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Related Concept Videos

Root Loci for Positive-Feedback Systems01:23

Root Loci for Positive-Feedback Systems

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The Hartley oscillator is a positive feedback system that sustains oscillations by feeding the output back to the input in phase, thereby reinforcing the signal. Positive feedback systems can be viewed as negative feedback systems with inverted feedback signals. In these systems, the root locus encompasses all points on the s-plane where the angle of the system transfer function equals 360 degrees.
The construction rules for the root locus in positive feedback systems are similar to those in...
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Linear time-invariant Systems01:23

Linear time-invariant Systems

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A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
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Stability01:28

Stability

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The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded...
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Plotting and Calibrating the Root Locus01:19

Plotting and Calibrating the Root Locus

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Root loci often diverge as system poles shift from the real axis to the complex plane. Key points in this transition are the breakaway and break-in points, indicating where the root locus leaves and reenters the real axis. The branches of the root locus form an angle of 180/n degrees with the real axis, where n is the number of branches at a breakaway or break-in point.
The maximum gain occurs at the breakaway points between open-loop poles on the real axis, while the minimum gain is...
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Construction of Root Locus01:15

Construction of Root Locus

238
The construction of a root locus involves several key steps to analyze and visualize the behavior of a system's poles with varying gain. The number of branches in the root locus equals the number of closed-loop poles and is symmetrical about the real axis.
For positive gain values, the root locus exists on the real axis to the left of an odd number of finite open-loop poles or zeros. The root locus starts at the open-loop poles and traces the paths of the closed-loop poles as the gain...
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Pole and System Stability01:24

Pole and System Stability

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The transfer function is a fundamental concept representing the ratio of two polynomials. The numerator and denominator encapsulate the system's dynamics. The zeros and poles of this transfer function are critical in determining the system's behavior and stability.
Simple poles are unique roots of the denominator polynomial. Each simple pole corresponds to a distinct solution to the system's characteristic equation, typically resulting in exponential decay terms in the system's...
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On discrete Lorenz-like attractors.

Sergey Gonchenko1, Alexander Gonchenko1, Alexey Kazakov2

  • 1Mathematical Center of Lobachevsky State University, 23 Prospekt Gagarina, 603950 Nizhny Novgorod, Russia.

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Summary
This summary is machine-generated.

This study explores chaotic discrete Lorenz-like attractors, revealing universal bifurcation scenarios that generate them. These scenarios can lead to new types of Lorenz attractors through period-2 attractors and crises.

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Area of Science:

  • Dynamical Systems
  • Chaos Theory
  • Nonlinear Dynamics

Background:

  • Discrete Lorenz-like attractors exhibit complex dynamical behaviors.
  • Understanding their formation is crucial for nonlinear dynamics.

Purpose of the Study:

  • Investigate geometrical and dynamical properties of discrete Lorenz-like attractors.
  • Describe universal bifurcation scenarios leading to these attractors.
  • Analyze the emergence of new attractor types through crises.

Main Methods:

  • Phenomenological description of bifurcation scenarios.
  • Demonstration using one-parameter families of 3D Hénon-like maps.
  • Analysis of period-2 Lorenz-like attractors and their crises.

Main Results:

  • Robustly chaotic (pseudohyperbolic) attractors arise from universal bifurcation scenarios.
  • Specific scenarios leading to period-2 Lorenz-like attractors are identified.
  • Crises of period-2 attractors generate novel discrete Lorenz shape attractors.

Conclusions:

  • Universal bifurcations provide a framework for understanding discrete Lorenz-like attractor formation.
  • Period-2 attractors and their crises are key mechanisms for generating complex dynamics.
  • The study expands the known types of discrete Lorenz attractors.