1,599 research outputs found
Can a Pseudo Periodic Orbit Avoid a Catastrophic Transition?
We propose a resilient control scheme to avoid catastrophic transitions associated with saddle-node bifurcations of periodic solutions. The conventional feedback control schemes related to controlling chaos can stabilize unstable periodic orbits embedded in strange attractors or suppress bifurcations such as period-doubling and Neimark-Sacker bifurcations whose periodic orbits are kept existing through bifurcation processes. However, it is impossible to apply these methods directly to a saddle-node bifurcation since the corresponding periodic orbit disappears after such a bifurcation. In this paper, we define a pseudo periodic orbit which can be obtained using transient behavior right after the saddle-node bifurcation, and utilize it as reference data to compose a control input. We consider a pseudo periodic orbit at a saddle-node bifurcation in the Duffing equations as an example, and show its temporary attraction. Then we demonstrate suppression control of this bifurcation, and show robustness of the control. As a laboratory experiment, a saddle-node bifurcation of limit cycles in the BVP oscillator is explored. A control input generated by a pseudo periodic obit can restore a stable limit cycle which disappeared after the saddle-node bifurcation
Bifurcation Analysis of Piecewise Smooth Ecological Models
The aim of this paper is the study of the long-term behavior of population communities described by piecewise smooth models (known as Filippov systems). Models of this kind are often used to describe populations with selective switching between alternative habitats or diets or to mimic the evolution of an exploited resource where harvesting is forbidden when the resource is below a prescribed threshold. The analysis is carried out by performing the bifurcation analysis of the model with respect to two parameters. A relatively simple method, called the puzzle method, is proposed to construct the complete bifurcation diagram step-by-step. The method is illustrated through four examples concerning the exploitation and protection of interacting populations
From travelling waves to mild chaos: a supercritical bifurcation cascade in pipe flow
We study numerically a succession of transitions in pipe Poiseuille flow that
leads from simple travelling waves to waves with chaotic time-dependence. The
waves at the origin of the bifurcation cascade possess a shift-reflect symmetry
and are both axially and azimuthally periodic with wave numbers {\kappa} = 1.63
and n = 2, respectively. As the Reynolds number is increased, successive
transitions result in a wide range of time dependent solutions that includes
spiralling, modulated-travelling, modulated-spiralling,
doubly-modulated-spiralling and mildly chaotic waves. We show that the latter
spring from heteroclinic tangles of the stable and unstable invariant manifolds
of two shift-reflect-symmetric modulated-travelling waves. The chaotic set thus
produced is confined to a limited range of Reynolds numbers, bounded by the
occurrence of manifold tangencies. The states studied here belong to a subspace
of discrete symmetry which makes many of the bifurcation and path-following
investigations presented technically feasible. However, we expect that most of
the phenomenology carries over to the full state-space, thus suggesting a
mechanism for the formation and break-up of invariant states that can sustain
turbulent dynamics.Comment: 38 pages, 35 figures, 1 tabl
Order out of Randomness : Self-Organization Processes in Astrophysics
Self-organization is a property of dissipative nonlinear processes that are
governed by an internal driver and a positive feedback mechanism, which creates
regular geometric and/or temporal patterns and decreases the entropy, in
contrast to random processes. Here we investigate for the first time a
comprehensive number of 16 self-organization processes that operate in
planetary physics, solar physics, stellar physics, galactic physics, and
cosmology. Self-organizing systems create spontaneous {\sl order out of chaos},
during the evolution from an initially disordered system to an ordered
stationary system, via quasi-periodic limit-cycle dynamics, harmonic mechanical
resonances, or gyromagnetic resonances. The internal driver can be gravity,
rotation, thermal pressure, or acceleration of nonthermal particles, while the
positive feedback mechanism is often an instability, such as the
magneto-rotational instability, the Rayleigh-B\'enard convection instability,
turbulence, vortex attraction, magnetic reconnection, plasma condensation, or
loss-cone instability. Physical models of astrophysical self-organization
processes involve hydrodynamic, MHD, and N-body formulations of Lotka-Volterra
equation systems.Comment: 61 pages, 38 Figure
Homogeneous nucleation of dislocations as a pattern formation phenomenon
Dislocation nucleation in homogeneous crystals initially unfolds as a linear
symmetry-breaking elastic instability. In the absence of explicit nucleation
centers, such instability develops simultaneously all over the crystal and due
to the dominance of long range elastic interactions it advances into the
nonlinear stage as a collective phenomenon through pattern formation. In this
paper we use a novel mesoscopic tensorial model (MTM) of crystal plasticity to
study the delicate role of crystallographic symmetry in the development of the
dislocation nucleation patterns in defect free crystals loaded in a hard
device. The model is formulated in 2D and we systematically compare lattices
with square and triangular symmetry. To avoid the prevalence of the
conventional plastic mechanisms, we consider the loading paths represented by
pure shears applied on the boundary of the otherwise unloaded body. These
loading protocols can be qualified as exploiting the 'softest' and the
'hardest' directions and we show that the associated dislocation patterns are
strikingly different
Bifurcation Analysis of Filippov's Ecological Models
The aim of this paper is the study of the long-term behavior of population communities described by a class of discontinuous models known as Filippov systems. The analysis is carried out by performing the bifurcation analysis of the model with respect to two parameters. A relatively simple method, called the puzzle method, is proposed to construct the complete bifurcation diagram step-by-step. The method is illustrated through four examples concerning the exploitation and protection of interacting populations
Asynchronous partial contact motion due to internal resonance in multiple degree-of-freedom rotordynamics
Sudden onset of violent chattering or whirling rotorstator contact motion in rotational machines can cause significant damage in many industrial applications. It is shown that internal resonance can lead to the onset of bouncing-type partial contact motion away from primary resonances. These partial contact limit cycles can involve any two modes of an arbitrarily high degree-of-freedom system, and can be seen as an extension of a synchronisation condition previously reported for a single disc system. The synchronisation formula predicts multiple drivespeeds, corresponding to different forms of mode-locked bouncing orbits. These results are backed up by a brute-force bifurcation analysis which reveals numerical existence of the corresponding family of bouncing orbits at supercritical drivespeeds, provided the dampingis sufficiently low. The numerics reveal many overlapping families of solutions, which leads to significant multi-stability of the response at given drive speeds. Further secondary bifurcations can also occur within each family, altering the nature of the response, and ultimately leading to chaos. It is illustrated how stiffness and damping of the stator have a large effect on the number and nature of the partial contact solutions, illustrating the extreme sensitivity that would be observed in practice
General Relativistic Magnetohydrodynamic Simulations of the Hard State as a Magnetically-Dominated Accretion Flow
(Abridged) We present one of the first physically-motivated two-dimensional
general relativistic magnetohydrodynamic (GRMHD) numerical simulations of a
radiatively-cooled black-hole accretion disk. The fiducial simulation combines
a total-energy-conserving formulation with a radiative cooling function, which
includes bremsstrahlung, synchrotron, and Compton effects. By comparison with
other simulations we show that in optically thin advection-dominated accretion
flows, radiative cooling can significantly affect the structure, without
necessarily leading to an optically thick, geometrically thin accretion disk.
We further compare the results of our radiatively-cooled simulation to the
predictions of a previously developed analytic model for such flows. For the
very low stress parameter and accretion rate found in our simulated disk, we
closely match a state called the "transition" solution between an outer
advection-dominated accretion flow and what would be a magnetically-dominated
accretion flow (MDAF) in the interior. The qualitative and quantitative
agreement between the numerical and analytic models is quite good, with only a
few well-understood exceptions. According to the analytic model then, at
significantly higher stress or accretion, we would expect a full MDAF to form.
The collection of simulations in this work also provide important data for
interpreting other numerical results in the literature, as they span the most
common treatments of thermodynamics, including simulations evolving: 1) the
internal energy only; 2) the internal energy plus an explicit cooling function;
3) the total energy without cooling; and 4) total energy including cooling. We
find that the total energy formulation is a necessary prerequisite for proper
treatment of radiative cooling in MRI accretion flows.Comment: 13 pages, 7 figures, submitted to Ap
Rate-induced transitions for parameter shift systems
Rate-induced transitions have recently emerged as an identifiable type of instability of attractors in nonautonomous dynamical systems. In most studies so far, these attractors can be associated with equilibria of an autonomous limiting system, but this is not necessarily the case. For a specific class of systems with a parameter shift between two autonomous systems, we consider how the breakdown of the quasistatic approximation for attractors can lead to rate-induced transitions, where nonautonomous instability can be characterised in terms of a critical rate of the parameter shift. We find a number of new phenomena for non-equilibrium attractors: weak tracking where the pullback attractor of the system limits to a proper subset of the attractor of the future limit system, partial tipping where certain phases of the pullback attractor tip and others track the quasistatic attractor, em invisible tipping where the critical rate of partial tipping is isolated and separates two parameter regions where the system exhibits end-point tracking. For a model parameter shift system with periodic attractors, we characterise thresholds of rate-induced tipping to partial and total tipping. We show these thresholds can be found in terms of certain periodic-to-periodic and periodic-to-equilibrium connections that we determine using Lin's method for an augmented system. Considering weak tracking for a nonautonomous Rossler system, we show that there are infinitely many critical rates at which a pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor
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