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Z. Zheng
numerous potential studies that revealed a series of interlocked TTFL’s together with
a complex network of reactions. These involve regulated protein phosphorylation
and degradation of TTFL components, protein complex assembly, nuclear translocation and other post-translational modifications, generating oscillations with a period
of approximately 24 h. Circadian oscillators within individual cells respond differently to entraining signals and control various physiological outputs, such as sleep
patterns, body temperature, hormone release, blood pressure, and metabolism. The
seminal discoveries by Hall, Rosbash and Young have revealed a crucial physiological mechanism explaining circadian adaptation, with important implications for
human health and disease.
Essentially, biological rhythms ubiquitously existing in various biological systems
are physically oscillatory behaviors, which are indications of temporally periodic
dynamics of nonlinear systems. On the other hand, rhythmic phenomena can be
extensively observed in various situations, ranging from physics, and chemistry to
biology [6, 43]. Therefore, these oscillatory behaviors with completely different
backgrounds can be universally studied in the framework of nonlinear dynamics.
4.3.1.1 Self-sustained Oscillation in Simple Nonlinear Systems
Self-sustained oscillation, also called the limit cycle, which is defined as a typical
time-periodic nonlinear behavior, has received much attention throughout the past
century [44]. The essential mechanism of sustained oscillation is the existence of
feedback in nonlinear systems, but the source of feedback depends strongly on
different situations.
First, the competition-balance mechanism between the positive feedback and the
negative feedback can maintain a stable oscillation. In mechanical systems, for
example, the conventional Van der Pol oscillator, the adjustable damping provides
the key mechanism for the energy compensation to sustain a stable limit cycle.
The damping becomes positive to dissipate energy when the amplitude is large and
becomes negative to consume energy when the amplitude is small. Stability analysis
can well present the dynamical mechanism of the limit-cycle motion. There are a lot
of historic examples and models in describing these typical oscillatory dynamics.
Let us study a simple nonlinear system with a limit cycle oscillation by adopting
the following complex dynamical equation:
˙
z = (λ + iω)z − bz|z|
2
,
(4.28)
where z(t) = x(t) + iy(t) is a complex order parameter. We use “i” to denote the
imaginary index throughout this chapter. Physically the order parameter z(t) can be
obtained via the reduction procedure by eliminating the fast modes. Equation (4.28)
presents a two-dimensional dynamics in real space, and the possible time-dependent
solution of this equation is the limit cycle.
Z. Zheng
numerous potential studies that revealed a series of interlocked TTFL’s together with
a complex network of reactions. These involve regulated protein phosphorylation
and degradation of TTFL components, protein complex assembly, nuclear translocation and other post-translational modifications, generating oscillations with a period
of approximately 24 h. Circadian oscillators within individual cells respond differently to entraining signals and control various physiological outputs, such as sleep
patterns, body temperature, hormone release, blood pressure, and metabolism. The
seminal discoveries by Hall, Rosbash and Young have revealed a crucial physiological mechanism explaining circadian adaptation, with important implications for
human health and disease.
Essentially, biological rhythms ubiquitously existing in various biological systems
are physically oscillatory behaviors, which are indications of temporally periodic
dynamics of nonlinear systems. On the other hand, rhythmic phenomena can be
extensively observed in various situations, ranging from physics, and chemistry to
biology [6, 43]. Therefore, these oscillatory behaviors with completely different
backgrounds can be universally studied in the framework of nonlinear dynamics.
4.3.1.1 Self-sustained Oscillation in Simple Nonlinear Systems
Self-sustained oscillation, also called the limit cycle, which is defined as a typical
time-periodic nonlinear behavior, has received much attention throughout the past
century [44]. The essential mechanism of sustained oscillation is the existence of
feedback in nonlinear systems, but the source of feedback depends strongly on
different situations.
First, the competition-balance mechanism between the positive feedback and the
negative feedback can maintain a stable oscillation. In mechanical systems, for
example, the conventional Van der Pol oscillator, the adjustable damping provides
the key mechanism for the energy compensation to sustain a stable limit cycle.
The damping becomes positive to dissipate energy when the amplitude is large and
becomes negative to consume energy when the amplitude is small. Stability analysis
can well present the dynamical mechanism of the limit-cycle motion. There are a lot
of historic examples and models in describing these typical oscillatory dynamics.
Let us study a simple nonlinear system with a limit cycle oscillation by adopting
the following complex dynamical equation:
˙
z = (λ + iω)z − bz|z|
2
,
(4.28)
where z(t) = x(t) + iy(t) is a complex order parameter. We use “i” to denote the
imaginary index throughout this chapter. Physically the order parameter z(t) can be
obtained via the reduction procedure by eliminating the fast modes. Equation (4.28)
presents a two-dimensional dynamics in real space, and the possible time-dependent
solution of this equation is the limit cycle.