stable fixed point

USAGE SUMMARY

The phrase "stable fixed point" is correct and usable in written English. You can use it in contexts related to mathematics, physics, or systems theory where stability and equilibrium are discussed. For example, "The system reached a stable fixed point after several iterations." Alternative expressions include "equilibrium point" and "steady state."

✓ Grammatically correct

Science

Human-verified examples from authoritative sources

Exact Expressions

60 human-written examples

The diagrams show that there is a stable fixed point for (ain(1.8, 2.69)), and the fixed point loses its stability as a increases.

Advances in Difference Equations

The diagrams show that there is a stable fixed point for (din(2.7, 2.9097)) and the fixed point loses its stability as d increases.

Advances in Difference Equations

When damped, the model has a stable fixed point for small values of chip thickness.

Journal of Sound and Vibration

Then T has a globally stable fixed point f ∗.

Fixed Point Theory and Applications

(40) Fig. 13 Poincaré map around a stable fixed point Fig. 14 Poincaré map around a stable fixed point.

ROBOMECH Journal

Any exponentially stable fixed point of the averaged system (7) is a stable fixed point of the original system (6) in the small-ϵ limit.

The Journal of Mathematical Neuroscience

Fig. 2(a) shows a stable fixed point, which is a focus.

Advances in Difference Equations

Thus, λ - 1 represents the stability of the stable fixed point.

The Journal of Mathematical Neuroscience

Within this region, the system possesses a single, stable, fixed point.

The Journal of Mathematical Neuroscience

This is precisely the stable fixed point used in the simulations.

The Journal of Mathematical Neuroscience

With the increase of the parameter a, the stable fixed point will become unstable.

Advances in Difference Equations

Expert writing Tips

Best practice

When discussing dynamic systems, clearly define the parameters under which the "stable fixed point" is observed, as stability can be conditional.

Common error

Avoid using "stable fixed point" when you actually mean the system is robust to perturbations. A stable point simply returns to its state after a small disturbance, while robustness implies maintaining functionality under significant changes.

Linguistic Context

The phrase "stable fixed point" functions as a noun phrase, primarily used to describe a specific state in a dynamic system where the system tends to return after a perturbation. Ludwig AI confirms its use in various scientific contexts.

Expression frequency: Very common

Frequent in

Science

100%

Less common in

News & Media

0%

Formal & Business

0%

Academia

0%

Ludwig's WRAP-UP

The phrase "stable fixed point" is a noun phrase denoting a crucial concept in dynamic systems, predominantly used within scientific and mathematical fields. As indicated by Ludwig AI, it describes an equilibrium state to which a system returns following a disturbance. Its primary function is to identify and describe this equilibrium condition, aiding in the analysis and prediction of system behaviors. Given its scientific origin, the term maintains a formal and scientific register, largely appearing in research articles and technical documents. Alternatives like "equilibrium point" or "steady state" may suit similar contexts while considering nuanced differences in meaning. Remember to define the conditions of stability clearly when employing the term.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

equilibrium point

Focuses on the state of balance rather than the stability of a specific point.

steady state

Emphasizes the unchanging nature of the system over time.

stable equilibrium

Highlights the equilibrium's characteristic of being stable.

asymptotically stable point

Describes stability in terms of approaching a point over time.

globally stable equilibrium

Specifies that the stability holds across all initial conditions.

attractor point

Focuses on the attracting nature of the point in a dynamic system.

stationary point

Describes a point where the system's variables are not changing.

fixed attractor

Highlights that the attractor is fixed in position, not oscillating.

resting state

Describes a state of inactivity or baseline condition.

balanced state

Focuses on the balance of forces or variables at the given point.

FAQs

How is a "stable fixed point" used in the context of differential equations?

In differential equations, a "stable fixed point" represents a solution where the system, after a small disturbance, returns to that point over time. It's a crucial concept for understanding the long-term behavior of dynamic systems.

What's the difference between a "stable fixed point" and an unstable one?

A "stable fixed point" is an equilibrium state to which a system returns after a small perturbation. An unstable fixed point, conversely, is one from which the system will diverge after even a slight disturbance.

What are some alternatives to "stable fixed point"?

You can use alternatives like "equilibrium point", "steady state", or "stable equilibrium" depending on the context.

How can I determine if a fixed point is stable?

Stability can often be determined by analyzing the eigenvalues of the Jacobian matrix of the system at the fixed point. If all eigenvalues have negative real parts, the fixed point is typically stable.