Related Experiment Video
Updated: Oct 29, 2025
09:36
Characterization of Anisotropic Leaky Mode Modulators for Holovideo
Published on: March 19, 2016
8.1K
Sliding homoclinic bifurcations in a Lorenz-type system: Analytic proofs
Vladimir N Belykh1, Nikita V Barabash1, Igor V Belykh2
1Department of Mathematics, Volga State University of Water Transport, 5A Nesterov Str., Nizhny Novgorod 603950, Russia.
Chaos (Woodbury, N.Y.)
|July 12, 2021
Summary
Non-smooth systems exhibit unique bifurcations. This study reveals how sliding motions in a Lorenz-type system can create stable limit cycles from unstable homoclinic orbits, unlike smooth systems.
Area of Science:
- Dynamical Systems
- Non-smooth Dynamics
- Bifurcation Theory
Background:
- Non-smooth systems display complex dynamics distinct from smooth systems.
- Homoclinic bifurcations are crucial in understanding system behavior.
- Previous studies on smooth systems show bifurcations yield only unstable dynamics.
Purpose of the Study:
- To investigate homoclinic bifurcations in a piecewise-smooth Lorenz-type system.
- To analyze the impact of sliding motions on bifurcation scenarios.
- To characterize bifurcations that lead to stable limit cycles from unstable orbits.
Main Methods:
- Analytical investigation of a piecewise-smooth Lorenz-type system.
- Construction of a Poincaré return map incorporating sliding motions.
- Rigorous characterization of sliding homoclinic bifurcations.
Main Results:
- Sliding motions enable novel bifurcations, producing stable limit cycles from unstable homoclinic orbits.
- Sliding homoclinic bifurcations were shown to destroy a chaotic Lorenz-type attractor.
- An explicit scaling factor for period-doubling bifurcations leading to quasi-attractors was derived.
Conclusions:
- The study reveals non-classical global bifurcation scenarios in non-smooth systems.
- Findings contrast with smooth system analogs, highlighting unique dynamics in non-smooth systems.
- This work provides a foundation for developing non-classical global bifurcation theory for non-smooth flows.
Related Concept Videos
01:30Fundamental Theorem of Algebra
19
The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as: with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the...
19
01:26Transmission-Line Differential Equations
481
Transmission lines are essential components of electrical power systems. They are characterized by the distributed nature of resistance (R), inductance (L), and capacitance (C) per unit length. To analyze these lines, differential equations are employed to model the variations in voltage and current along the line.
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured...
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured...
481
01:18Castigliano's Theorem
642
Castigliano's theorem analyzes displacements and rotations in elastic structures. It relates the derivative of elastic strain energy to the applied forces or moments, allowing for the calculation of deformations. The theorem states that the partial derivative of the total strain energy of a system with respect to a specific load results in the displacement at the point where the load is applied. This principle applies to both forces and moments.
642
01:23Root Loci for Positive-Feedback Systems
181
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...
The construction rules for the root locus in positive feedback systems are similar to those in...
181
01:19Routh-Hurwitz Criterion II
537
In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
537
01:18Second Order systems II
233
In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
233