “collision” unchanged except possibly for a phase shift.

What is a nonlinear problem?

A smooth nonlinear programming (NLP) or nonlinear optimization problem is one in which the objective or at least one of the constraints is a smooth nonlinear function of the decision variables. An example of a smooth nonlinear function is: 2 X12 + X23 + log X3. …

Why do we need to learn nonlinear systems?

The dynamics of a nonlinear system are richer; consequently the process of formulating a control law through a standard design process entails deeper understanding and more sophistication.

Why nonlinear control is required?

Even if the plant is linear, a nonlinear controller can often have attractive features such as simpler implementation, faster speed, more accuracy, or reduced control energy, which justify the more difficult design procedure. An example of a nonlinear control system is a thermostat-controlled heating system.

What is the difference between linear and nonlinear functions?

linear functions have no exponents higher than 1, and a graph that looks like a straight line. non-linear functions have at least one exponent higher than 1, and a graph that isn’t a straight line.

What are solitons used for?

The optical waveguide the soliton creates while propagating is not only a mathematical model, but it actually exists and can be used to guide other waves at different frequencies. This way it is possible to let light interact with light at different frequencies (this is impossible in linear media).

How do solitons form?

Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.)

What is nonlinear process?

Nonlinear process plans are the basis for a flexible reaction to changes of the current state in production systems. Adaptive processes planning requires nonlinear process plans. In order to identify these nonlinear process plans alternative processing steps have to be defined in a first step.

What are solitons and what causes them?

Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.

Why are nonlinear problems difficult to solve?

One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle.

Are there any soliton solutions for the Schrodinger equation?

See soliton (optics) for a more detailed description. Many exactly solvable models have soliton solutions, including the Korteweg–de Vries equation, the nonlinear Schrödinger equation, the coupled nonlinear Schrödinger equation, and the sine-Gordon equation.

What is a pressure soliton model in neuroscience?

The recent and not widely accepted soliton model in neuroscience proposes to explain the signal conduction within neurons as pressure solitons. A topological soliton, also called a topological defect, is any solution of a set of partial differential equations that is stable against decay to the “trivial solution”.